The Steklov problem and Remainder Estimates for Krein Systems generated by a Muckenhoupt weight

Michel Alexis Michel Alexis: [email protected] McMaster University Department of Mathematics & Statistics 1280 Main St West, Hamilton, ON, L8S 4L8, Canada

Abstract.

We show that solutions to Krein systems, the continuous frequency analogue of orthogonal polynomials on the unit circle, generated by an $A_{2}(\mathbb{R})$ weight $w$ satisfying $w-1\in L^{1}(\mathbb{R})+L^{2}(\mathbb{R})$ , are uniformly bounded in $L^{p}_{\mathrm{loc}}(w,\mathbb{R})$ for $p$ sufficiently close to $2$ . This provides a positive answer to the Steklov problem for Krein systems. Furthermore, we define a “remainder” which measures the difference between the solution to a Krein system and a polynomial-like approximant, and we estimate these remainders in $L^{p}_{w}(\mathbb{R})$ for $w\in A_{2}(\mathbb{R})$ satisfying some additional conditions. Such polynomial-like approximants, and hence remainder estimates, seem unique to Krein systems, with no analogue for orthogonal polynomials on the unit circle.

I am thankful to Sergey Denisov for helpful discussions and advising on this topic.

1. Introduction

Let $d\sigma\stackrel{{\scriptstyle\rm def}}{{=}}w(\lambda)\frac{d\lambda}{2\pi}$ be a measure on $\mathbb{R}$ with weight $w\geq 0$ satisfying $w-1\in L^{1}(\mathbb{R})+L^{2}(\mathbb{R})$ . Then one can define a family of “orthonormal continuous polynomials” $\{P(r,\lambda;\sigma)\}_{r\geq 0}$ with the following properties.

(i)

$P(r,\lambda;\sigma)$ is a “continuous polynomial” in $\mathrm{exp}\left(i\lambda\right)$ of “degree” $r$ , where $\mathrm{exp}\left(x\right)\stackrel{{\scriptstyle\rm def}}{{=}}e^{x}$ : for each $r\geq 0$ there exists $B(r,\cdot)\in L^{2}([0,r])$ so that

P(r,\lambda;\sigma)=\mathrm{exp}\left(i\lambda r\right)+\int\limits_{0}^{r}B(r,s)\mathrm{exp}\left(i\lambda s\right)\,ds\,.

(ii)

$\{P(r,\lambda;\sigma)\}_{r\geq 0}$ is an orthonormal system with respect to $\sigma$ , i.e. for all $f,g\in L^{2}(\mathbb{R}^{+})$ we have equality of the inner-products

\displaystyle\langle f(s),g(s)\rangle_{(ds\,,\,\mathbb{R}_{+})}

\displaystyle=\langle\int\limits_{0}^{\infty}f(s)P(s,\lambda;\sigma)\,ds,\int\limits_{0}^{\infty}g(s)P(s,\lambda;\sigma)\,ds\rangle_{(d\sigma(\lambda),\mathbb{R})}\,.

(1)

Equivalently, the generalized Fourier transform $\mathcal{O}$ is an isometry from $L^{2}(\mathbb{R}^{+})$ to $L^{2}_{\sigma}(\mathbb{R})$ , where

\mathcal{O}f(\lambda)\stackrel{{\scriptstyle\rm def}}{{=}}\int\limits_{0}^{\infty}f(s)P(s,\lambda;\sigma)\,ds\,.

(iii)

$P(r,\lambda;\sigma)$ is the solution to the Krein differential system (20), whose coefficients $A(r)$ uniquely determine $P(r,\lambda;\sigma)$ . As such $P(r,\lambda;\sigma)$ is referred to as “the solution to the Krein system.”

For instance, when $d\sigma=\frac{d\lambda}{2\pi}$ , then $P(r,\lambda;\sigma)=\mathrm{exp}\left(i\lambda r\right)$ , so $\{P(r,\lambda;\sigma)\}_{r\geq 0}$ is simply the Fourier orthonormal system $\{\mathrm{exp}\left(i\lambda r\right)\}_{r\geq 0}$ . See e.g. references [8, 3] for more information on Krein systems.

Notation. When the measure $\sigma$ is clear from context, we simply write $P(r,\lambda)$ . And given a weight $w$ we write $P(r,\lambda;w)\stackrel{{\scriptstyle\rm def}}{{=}}P\left(r,\lambda;w\frac{d\lambda}{2\pi}\right)$ .

First introduced in 1954 by Krein in [8], solutions $P(r,\lambda)$ to Krein systems are the continuous frequency analogue of Orthogonal Polynomials on the Unit Circle (OPUC): given a finite measure $\mu$ on the unit circle $\mathbb{T}\subset\mathbb{C}$ with infinite support, the OPUC consist of an orthonormal sequence $\{\varphi_{n}\}_{n\geq 0}$ in $L^{2}_{\mu}(\mathbb{T})$ , where each $\varphi_{n}$ is a degree $n$ polynomial over $\mathbb{C}$ with positive leading coefficient. One can obtain this sequence by, e.g., applying the Gram-Schmidt algorithm to the polynomials $\{1,z,z^{2},\ldots\}$ . See [12] for a robust reference on OPUC.

Consider the following problem for Krein systems.

Problem 1.1 (The Steklov problem for Krein systems).

If $d\sigma=\frac{w(\lambda)}{2\pi}\,d\lambda$ , what conditions on $w$ guarantee there exists $p>2$ such that $\{P(r,\lambda;\sigma)\}_{r\geq 0}$ is bounded in $L^{p}_{loc}(\sigma,\mathbb{R})$ ? More precisely, when does there exist $p>2$ such that for each compact $\Delta\subset\mathbb{R}$ ,

\sup\limits_{r\geq 0}\|P(r,\cdot;\sigma)\|_{L^{p}_{w}(\Delta)}<\infty\,?

(2)

The Steklov problem and estimates like (2) are related to the following problem: for which measures $\sigma$ is the maximal function

M_{\sigma}(\lambda)\stackrel{{\scriptstyle\rm def}}{{=}}\sup\limits_{r\geq 0}|P(r,\lambda;\sigma)|

finite at almost every $\lambda\in\mathbb{R}$ ? Known as nonlinear Carleson’s Theorem, this was recently proved in [11] for a conjectured class of measures.

The Steklov problem was originally posed for orthogonal polynomials: given the OPUC $\{\varphi_{n}\}_{n\geq 0}$ generated by the measure $d\mu=w\frac{d\theta}{2\pi}$ on the unit circle $\mathbb{T}$ , what conditions on $w$ guarantee there exists $p>2$ for which

\sup\limits_{n\geq 0}\|\varphi_{n}\|_{L^{p}_{w}(\mathbb{T})}<\infty?

(3)

Nazarov first showed that when $w,w^{-1}\in L^{\infty}(\mathbb{T})$ , then (3) holds for some $p>2$ ([4]). Then in [5], Denisov-Rush generalized Nazarov’s result to the case when $w,w^{-1}\in\mathrm{BMO}(\mathbb{T})$ . And most recently in [1], Alexis-Aptekarev-Denisov further generalized these results to the case when $w\in A_{2}(\mathbb{T})$ . Since Krein systems are the continuous analogue of OPUC, we adapt the most recent successes in [1] to solutions of Krein systems generated by a weight $w\in A_{2}(\mathbb{R})$ with $A_{2}(\mathbb{R})$ as defined below (see [13, p.194]).

Definition. Let $p\in(1,\infty)$ . The weight $w\in A_{p}(\mathbb{R})$ if

[w]_{A_{p}(\mathbb{R})}\stackrel{{\scriptstyle\rm def}}{{=}}\sup_{I}\,\left\langle w\right\rangle_{I}\,\left\langle w^{\frac{1}{1-p}}\right\rangle_{I}^{{p-1}}<\infty,\,

where $I$ ranges over the finite intervals in $\mathbb{R}$ .

Note (2) will immediately follow if we can show there exists $\delta=\delta(w)>0$ such that if $p\in[2,2+\delta)$ , then

\sup\limits_{r\geq 0}\|P(r,\lambda;w)-\mathrm{exp}\left(i\lambda r\right)\|_{L^{p}_{w}(\mathbb{R})}<\infty\,.

(4)

Whence the first main theorem of this paper below, for which we recall $L^{p_{1}}(\mathbb{R})+L^{p_{2}}(\mathbb{R})\subset L^{1}(\mathbb{R})+L^{2}(\mathbb{R})$ when $p_{1},p_{2}\in[1,2]$ .

Theorem 1.2.

Suppose that $w-1=u_{1}+u_{2}$ where $u_{1}\in L^{p_{1}}(\mathbb{R}),u_{2}\in L^{p_{2}}(\mathbb{R})$ with $1\leq p_{1}\leq p_{2}\leq 2$ . If $[w]_{A_{2}(\mathbb{R})}\leq\gamma$ , then

(a)

there exists $\epsilon(\gamma)\in(0,\frac{1}{2})$ with $\lim\limits_{\gamma\to 1}\epsilon(\gamma)=\frac{1}{2}$ such that for any $p\in[p_{2},\infty)$ satisfying $\left|\frac{1}{2}-\frac{1}{p}\right|<\epsilon(\gamma)$ , we have

$\sup\limits_{r\geq 0}\|P(r,\lambda;w)-\mathrm{exp}\left(i\lambda r\right)\|_{L^{p}_{w}(\mathbb{R})}<\infty\,.$

(b)

for any $p\in[p_{2},\infty)$ there exists $\tau_{0}(p)\in(0,1)$ such that whenever $\tau\stackrel{{\scriptstyle\rm def}}{{=}}[w]_{A_{2}(\mathbb{R})}-1\leq\tau_{0}(p)$ , we have

\sup\limits_{r\geq 0}\|P(r,\lambda;w)-\mathrm{exp}\left(i\lambda r\right)\|_{L^{p}_{w}(\mathbb{R})}\lesssim_{p}\tau^{\frac{p-p_{2}}{2p}}(\tau^{\frac{p_{2}-p_{1}}{2}}\|u_{1}\|_{L^{p_{1}}(\mathbb{R})}^{p_{1}}+\|u_{2}\|_{L^{p_{2}}(\mathbb{R})}^{p_{2}})^{\frac{1}{p}}\,.

(5)

Solutions to the Krein system generate generalized eigenfunctions for a unique Dirac operator $\cal{D}$ with spectral measure $2\,d\sigma$ . Furthermore, if $A(r)$ , the coefficient of the Krein system (20), is real-valued and in $L^{2}(\mathbb{R}^{+})$ , then when written in its canonical form, $\cal{D}^{2}$ is a diagonal operator whose entries are Schrödinger operators. Thus, Theorem 1.2 allows one to obtain $L^{p}_{w}(\mathbb{R})$ -information on generalized eigenfunctions for the Dirac and Schrödinger operators. See e.g. [3, Sections 14-16] for more details.

We note the following corollaries of Theorem 1.2.

Corollary 1.3.

If $w-1\in L^{p_{1}}(\mathbb{R})+L^{p_{2}}(\mathbb{R})$ for $1\leq p_{1}\leq p_{2}\leq 2$ , then for any $p\in(p_{2},\infty)$ , we have

\lim\limits_{[w]_{A_{2}(\mathbb{R})}\to 1}\sup\limits_{r\geq 0}\|P(r,\lambda;w)-\mathrm{exp}\left(i\lambda r\right)\|_{L^{p}_{w}(\mathbb{R})}=0

where the limit is taken among weights satisfying $\|w-1\|_{L^{p_{1}}(\mathbb{R})+L^{p_{2}}(\mathbb{R})}\leq C$ , with $C\geq 0$ an arbitrary absolute constant.

Proof.

Take $\tau\to 0$ in (5). ∎

Corollary 1.3 states that as $w$ becomes flatter, $P(r,\lambda)$ tends to standard exponential, demonstrating some continuity of solutions to Krein systems with respect to the measure.

Corollary 1.4.

Suppose $w-1\in L^{p_{1}}(\mathbb{R})+L^{p_{2}}(\mathbb{R})$ with $1\leq p_{1}\leq p_{2}<2$ . If $[w]_{A_{2}(\mathbb{R})}\leq\gamma$ , then there exists $\delta(\gamma,p_{2})>0$ such that

P(r,\lambda;w)-\mathrm{exp}\left(i\lambda r\right)\in L^{p}_{w}(\mathbb{R})

for all $r>0$ , $p\in[2-\delta,\infty)$ .

Proof.

Apply Theorem 1.2 (a). ∎

Corollary 1.4 is non-trivial, since as we discuss in Section 3, a priori all we can say about $P(r,\lambda;w)-\mathrm{exp}\left(i\lambda r\right)$ is that it belongs in $L^{p}_{w}(\mathbb{R})$ for $p\in[2,\infty)$ .

The requirement that $p\geq p_{2}$ in Theorem 1.2 is sharp in the sense of Proposition 1.5 below, which morally says that the decay of $\lambda\mapsto P(r,\lambda;w)-\mathrm{exp}\left(i\lambda r\right)$ cannot exceed the decay of $\lambda\mapsto w(\lambda)-1$ .

Proposition 1.5.

If $p_{2}\in(1,2]$ , then for every $\delta>0$ , there exists a weight $w$ such that $w-1\in L^{p_{2}}(\mathbb{R})$ , $[w]_{A_{2}(\mathbb{R})}\leq 1+\delta$ and

\sup\limits_{r\geq 0}\|P(r,\lambda;w)-\mathrm{exp}\left(i\lambda r\right)\|_{L^{p}_{w}(\mathbb{R})}=\infty

for every $p<p_{2}$ .

In the same vein as Theorem 1.2 (a), one may also estimate a “remainder” term, which we define precisely in Section 6. Let $\langle\lambda\rangle\stackrel{{\scriptstyle\rm def}}{{=}}(1+\lambda^{2})^{1/2}$ . If $\langle\lambda\rangle^{k}(w-1)\in L^{1}(\mathbb{R})$ then the remainder function $R_{k,r}(\lambda)$ exists and satisfies

R_{k,r}(\lambda)=\lambda^{k}(P(r,\lambda)-\mathrm{exp}\left(i\lambda r\right))-\sum\limits_{l=0}^{k-1}\lambda^{l}a_{k-l,r}(\lambda)\,,

where $a_{l,r}(\lambda)\in L^{\infty}(\mathbb{R})$ , $l=1,\dots,k$ . Note that $R_{k,r}$ measures how much $\lambda^{k}(P(r,\lambda)-\mathrm{exp}\left(i\lambda r\right))$ deviates from a polynomial-like term of “degree” $k-1$ . In Theorem 1.6 below, we estimate $\|R_{1,r}\|_{L^{p}_{w}(\mathbb{R})}$ , which quantifies the decay of $P(r,\lambda;w)-\mathrm{exp}\left(i\lambda r\right)$ , providing information with no analogue for OPUC. Furthermore, the assumptions of Theorem 1.6 below imply

a_{1,r}(\lambda)=\mathrm{exp}\left(i\lambda r\right)\alpha_{\infty}(r)+\alpha_{2}(r)\,,

where each $\alpha_{q}(r)$ does not depend on $\lambda$ , $\alpha_{q}\in L^{q}(\mathbb{R}^{+})$ , and $\lim\limits_{r\to\infty}\alpha_{\infty}(r)$ exists (see Lemma 6.2); as such, the estimate on $\|R_{1,r}\|_{L^{p}_{w}(\mathbb{R})}$ below also qualifies the behavior of the “continuous polynomials” $P(r,\lambda;w)$ for $r$ large. We will need the following definitions for what follows.

•

Let $H^{2}(\Omega)$ denotes the Hardy space in the domain $\Omega\subset\mathbb{C}$ .

•

If $B$ is a Banach space with norm $\|\cdot\|_{B}$ , $(\mu,X)$ is a measure space and $p\in[1,\infty]$ , let

L^{p}_{\mu}(X;B)\stackrel{{\scriptstyle\rm def}}{{=}}\left\{f:X\to B~{}\biggr{|}~{}\|f\|_{L^{p}_{\mu}(X;B)}\stackrel{{\scriptstyle\rm def}}{{=}}\left\|\|f\|_{B}\right\|_{L^{p}_{\mu}(X)}<\infty\right\}\,.

•

Given two Banach spaces $B_{1},B_{2}$ with norms $\|\cdot\|_{B_{1}}$ and $\|\cdot\|_{B_{2}}$ , for each $f\in B_{1}+B_{2}$ define

$\|f\|_{B_{1}+B_{2}}\stackrel{{\scriptstyle\rm def}}{{=}}\inf\limits_{f=f_{1}+f_{2}}\|f_{1}\|_{B_{1}}+\|f_{2}\|_{B_{2}}\,.$

Theorem 1.6.

Suppose that $\langle\lambda\rangle^{q}(w-1)\in L^{1}(\mathbb{R})$ for some $q>2$ and

\mathrm{exp}\left(\frac{1}{2\pi i}\int\limits_{-\infty}^{\infty}\frac{\log(w(t))}{t-\lambda}\,dt\right)-1=\int\limits_{0}^{\infty}h(x)\mathrm{exp}\left(i\lambda x\right)\,dx\in H^{2}(\mathbb{C}^{+})\,.

(6)

If $[w]_{A_{2}(\mathbb{R})}\leq\gamma$ , then there exists $\epsilon(\gamma)>0$ such that whenever $p\in(1,q]$ satisfies $\left|\frac{1}{p}-\frac{1}{2}\right|<\epsilon(\gamma)$ , then

\|R_{1,r}(\lambda)\|_{L^{2}_{dr}(\mathbb{R}^{+};L^{p}_{w}(\mathbb{R}))+L^{\infty}_{dr}(\mathbb{R}^{+};L^{p}_{w}(\mathbb{R}))}<\infty\,.

Compare Theorem 1.6 and Theorem 1.2 (a), the latter of which can be understood as stating that the $L^{\infty}_{dr}(\mathbb{R}^{+};L^{p}_{w}(\mathbb{R}))$ -norm of $0$ th order remainder $R_{0,r}$ is finite.

We can also estimate higher order remainders when the weight $w$ is very close to $1$ .

Theorem 1.7.

Suppose $(w-1)\langle\lambda\rangle^{k}\in L^{1}(\mathbb{R})$ for some integer $k\geq 0$ . If $\|(w-1)\langle\lambda\rangle^{k}\|_{L^{\infty}(\mathbb{R})}\leq\delta$ , for some $\delta\in[0,1)$ , then

(a)

there exists $\epsilon(\delta)\in(0,\frac{1}{2})$ , with $\lim\limits_{\delta\to 0}\epsilon(\delta)=\frac{1}{2}$ , such that for all $p$ satisfying $\left|\frac{1}{2}-\frac{1}{p}\right|<\epsilon(\delta)$ ,

$\sup\limits_{r>0}\|R_{k,r}(\lambda)\|_{L^{p}(\mathbb{R})}<\infty\,.$

(b)

for any $p\in(1,\infty)$ , there exists $\delta_{0}(p)>0$ such that for all $\delta\in(0,\delta_{0}(p))$ , we have

\sup\limits_{r>0}\|R_{k,r}(\lambda)\|_{L^{p}(\mathbb{R})}\lesssim_{p,k}\delta^{1-\frac{1}{p}}\left(\|\langle\lambda\rangle^{k}(w-1)\|_{L^{1}(\mathbb{R})}^{\frac{1}{p}}+\|\langle\lambda\rangle^{k}(w-1)\|_{L^{1}(\mathbb{R})}^{1+\frac{1}{p}}\right)\,.

In particular Theorem 1.7 (b) has the following corollary, which shows some continuity of $R_{k,r}$ with respect to the measure.

Corollary 1.8.

If $\|(w-1)\langle\lambda\rangle^{k}\|_{L^{1}(\mathbb{R})}\leq C$ , where $C$ is any fixed constant and $k$ a nonnegative integer, then for any fixed $1<p<\infty$ we have

\lim\limits_{\|(w-1)\langle\lambda\rangle^{k}\|_{L^{\infty}(\mathbb{R})}\to 0}\sup\limits_{r>0}\|R_{k,r}(\lambda)\|_{L^{p}(\mathbb{R})}=0\,.

While examples of weights to which we can apply Theorem 1.7 are easy to find, e.g. any weight satisfying $|w-1|\leq\frac{\delta}{\langle\lambda\rangle^{k+2}}$ , let’s mention some weights to which we can apply Theorems 1.2 and 1.6. Define

w_{\beta}(\lambda)\stackrel{{\scriptstyle\rm def}}{{=}}\begin{cases}|\lambda|^{\beta}&\text{when }|\lambda|\leq 1\\ 1&\text{when }|\lambda|>1\end{cases}\,,

which is an $A_{2}(\mathbb{R})$ weight when $|\beta|<1$ . Then finite products $w(\lambda)\stackrel{{\scriptstyle\rm def}}{{=}}\prod\limits_{j=1}^{n}w_{\beta_{j}}(\lambda-\lambda_{j})$ , where $|\beta_{j}|<1$ and $\lambda_{1},\ldots,\lambda_{n}$ are all distinct, satisfy the conditions of Theorems 1.2 and 1.6.

In practice, condition (6) is difficult to check, thus making it difficult to specify weights with infinitely many singularities to which Theorem 1.6 applies. However, if we require the singularities be well spaced out, one can generate other weights that satisfy the assumptions of Theorem 1.2. Consider for instance

w(\lambda)\stackrel{{\scriptstyle\rm def}}{{=}}\,\prod\limits_{j=0}^{\infty}w_{\beta_{j}}\left(\frac{\lambda-2^{j}}{\epsilon_{j}}\right)\,,

where $\sup\limits_{j}|\beta_{j}|<1$ , and $\{\epsilon_{j}\}_{j\geq 0}$ is a sequence of positive reals such that $\sum\limits_{j}\epsilon_{j}<\infty$ .

As far as methods are concerned, the proofs of Theorems 1.2 and 1.6 are based on [1]’s proof of a positive answer to the OPUC Steklov problem for $A_{2}(\mathbb{T})$ weights: this involves the theory of weighted operators, some basic spectral theory and complex interpolation. Meanwhile Theorem 1.7 is based on the original, elegantly simply method of Nazarov used to prove [4, Theorem 2.1], which involves just basic knowledge of the Hilbert transform.

1.1. Lingering questions: some potentially open problems

Regarding lingering questions, let us first comment on our methods. Essential to our proof of Theorem 1.2 is the orthogonality

P(r,\lambda;w)\perp_{w}\mathrm{Range}\,\mathcal{P}_{[0,r]}\,,

where $\mathcal{P}_{[0,r]}$ is Fourier projection onto the frequency band $[0,r]$ . But using for instance Lemma 4.1, we end up showing

P(r,\lambda;w)\perp_{wq(\lambda)}\mathrm{Range}\,\mathcal{P}_{[0,r]}\,,

where $q(\lambda)$ is any nonnegative polynomial. Thus, as far as the algebra in the proof of Theorem 1.2 is concerned, one is tempted to consider the case $\widetilde{w}\stackrel{{\scriptstyle\rm def}}{{=}}wq(\lambda)\in A_{2}(\mathbb{R})$ and prove another version of this result. But issues arise: for Krein systems and their solutions to be well-defined, the weight $w$ needs to be “centered” near $1$ , with e.g. $w-1\in L^{1}(\mathbb{R})+L^{2}(\mathbb{R})$ . This directly conflicts with $\widetilde{w}\in A_{2}(\mathbb{R})$ . Indeed, heuristically $\widetilde{w}\in A_{2}(\mathbb{R})$ means $\widetilde{w}$ is more or less constant, which would then mean $w$ decays to $0$ , which would contradict the required decay of $w-1$ . A possible remedy here is to study the more general de Branges canonical systems, wherein the weight $w$ can deviate from $1$ in more exotic ways. I have not investigated this potential framing of the problem and thus so far it remains an open question.

While Proposition 1.5 shows that the requirement that $p\geq p_{2}$ is sharp in Theorem 1.2, it is unclear to me whether the requirement that $p\leq q$ is sharp in the statement of Theorem 1.6. Furthermore, while Proposition 1.5 had a nice interpretation — that $P(r,\lambda;w)-\mathrm{exp}\left(i\lambda r\right)$ cannot decay faster that $w-1$ — the requirement that $p\leq q$ is more difficult to interpret. in elucidating this part of Theorem 1.6.

Finally, if $w\in A_{2}(\mathbb{R})$ and $w-1\in L^{1}(\mathbb{R})+L^{2}(\mathbb{R})$ , then Theorem 1.2 implies

\sup\limits_{r>0}\|P(r,\lambda)-\mathrm{exp}\left(i\lambda r\right)\|_{L^{p}_{w}(\mathbb{R})}<\infty\,

(7)

for some $p>2$ . While this in turn implies

\sup\limits_{r>0}\|P(r,\lambda)\|_{L^{p}_{w(\lambda)\frac{d\lambda}{1+\lambda^{2}}}(\mathbb{R})}<\infty

(8)

for some $p>2$ , I feel the estimate (8) is more natural to try and prove directly. I wonder if there are other methods or assumptions which would more readily yield (8) without going through (7).

1.2. Organization of this Paper

This paper is organized as follows. First, we outline the basic theory of Krein systems in Section 2. In Section 3, we discuss a priori which $L^{p}_{w}(\mathbb{R})$ spaces $P(r,\lambda;w)-\mathrm{exp}\left(i\lambda r\right)$ belongs to. In Section 4 we prove a useful orthogonality lemma. We spend Sections 5, 6 and 7 proving the main results of this paper, i.e. Theorem 1.2 and Proposition 1.5, Theorem 1.6, and Theorem 1.7. In Sections 8 and 9 we prove the technical Lemmas 5.3 and 5.4 essential in the proofs of the results in this paper. And finally, in Appendix A, we prove Proposition 5.5.

1.3. Notation

If $B$ is a Banach space, we denote its dual by $B^{*}$ , and the space of bounded operators on $B$ by $\cal{L}(B)$ .

If $X\subset\mathbb{R}^{d}$ and $d\mu=w(x)\,dx$ , we define $L^{p}_{w}(X)\stackrel{{\scriptstyle\rm def}}{{=}}L^{p}_{\mu}(X)$ . If $\mu$ is Lebesgue measure, we omit $\mu$ , i.e. $L^{p}(X)\stackrel{{\scriptstyle\rm def}}{{=}}L^{p}_{dx}(X)$ . If $T$ is a bounded linear operator between two Banach spaces $L^{p}_{\mu}(X)$ , $L^{q}_{\nu}(Y)$ , we write its norm as $\|T\|_{p,q}$ .

If $p\in[1,\infty]$ , the dual exponent is denoted by $p^{\prime}=p/(p-1)$ .

For $f\in L^{1}_{\mu}(X)$ , denote the average of $f$ over the measure space $(\mu,X)$ by

\langle f\rangle_{X,\mu}\stackrel{{\scriptstyle\rm def}}{{=}}\frac{1}{\mu(X)}\int_{X}fd\mu\,.

If $X\subset\mathbb{R}^{d}$ and $\mu$ is Lebesgue measure, we will simply write $\langle f\rangle_{X}$ .

In this paper we work with spatial variable $\lambda\in\mathbb{R}$ and frequency variable $r\in\mathbb{R}^{+}$ . Thus for Banach spaces like $L^{p}_{\sigma}(\mathbb{R})$ or $L^{p}_{w}(\mathbb{R})$ , the variable of integration is understood to be $\lambda$ , i.e. $\|f(r,\lambda)\|_{L^{p}_{w}(\mathbb{R})}\stackrel{{\scriptstyle\rm def}}{{=}}\|f(r,\lambda)\|_{L^{p}_{w(\lambda)d\lambda}}$ . And given a measure, weight or function over $\mathbb{R}$ , we write them with respect to variable $\lambda$ , i.e. $w=w(\lambda)$ .

Given a measure space $(\mu,X)$ , we denote the $L^{2}_{\mu}(X)$ inner product by

\langle f,g\rangle_{(\mu,X)}\stackrel{{\scriptstyle\rm def}}{{=}}\int\limits_{X}f(x)\overline{g(x)}\,d\mu(x)\,.

If it is clear from context that $X=\mathbb{R}^{d}$ , we will simply write $\langle f,g\rangle_{\mu}$ .

If $\mathrm{exp}\left(x\right)\stackrel{{\scriptstyle\rm def}}{{=}}e^{x}$ , then the Fourier transform $\hat{f}$ , or $\cal{F}f$ , of a function $f$ and its inverse transform $\check{f}$ , or $\cal{F}^{-1}f$ are given by

\widehat{f}(\xi)\stackrel{{\scriptstyle\rm def}}{{=}}\int\limits_{-\infty}^{\infty}f(x)\mathrm{exp}\left(-2\pi ix\xi\right)\,dx\,,\quad\widecheck{f}(x)\stackrel{{\scriptstyle\rm def}}{{=}}\int\limits_{-\infty}^{\infty}f(\xi)\mathrm{exp}\left(2\pi ix\xi\right)d\xi\,.

Given a set $A\subseteq X$ where $X=\mathbb{R}^{d}$ , we will use notation $A^{c}$ for its complement, i.e., $A^{c}=X\backslash A$ .

Let $C_{c}^{\infty}(U)$ denote the space of smooth functions compactly supported in $U$ , an open subset of $\mathbb{R}^{d}$ .

For two non-negative functions $f_{1}$ and $f_{2}$ , we write $f_{1}\lesssim f_{2}$ if there is an absolute constant $C$ such that

f_{1}\leqslant Cf_{2}

for all values of the arguments of $f_{1}$ and $f_{2}$ . If the constant depends on a parameter $\alpha$ , we will write $f_{1}\lesssim_{\alpha}f_{2}$ . We define $\gtrsim$ similarly and write $f_{1}\sim f_{2}$ if $f_{1}\lesssim f_{2}$ and $f_{2}\lesssim f_{1}$ simultaneously.

If a constant $C$ depends on parameter $\alpha$ , we will express this by writing $C(\alpha)$ or $C_{\alpha}$ .

2. Krein Systems Basics

In this section we explain the basics of Krein systems following [3], listing most facts needed for this paper without proof; see [3] for an accessible in-depth survey on Krein systems.

Suppose $\sigma$ is a Poisson-finite measure over $\mathbb{R}$ , i.e.

\int\limits_{\mathbb{R}}\frac{d\sigma(\lambda)}{1+\lambda^{2}}<\infty\,.

Definition. A function $H:\mathbb{R}\to\mathbb{C}$ is Hermitian if $H(-x)=-\overline{H(x)}$ .

Definition. A Hermitian function $H\in L^{2}_{loc}(\mathbb{R})$ is the accelerant associated to the measure $\sigma$ if there exists a real constant $\beta$ such that for all $x\in\mathbb{R}$ ,

\int\limits_{0}^{x}(x-s)H(s)\,ds=i\beta x+\int\limits_{-\infty}^{\infty}\left(1+\frac{i\lambda x}{1+\lambda^{2}}-\mathrm{exp}\left(i\lambda x\right)\right)\frac{1}{\lambda^{2}}\,\left(d\sigma(\lambda)-\frac{d\lambda}{2\pi}\right)\,.

(9)

Formally differentiating (9) twice yields “ $H(2\pi x)=\widecheck{(d\sigma-\frac{d\lambda}{2\pi})}(x)$ .” Thus intuitively the accelerant captures the moments of the signed measure $d\sigma-\frac{d\lambda}{2\pi}$ .

If an accelerant $H$ exists, then for each $r>0$ , define the operator $\mathcal{H}_{r}$ on $L^{2}([0,r])$ by

\mathcal{H}_{r}f(x)\stackrel{{\scriptstyle\rm def}}{{=}}\int\limits_{0}^{r}H(x-y)f(y)\,dy\,.

(10)

Of interest is when $I+\mathcal{H}_{r}>0$ , which occurs for a large class of measures.

Lemma 2.1.

Suppose $d\sigma=w\frac{d\lambda}{2\pi}$ . If $w-1\in L^{1}(\mathbb{R})+L^{2}(\mathbb{R})$ , then

H(x)\stackrel{{\scriptstyle\rm def}}{{=}}\frac{1}{2\pi}(\widecheck{w-1})\left(\frac{x}{2\pi}\right)=\mathcal{F}^{-1}(w(2\pi\cdot)-1)(x)

is the accelerant associated to $d\sigma$ , and $I+\mathcal{H}_{r}>0$ .

Proof.

Let $a\in L^{1}(\mathbb{R})+L^{2}(\mathbb{R})$ and first define $H\stackrel{{\scriptstyle\rm def}}{{=}}\frac{1}{2\pi}\widecheck{a}\left(\frac{x}{2\pi}\right)\in C_{0}(\mathbb{R})+L^{2}(\mathbb{R})\subset L^{2}_{loc}(\mathbb{R})$ and

\beta\stackrel{{\scriptstyle\rm def}}{{=}}\frac{1}{2\pi}\int\limits_{\mathbb{R}}\frac{\lambda}{1+\lambda^{2}}a(\lambda)\,d\lambda\,.

(11)

Suppose we can show that

\int\limits_{0}^{x}(x-s)H(s)\,ds=i\beta x+\frac{1}{2\pi}\int\limits_{-\infty}^{\infty}\left(1+\frac{i\lambda x}{1+\lambda^{2}}-\mathrm{exp}\left(i\lambda x\right)\right)\frac{1}{\lambda^{2}}a(\lambda)\,d\lambda\,

(12)

holds for all $x\in\mathbb{R}$ . Then setting $a=w-1$ , we will show that $H(x)\stackrel{{\scriptstyle\rm def}}{{=}}\frac{1}{2\pi}\widecheck{(w-1)}\left(\frac{x}{2\pi}\right)$ is the accelerant associated to $d\sigma=\frac{w(\lambda)}{2\pi}d\lambda$ , i.e. we will show (12) holds.

We now note that $I+\mathcal{H}_{r}>0$ : take $f\in L^{2}([0,r])$ and extend $f$ to a function over $\mathbb{R}$ by defining $f(x)=0$ for $x\notin[0,r]$ . Then Fourier inversion for distributions yields

	$\displaystyle\langle(I+\mathcal{H}_{r})f,f\rangle_{(dx,\mathbb{R})}=\langle f,f\rangle_{(dx,\mathbb{R})}+\langle H\ast f,f\rangle_{(dx,\mathbb{R})}$	$\displaystyle=\langle\hat{f},\hat{f}\rangle_{(d\lambda,\mathbb{R})}+\langle(w(2\pi\cdot)-1)\hat{f},\hat{f}\rangle_{(d\lambda,\mathbb{R})}$
		$\displaystyle=\\|\hat{f}\\|_{L^{2}_{w(2\pi\cdot)}(\mathbb{R})}^{2}$
		$\displaystyle>0$

whenever $f\neq 0$ , i.e. $I+\mathcal{H}_{r}>0$ .

It remains to show (12) holds for $H,\beta,a$ as above. If we can show (12) holds for $a\in L^{1}(\mathbb{R})$ and for $a\in L^{2}(\mathbb{R})$ , then by linearity (12) holds for all $a\in L^{1}(\mathbb{R})+L^{2}(\mathbb{R})$ .

The case that $a\in L^{1}(\mathbb{R})$ : First note that both sides of (12) are well-defined functions for $a\in L^{1}$ . Next note that the second derivative (with respect to variable $x$ ) of the left side of (12) equals the second derivative of the right side of (12). Indeed, this follows from the dominated convergence theorem and that $a\in L^{1}(\mathbb{R})$ implies $H\in C(\mathbb{R})$ .

Thus the first derivatives of each side of (12) will be identical so long as we can verify they agree at a point, say $x=0$ . But $\beta$ is defined precisely to make this occur; one can check this using the dominated convergence theorem.

Thus to verify (12) holds for all $x$ , it suffices to verify each side of (12) agrees at a point, say $x=0$ . Direct computation shows each side of (12) is $0$ at $x=0$ . This completes the proof of this case.

The case that $a\in L^{2}(\mathbb{R})$ : First let $a_{n}\in L^{1}(\mathbb{R})\cap L^{2}(\mathbb{R})$ approximate $a$ in $L^{2}(\mathbb{R})$ , e.g. define

a_{n}\stackrel{{\scriptstyle\rm def}}{{=}}\chi_{[-n,n]}a\,.

By the previous case, if $H_{n}(x)\stackrel{{\scriptstyle\rm def}}{{=}}\frac{1}{2\pi}\widecheck{a_{n}}\left(\frac{x}{2\pi}\right)$ , and $\beta_{n}=\frac{1}{2\pi}\int\limits_{\mathbb{R}}\frac{\lambda}{1+\lambda^{2}}a_{n}(\lambda)\,d\lambda$ , then (12) holds for $(H_{n},\beta_{n},a_{n})$ . We note that $H_{n}\to H$ in $L^{2}(\mathbb{R})$ by Plancherel’s theorem; we also have $\beta_{n}\to\beta$ for $\beta$ given by (11). Now consider (12) for $(H_{n},\beta_{n},a_{n})$ and take $n\to\infty$ to get it for $(H,\beta,a)$ . This completes the proof. ∎

Assume $H\in L^{2}_{loc}(\mathbb{R})$ is an accelerant for which $I+\mathcal{H}_{r}>0$ for each $r>0$ . Then the resolvent operator

\mathcal{G}_{r}\stackrel{{\scriptstyle\rm def}}{{=}}I-(I+\mathcal{H}_{r})^{-1}

(13)

is an integral operator on $L^{2}([0,r])$ with kernel $\Gamma_{r}(s,t)\in L^{2}_{ds\times dt}([0,r]^{2})$ possessing the following properties.

Properties of the Resolvent kernel $\Gamma$

(i)

Resolvent Identity: from (13) we have

\displaystyle\Gamma_{r}(t,s)+\int\limits_{0}^{r}H(t-u)\Gamma_{r}(u,s)\,du

\displaystyle=H(t-s)\,,\;0\leq s,t\leq r\,.

(14)

(ii)

Symmetries: $\Gamma$ has following symmetries:

$\Gamma_{r}(s,t)=\overline{\Gamma_{r}(t,s)}\,,\quad\Gamma_{r}(s,t)=\Gamma_{r}(r-t,r-s)\,,\quad 0\leq s,t\leq r\,.$ (15)
(iii)

Regularity inherited from accelerant: if $H\in C^{k}(\mathbb{R})$ for some $k\geq 0$ , then for each $r>0$ , $\Gamma_{r}(\cdot,\cdot)\in C^{k}([0,r]^{2})$ . Furthermore, $\Gamma_{r}(s,t)$ is continuously differentiable in $r$ and

$\partial_{r}\Gamma_{r}(s,t)=-\Gamma_{r}(s,r)\Gamma_{r}(r,t)\,,\quad 0\leq s,t\leq r\,.$ (16)
(iv)

Continuity in $L^{2}$ : Define

$g_{r}(s)\stackrel{{\scriptstyle\rm def}}{{=}}\begin{cases}\Gamma_{r}(s,0)&\text{ if }0\leq s\leq r\\ 0&\text{ if }s>r\end{cases}\,.$ (17)

If $H\in L^{2}_{loc}(\mathbb{R})$ , then the mapping $r\mapsto g_{r}$ is an element of $C([0,R],L^{2}([0,R]))$ for all $R>0$ (see [3, Chapter 6]).

Using $\Gamma_{r}$ , one can define the “continuous polynomials” $\{P(r,\lambda;\sigma)\}_{r\geq 0}$ . These are entire functions $P(r,\cdot\,;\sigma)$ , each of “degree” $r$ , given by

\displaystyle P(r,\lambda;\sigma)

\displaystyle\stackrel{{\scriptstyle\rm def}}{{=}}\mathrm{exp}\left(i\lambda r\right)-\int\limits_{0}^{r}\Gamma_{r}(r,t)\,\mathrm{exp}\left(i\lambda t\right)\,dt=(I+\mathcal{H}_{r})^{-1}(\mathrm{exp}\left(i\lambda\cdot\right))(r)\,.

(18)

This definition is motivated by analogy to the OPUC: if one considers the Toeplitz matrix

\mathbf{T_{n}}\stackrel{{\scriptstyle\rm def}}{{=}}\begin{pmatrix}c_{0}&c_{1}&\ldots&c_{n}\\ c_{-1}&c_{0}&\ldots&c_{n-1}\\ \vdots&\vdots&\ddots&\vdots\\ c_{-n}&c_{-n+1}&\ldots&c_{0}\end{pmatrix}\,,

where $c_{j}\stackrel{{\scriptstyle\rm def}}{{=}}\int\limits_{\mathbb{T}}z^{-j}\,d\mu$ are the moments of a measure $\mu$ on $\mathbb{T}$ , then the orthogonal polynomial $\varphi_{n}(z)$ associated to $\mu$ is given by last row of the column-vector

\sqrt{\frac{\det\mathbf{T_{n}}}{\det\mathbf{T_{n-1}}}}\,\,\mathbf{T_{n}}^{-1}\begin{pmatrix}1&z&z^{2}&\ldots&z^{n}\end{pmatrix}^{\intercal}\,.

And so similar to the OPUC, $\{P(r,\lambda;\sigma)\}_{r\geq 0}$ is an orthonormal system with respect to $d\sigma$ , i.e. (1) holds for all $f,g\in L^{2}(\mathbb{R}^{+})$ . Furthermore, like how each $z^{n}$ may be expressed as a sum of the OPUC $\{\varphi_{k}\}_{0\leq k\leq n}$ , each exponential $\mathrm{exp}\left(i\lambda r\right)$ is a “continuous sum” of the polynomials $\{P(s,\lambda;\sigma\}_{0\leq s\leq r}$ , i.e. given $r\in(0,R)$ we have

\mathrm{exp}\left(i\lambda r\right)=P(r,\lambda;\sigma)+\int\limits_{0}^{r}L_{R}(r,s)P(s,\lambda;\sigma)\,ds=(I+\cal{L}_{R})P(\cdot,\lambda;\sigma)(r)\,,

(19)

for some Volterra operator $\cal{L}_{R}$ bounded on $L^{2}([0,R])$ .

Similar to the role the Verblunsky coefficients $\{\alpha_{n}\}$ play for OPUC (see e.g. [12, Theorem 1.5.2]), there exists $A(r)\in L^{2}_{loc}(\mathbb{R}^{+})$ such that $\begin{pmatrix}P(r,\lambda)&P_{*}(r,\lambda)\end{pmatrix}^{\intercal}$ is the solution to the Krein (differential) system

\begin{cases}P^{\prime}&=i\lambda P-\overline{A}P_{*}\,,\quad P(0,\lambda)=1\\ P_{*}^{\prime}&=-AP\,,\quad P_{*}(0,\lambda)=1\end{cases}\,,\qquad\lambda\in\mathbb{C}\,,

(20)

where the derivative is taken with respect to $r$ and $P_{*}(r,\lambda;\sigma)\stackrel{{\scriptstyle\rm def}}{{=}}\mathrm{exp}\left(i\lambda r\right)\overline{P(r,\overline{\lambda};\sigma)}$ . If $H$ is continuous, then $A(r)$ is given by the $0^{\text{th}}$ “coefficient” of $P(r,\lambda)$ , i.e.

\overline{A(r)}=\Gamma_{r}(r,0)\,.

(21)

3. A priori estimates: when does $P(r,\lambda)-\mathrm{exp}\left(i\lambda r\right)\in L^{p}(\mathbb{R})$ ?

Suppose, just within the scope of this paragraph, that $w$ is very regular, e.g. $w,w^{-1}\in L^{\infty}(\mathbb{R})$ . For (4) to hold for a fixed $p$ , it is then necessary that $P(r,\lambda)-\mathrm{exp}\left(i\lambda r\right)\in L^{p}(\mathbb{R})$ for each $r>0$ to begin with. But this is not immediate for arbitrary weights. How do properties of $w$ determine which $L^{p}$ spaces $P(r,\lambda)-\mathrm{exp}\left(i\lambda r\right)$ belongs to?

Proposition 3.1.

If $w-1\in L^{1}(\mathbb{R})+L^{2}(\mathbb{R})$ , then $P(r,\lambda;w)-\mathrm{exp}\left(i\lambda r\right)\in L^{p}(\mathbb{R})$ for $2\leq p\leq\infty$ , for each $r>0$ . In particular, $P(r,\lambda;w)-\mathrm{exp}\left(i\lambda r\right)\in L^{p}_{w}(\mathbb{R})$ for $2\leq p<\infty$ , for each $r>0$ .

Thus a priori, if $w-1\in L^{1}(\mathbb{R})+L^{2}(\mathbb{R})$ all we can say is that $P(r,\lambda)-\mathrm{exp}\left(i\lambda r\right)\in L^{p}_{w}(\mathbb{R})$ for $2\leq p<\infty$ .

We spend the rest of this section proving Proposition 3.1.

Lemma 3.2.

If an accelerant $H\in L^{2}_{loc}(\mathbb{R})$ satisfies $I+\mathcal{H}_{r}>0$ for each $r>0$ , where $\mathcal{H}_{r}$ is as in (10), then

(a)

$P(r,\lambda)-\mathrm{exp}\left(i\lambda r\right)\in L^{p}(\mathbb{R})$ for all $2\leq p\leq\infty$ .

(b)

for each $R>0$ , we have

\sup\limits_{0\leq r\leq R}|P(r,\lambda)-\mathrm{exp}\left(i\lambda r\right)|\leq R^{1/2}\|g_{r}\|_{C([0,R],L^{2}([0,R]))}<\infty\,,

(22)

where $g_{r}$ is as in (17). Furthermore, $P(r,\lambda)-\mathrm{exp}\left(i\lambda r\right)$ is continuous in each variable.

Proof.

Let us first focus on part (a). By Hölder’s inequality it suffices to show $P(r,\lambda)-\mathrm{exp}\left(i\lambda r\right)$ is an element of $L^{p}(\mathbb{R})$ for $p=2,\infty$ . By (15) we have

P(r,\lambda)-\mathrm{exp}\left(i\lambda r\right)=-\mathrm{exp}\left(i\lambda r\right)\int\limits_{0}^{r}\Gamma_{r}(s,0)\mathrm{exp}\left(-i\lambda s\right)\,ds=-\mathrm{exp}\left(i\lambda r\right)\int\limits_{0}^{r}g_{r}(s)\mathrm{exp}\left(-i\lambda s\right)\,ds\,,

(23)

where $g_{r}$ is as in (17). Since $g_{r}$ is an element of $L^{2}([0,r])$ , then applying Plancherel to (23) yields $P(r,\lambda)-\mathrm{exp}\left(i\lambda r\right)\in L^{2}(\mathbb{R})$ .

Applying Cauchy-Schwarz to the right side of (23) gives the $L^{\infty}(\mathbb{R})$ estimate needed to complete the proof of (a); in fact Cauchy-Schwarz yields the estimate (22). The continuity in part 22 follows from $r\mapsto g_{r}$ being an element of $C([0,R],L^{2}([0,R]))$ for each $R>0$ (see Property (iv) of the resolvent kernel). ∎

Lemma 3.3.

If $\langle\lambda\rangle^{k}(w-1)\in L^{1}(\mathbb{R})$ for $k\geq 0$ an integer, then $w\frac{d\lambda}{2\pi}$ has accelerant $\frac{1}{2\pi}(\widecheck{w-1})\left(\frac{x}{2\pi}\right)\in C^{k}(\mathbb{R})$ , the resolvent kernel $\Gamma$ exists, and $\Gamma_{r}(\cdot,\cdot)\in C^{k}([0,r]^{2})$ for each $r>0$ .

Proof.

By Lemma 2.1, $H(x)\stackrel{{\scriptstyle\rm def}}{{=}}\frac{1}{2\pi}(\widecheck{w-1})\left(\frac{x}{2\pi}\right)$ is the accelerant, and $I+\mathcal{H}_{r}>0$ . The latter conclusion $\Gamma$ exists. Since $\langle\lambda\rangle^{k}(w-1)\in L^{1}(\mathbb{R})$ , then $H\in C^{k}(\mathbb{R})$ and so $\Gamma_{r}(\cdot,\cdot)\in C^{k}([0,r]^{2})$ by Property (iii) of the Resolvent kernel $\Gamma$ . ∎

Proof of Proposition 3.1.

Lemma 2.1 implies that the measure $d\sigma=w(\lambda)\frac{d\lambda}{2\pi}$ has accelerant $H\in L^{2}_{loc}(\mathbb{R})$ . It follows from Lemma 3.2 (a) that

P(r,\lambda)-\mathrm{exp}\left(i\lambda r\right)\in L^{p}(\mathbb{R})\text{ for }p\in[2,\infty]\,.

To get the weighted-norm estimate, write

	$\displaystyle w\|P(r,\lambda)-\mathrm{exp}\left(i\lambda r\right)\|^{p}$	$\displaystyle=(w-1)\|P(r,\lambda)-\mathrm{exp}\left(i\lambda r\right)\|^{p}+\|P(r,\lambda)-\mathrm{exp}\left(i\lambda r\right)\|^{p}$
		$\displaystyle=u_{1}\|P(r,\lambda)-\mathrm{exp}\left(i\lambda r\right)\|^{p}+u_{2}\|P(r,\lambda)-\mathrm{exp}\left(i\lambda r\right)\|^{p}+\|P(r,\lambda)-\mathrm{exp}\left(i\lambda r\right)\|^{p}\,,$

where $w-1=u_{1}+u_{2}$ , with $u_{1}\in L^{1}(\mathbb{R})$ and $u_{2}\in L^{2}(\mathbb{R})$ . We are left with checking the sum is integrable.

We just showed $|P(r,\lambda)-\mathrm{exp}\left(i\lambda r\right)|^{p}$ is integrable for any $p\in[2,\infty]$ . Then $u_{1}|P(r,\lambda)-\mathrm{exp}\left(i\lambda r\right)|^{p}$ is integrable since $P(r,\lambda)-\mathrm{exp}\left(i\lambda r\right)\in L^{\infty}(\mathbb{R})$ and $u_{1}\in L^{1}(\mathbb{R})$ . To see that $u_{2}|P(r,\lambda)-\mathrm{exp}\left(i\lambda r\right)|^{p}$ is integrable, apply the Cauchy-Schwarz inequality, and use the fact that $u_{2}\in L^{2}(\mathbb{R})$ and $|P(r,\lambda)-\mathrm{exp}\left(i\lambda r\right)|\in L^{2p}(\mathbb{R})$ . ∎

4. Krein system solutions are orthogonal to lower Fourier frequencies

Definition. Let $\mathcal{P}_{[{a},{b}]}$ denote the Fourier projection onto the frequency band $[a,b]$ , i.e.

\mathcal{P}_{[{a},{b}]}\left(\int\limits_{\mathbb{R}}f(\xi)\mathrm{exp}\left(i(\cdot)\xi\right)\,d\xi\right)(\lambda)=\int\limits_{a}^{b}f(\xi)\mathrm{exp}\left(i\lambda\xi\right)\,d\xi

for all $f\in L^{2}(\mathbb{R})$ . Note $\mathcal{P}_{[{a},{b}]}=\mathcal{F}^{-1}\chi_{[\frac{a}{2\pi},\frac{b}{2\pi}]}\mathcal{F}$ .

Remark. Note that $\mathcal{P}_{[{0},{r}]}g:L^{1}(\mathbb{R})\to L^{2}(\mathbb{R})$ . Indeed, by Plancherel’s theorem, it suffices to show

\chi_{[0,\frac{b}{2\pi}]}\mathcal{F}:L^{1}(\mathbb{R})\to L^{2}(\mathbb{R}),

which follows from the fact that

\chi_{[0,\frac{b}{2\pi}]}:L^{\infty}(\mathbb{R})\to L^{2}(\mathbb{R})\,,\quad\mathcal{F}:L^{1}(\mathbb{R})\to L^{\infty}(\mathbb{R})\,,

where the last boundedness property follows from the Riemann-Lebesgue lemma.

Solutions to the Krein system satisfy the following orthogonality Lemma.

Lemma 4.1.

If $w-1\in L^{1}(\mathbb{R})+L^{2}(\mathbb{R})$ , then for each nonnegative integer $k$ , we have

\langle P(r,\lambda;w),\lambda^{k}\int\limits_{0}^{r}f(s)\mathrm{exp}\left(i\lambda s\right)\,ds\rangle_{w(\lambda)d\lambda}=\langle P(r,\lambda;w)w,\lambda^{k}\int\limits_{0}^{r}f(s)\mathrm{exp}\left(i\lambda s\right)\,ds\rangle_{d\lambda}=0\,

(24)

for all $f\in C_{c}^{\infty}((0,r))$ .

Furthermore, we also have

	$\displaystyle\langle\mathrm{exp}\left(i\lambda r\right),\lambda^{k}\int\limits_{0}^{r}f(s)\mathrm{exp}\left(i\lambda s\right)\,ds\rangle_{d\lambda}=0\,,$		(25)
	$\displaystyle\langle 1,\lambda^{k}\int\limits_{0}^{r}f(s)\mathrm{exp}\left(i\lambda s\right)\,ds\rangle_{d\lambda}=0.$		(26)

Remark. The main statement of interest in this lemma is (24). Ignoring issues of integrability, if $k=0$ , then this is the intuitive statement that

P(r,\lambda)\perp_{w}\,\mathrm{Range}\,\mathcal{P}_{[0,r]}\,,

whose analogue for OPUC was used in [5, 1] in addressing the Steklov problem for OPUC.

For $k\geq 1$ , this gives us the more surprising statement

P(r,\lambda)\perp_{\lambda^{k}w}\,\mathrm{Range}\,\mathcal{P}_{[0,r]}\,,

possessing no analogue for OPUC.

Proof.

We focus first on (24). Use integration by parts to write

\langle P(r,\lambda)w,\lambda^{k}\int\limits_{0}^{r}f(s)\mathrm{exp}\left(i\lambda s\right)\,ds\rangle_{d\lambda}=i^{k}\langle P(r,\lambda)w,\int\limits_{0}^{r}f^{(k)}(s)\mathrm{exp}\left(i\lambda s\right)\,ds\rangle_{d\lambda}\,.

Since $w-1\in L^{1}(\mathbb{R})+L^{2}(\mathbb{R})$ , then it follows from Lemma 2.1 and then Lemma 3.2 that the accelerant $H$ associated to $w$ is in $L^{2}_{loc}(\mathbb{R})$ and $P$ is continuous in both variables. This last property allows us to write $P(r,\lambda)$ as the limit of its averages in $r$ and so we may write the inner-product as

i^{k}\langle\left(\lim\limits_{\epsilon\to 0}\frac{1}{\epsilon}\int\limits_{r}^{r+\epsilon}P(s,\lambda)\,ds\,\right)w,\int\limits_{0}^{r}f^{(k)}(s)\mathrm{exp}\left(i\lambda s\right)\,ds\rangle_{d\lambda}\,.

Since $f^{(k)}(s)\in C_{c}^{\infty}(0,r)$ and $w-1\in L^{1}(\mathbb{R})+L^{2}(\mathbb{R})$ , then $w(\lambda)\int\limits_{0}^{r}f^{(k)}(s)\mathrm{exp}\left(i\lambda s\right)\,ds\in L^{1}(\mathbb{R})$ . Apply the dominated convergence theorem, using e.g. (22) as justification (which holds thanks to Lemma 2.1), to pull $\lim\limits_{\epsilon\to 0}$ outside the inner-product, yielding

i^{k}\lim\limits_{\epsilon\to 0}\langle\frac{1}{\epsilon}\int\limits_{r}^{r+\epsilon}P(s,\lambda)\,ds\,,\int\limits_{0}^{r}f^{(k)}(s)\mathrm{exp}\left(i\lambda s\right)\,ds\rangle_{w(\lambda)d\lambda}\,.

(27)

Write

\int\limits_{0}^{r}f^{(k)}(s)\mathrm{exp}\left(i\lambda s\right)\,ds=\langle\mathrm{exp}\left(i\lambda s\right),\overline{f^{(k)}(s)}\rangle_{(ds,[0,r])}\,,

which, by the change of basis formula (19), equals

\langle(I+\cal{L}_{r})P(\cdot,\lambda;\sigma)(s),\overline{f^{(k)}(s)}\rangle_{(ds,[0,r])}=\langle P(s,\lambda;\sigma),(I+\cal{L}_{r}^{*})(\overline{f^{(k)}})(s)\rangle_{(ds,[0,r])}=\int\limits_{0}^{r}P(s,\lambda)g(s)\,ds\,,

where $g(s)\stackrel{{\scriptstyle\rm def}}{{=}}\overline{(I+\cal{L}_{r}^{*})(\overline{f^{(k)}})(s)}\in L^{2}([0,r])$ . Thus we can rewrite (27) as

i^{k}\lim\limits_{\epsilon\to 0}\langle\frac{1}{\epsilon}\int\limits_{r}^{r+\epsilon}P(s,\lambda)\,ds\,,\int\limits_{0}^{r}P(s,\lambda)g(s)\,ds\rangle_{w(\lambda)d\lambda}\,.

By the orthogonality of Krein system solutions (1) this equals $0$ , thereby completing the proof of (24).

By applying (24) with $w=1$ and using $P(r,\lambda;1)=\mathrm{exp}\left(i\lambda r\right)$ , we get (25).

As for (26), use integration by parts to write

\langle 1,\lambda^{k}\int\limits_{0}^{r}f(s)\mathrm{exp}\left(i\lambda s\right)\,ds\rangle_{d\lambda}=i^{k}\langle 1,\int\limits_{0}^{r}f^{(k)}(s)\mathrm{exp}\left(i\lambda s\right)\,ds\rangle_{d\lambda}\,.

Use Fourier inversion to write the inner-product as

\langle 1,\int\limits_{0}^{r}f^{(k)}(s)\mathrm{exp}\left(i\lambda s\right)\,ds\rangle_{d\lambda}=\overline{\int\limits_{\mathbb{R}}\int\limits_{0}^{r}f^{(k)}(s)\mathrm{exp}\left(i\lambda s\right)\,ds\,d\lambda}=\overline{f^{(k)}(0)}=0\,,

where the last equality follows from $\mathrm{supp}f\subseteq(0,r)$ . ∎

The above lemma will often be paired with the one below.

Lemma 4.2.

Let $r>0$ . Then given $g\in L^{1}(\mathbb{R})+L^{2}(\mathbb{R})$ , we have $\mathcal{P}_{[{0},{r}]}g\in L^{2}(\mathbb{R})$ , which equals $0$ if and only if

\langle g(\lambda),\int\limits_{0}^{r}f(s)\mathrm{exp}\left(i\lambda s\right)\,ds\rangle_{d\lambda}=0\,

for all $f\in C^{\infty}_{c}(0,r)$ .

Proof.

Let $g=g_{1}+g_{2}$ with $g_{1}\in L^{1}(\mathbb{R}),\,g_{2}\in L^{2}(\mathbb{R})$ . By the remark at the beginning of the section and Plancherel’s theorem, we have $\mathcal{P}_{[{0},{r}]}g\in L^{2}(\mathbb{R})$ .

As for the if and only if, note $\mathcal{P}_{[{0},{r}]}g=0$ if and only if $\chi_{[0,r]}\cal{F}g\in L^{2}(\mathbb{R})$ if and only if for all $f\in C_{c}^{\infty}(0,r)$ , we have

\langle\cal{F}g(s),f(s)\rangle_{ds}=0\,,

which, by taking Fourier inverses of both entries in the inner product, holds if and only if

\langle g(\lambda),\int\limits_{0}^{r}f(s)\mathrm{exp}\left(i\lambda s\right)\,ds\rangle_{d\lambda}=0\,.

∎

In the sections that follow, we will consider linear operators $T$ which satisfy

\|T\|_{L^{p}_{w}(\mathbb{R}^{d}),L^{p}_{w}(\mathbb{R}^{d})}=\|w^{1/p}Tw^{-1/p}\|_{p,p}\leq\cal{F}([w]_{A_{p}(\mathbb{R})},p)\,,

(28)

where the function $\cal{F}(t,p)$ on $[1,\infty)\times(1,\infty)$ is continuous in $t$ for every fixed $p\in(1,\infty)$ . In what follows, we do not need to know $\cal{F}$ explicitly. However, $\cal{F}$ is known in many applications. For example, the Hunt-Muckenhoupt-Wheeden theorem [13, p.205] shows that $T$ can be taken as a singular integral operator and recent breakthrough on domination of singular integrals by sparse operators provides the sharp dependence of $\cal{F}$ on $[w]_{A_{p}}$ . In particular, for a large class of singular integral operators, one can take $\cal{F}(t,p)=C(p)t^{\max(1,(p-1)^{-1})},$ (see, e.g., [9, p.264]).

Lemma 4.3.

The Fourier projections $\mathcal{P}_{[{0},{r}]}$ satisfy (28) for some $\cal{F}$ independent of $r$ .

Proof.

By e.g. [9, p.264], (28) is satisfied by the Hilbert transform $\cal{H}$ , which has Fourier multiplier $-i\,\mathrm{sign}(\xi)$ [13, p.26]. Now note each $\mathcal{P}_{[{0},{r}]}$ is a linear combination of modulated Hilbert transforms, i.e.

\mathcal{P}_{[{0},{r}]}=i\left(\frac{\cal{H}-\mathrm{exp}\left(i\lambda\frac{r}{2\pi}\right)\cal{H}\mathrm{exp}\left(-i\lambda\frac{r}{2\pi}\right)}{2}\right)\,.

(29)

One can check this by e.g. looking at the Fourier multipliers of all the operators involved. The triangle inequality then yields (28) for $T=\mathcal{P}_{[{0},{r}]}$ and function $\cal{F}$ independent of $r$ . ∎

5. The Steklov problem for an $A_{2}(\mathbb{R})$ weight: proof of Theorem 1.2 and Proposition 1.5

In this section we prove Theorem 1.2, and demonstrate its sharpness by Proposition 1.5. We need the following result, which follows from e.g. [15, Theorem 1, Corollary to Theorem 1].

Lemma 5.1 (Reverse Hölder inequality, open inclusion of $A_{p}$ weights).

Suppose $[w]_{A_{p}(\mathbb{R})}\leq\gamma$ , $p\in(1,\infty)$ . Then there exists $q(\gamma,p)>1$ , satisfying $\lim\limits_{\gamma\to 1}q(\gamma,p)=+\infty$ , such that for all $t\in[0,q]$ , we have

\langle w^{t}\rangle_{I}\lesssim_{\gamma}\langle w\rangle_{I}^{t}\,

for all intervals $I$ , and $[w^{t}]_{A_{p}(\mathbb{R})}\lesssim_{\gamma,p}1$ .

Furthermore, there exists $s(\gamma,p)\in(1,p)$ such that for all $t\in[s,\infty)$ , we have $[w]_{A_{t}(\mathbb{R})}\leq\eta(\gamma,p)$ , where $s$ and $\eta$ satisfy $\lim\limits_{\gamma\to 1}s(\gamma,p)=1$ and $\lim\limits_{\gamma\to 1}\eta(\gamma,p)=1$ .

Assume $[w]_{A_{2}(\mathbb{R})}\leq\gamma$ and define $\widehat{p}_{\gamma}^{\prime}=s(\gamma,2)$ , where $s$ is as in Lemma 5.1. Then $\lim\limits_{\gamma\to 1}\widehat{p}_{\gamma}=\infty$ , and

[w]_{A_{{\widehat{p}_{\gamma}}^{\prime}}(\mathbb{R})}\leq\eta(\gamma,2)\,,\quad\widehat{p}_{\gamma}>2\,.

Define

\epsilon(\gamma)\stackrel{{\scriptstyle\rm def}}{{=}}\frac{1}{p_{\gamma}^{\prime}}-\frac{1}{2}\,.

(30)

Thus, if $p\in[\widehat{p}_{\gamma}^{\prime},\widehat{p}_{\gamma}]=\{p~{}:~{}|\frac{1}{p}-\frac{1}{2}|\leq\epsilon(\gamma)\}$ , then

[w]_{A_{p}(\mathbb{R})},\,[w]_{A_{p^{\prime}}(\mathbb{R})}\leq\eta(\gamma,2)\,,

which in particular implies that for all such $p$ , we have

[w]_{A_{p}(\mathbb{R})},[w^{-p/p^{\prime}}]_{A_{p}(\mathbb{R})}\lesssim_{\gamma}1\,.

(31)

Note $\epsilon(\gamma)\in(0,\frac{1}{2})$ , with $\lim\limits_{\gamma\to 1}\epsilon(\gamma)=\frac{1}{2}$ .

If we additionally assume $w-1\in L^{1}(\mathbb{R})+L^{2}(\mathbb{R})$ , then Proposition 3.1 implies $P(r,\lambda;w)-\mathrm{exp}\left(i\lambda r\right)\in L^{2}(\mathbb{R})\cap L^{\infty}(\mathbb{R})$ . Together, these estimates yield

wP(r,\lambda;w)-\mathrm{exp}\left(i\lambda r\right)=(w-1)(P(r,\lambda;w)-\mathrm{exp}\left(i\lambda r\right))\\ +(w-1)\mathrm{exp}\left(i\lambda r\right)+(P(r,\lambda;w)-\mathrm{exp}\left(i\lambda r\right))\in L^{1}(\mathbb{R})+L^{2}(\mathbb{R})\,.

(32)

Proposition 3.1 and the fact that $w-1\in L^{1}(\mathbb{R})+L^{2}(\mathbb{R})$ also imply

X_{p}\stackrel{{\scriptstyle\rm def}}{{=}}w^{1/p}\left(P(r,\lambda;w)-\mathrm{exp}\left(i\lambda r\right)\right)\in L^{p}(\mathbb{R})\,,\quad 2\leq p<\infty\,.

(33)

Since

\|P(r,\lambda;w)-\mathrm{exp}\left(i\lambda r\right)\|_{L^{p}_{w}(\mathbb{R})}=\|X_{p}\|_{L^{p}(\mathbb{R})}\,,

it suffices to estimate $X_{p}$ in $L^{p}(\mathbb{R})$ , which we’ll do using functional analysis methods. Note (33) means we can do functional analysis on $X_{p}$ in the space $L^{p}(\mathbb{R})$ for any $p\in[2,\infty)$ when $w-1\in L^{1}(\mathbb{R})+L^{2}(\mathbb{R})$ .

Lemma 5.2.

If $[w]_{A_{2}(\mathbb{R})}\leq\gamma$ and $w-1\in L^{1}(\mathbb{R})+L^{2}(\mathbb{R})$ , then there exists $\epsilon(\gamma)\in(0,\frac{1}{2})$ , with $\lim\limits_{\gamma\to 1}\epsilon(\gamma)=\frac{1}{2}$ , such that for all $p\in[2,\infty)$ satisfying $|\frac{1}{p}-\frac{1}{2}|<\epsilon(\gamma)$ , we have

	$\displaystyle X_{p}$	$\displaystyle=w^{1/p}\mathcal{P}_{[0,r]}w^{-1/p}X_{p}\,,$		(34)
	$\displaystyle 0$	$\displaystyle=w^{-1/p^{\prime}}\mathcal{P}_{[0,r]}w^{1/p^{\prime}}X_{p}+w^{-1/p^{\prime}}\mathcal{P}_{[0,r]}w^{1/p^{\prime}}(w^{1/p}-w^{-1/p^{\prime}})\mathrm{exp}\left(i\lambda r\right)\,$		(35)

in $L^{p}(\mathbb{R})$ . In particular, for all such values of $p$ we have

(I-Q_{w,p})X_{p}=-w^{-1/p^{\prime}}\mathcal{P}_{[0,r]}w^{1/p^{\prime}}(w^{1/p}-w^{-1/p^{\prime}})\mathrm{exp}\left(i\lambda r\right)\,,

(36)

where

Q_{w,p}\stackrel{{\scriptstyle\rm def}}{{=}}w^{1/p}\mathcal{P}_{[0,r]}w^{-1/p}-w^{-1/p^{\prime}}\mathcal{P}_{[0,r]}w^{1/p^{\prime}}\,.

Remark. One might wonder if $(w^{1/p}-w^{-1/p^{\prime}})\,\mathrm{exp}\left(i\lambda r\right)\in L^{p}(\mathbb{R})$ ; it is by taking $q=0$ , and both $\widetilde{p}$ and $p$ equal in (37) of the following lemma.

Lemma 5.3 (Integrability of $A_{p}$ weights).

Let $d\mu(\lambda)=\langle\lambda\rangle^{q}d\lambda$ for some $q\geq 0$ . Suppose that

•

$[w]_{A_{p^{\prime}}(\mathbb{R})}\leq\gamma$ for some $p^{\prime}\in(1,\infty)$ .
•

$w-1=u_{1}+u_{2}$ with $u_{1}\in L^{p_{1}}_{\mu}(\mathbb{R}),u_{2}\in L^{p_{2}}_{\mu}(\mathbb{R})$ .
•

$1\leq p_{1}\leq p_{2}\leq p$ .

Then there exists $\epsilon(\gamma,p)>0$ , with $\lim\limits_{\gamma\to 1}\epsilon(\gamma,p)=\min\{1/p,1/p^{\prime}\}$ , such that for all $\widetilde{p}$ satisfying $|\frac{1}{p}-\frac{1}{\widetilde{p}}|<\epsilon(\gamma,p)$ , we have

\|w^{1/\widetilde{p}}-w^{-1/\widetilde{p}^{\prime}}\|_{L^{p}_{\mu}(\mathbb{R})}<\infty\,.

(37)

If $q=0$ and $\widetilde{p}=p$ , then there exists $\tau_{0}(p)$ a small constant such that whenever

\tau\stackrel{{\scriptstyle\rm def}}{{=}}[w]_{A_{\infty}(\mathbb{R})}-1\leq\tau_{0}(p)\,,

we have the perturbative estimate

\|w^{1/p}-w^{-1/p^{\prime}}\|_{L^{p}(\mathbb{R})}\lesssim_{p}\tau^{(p-p_{2})/(2p)}(\tau^{\frac{p_{2}-p_{1}}{2}}\|u_{1}\|_{L^{p_{1}}(\mathbb{R})}^{p_{1}}+\|u_{2}\|_{L^{p_{2}}(\mathbb{R})}^{p_{2}})^{1/p}\,.

(38)

We defer the proof of Lemma 5.3 till Section 8.

Proof of Lemma 5.2.

Take $\epsilon(\gamma)$ as in (30). Then note all relevant quantities are well-defined as elements of, or operators on, $L^{p}(\mathbb{R})$ : $X_{p}\in L^{p}(\mathbb{R})$ by (33), $w^{-1/p^{\prime}}\mathcal{P}_{[0,r]}w^{1/p^{\prime}}$ and $w^{-1/p^{\prime}}\mathcal{P}_{[0,r]}w^{1/p^{\prime}}$ are bounded on $L^{p}(\mathbb{R})$ by the Hunt-Muckenhoupt-Wheeden theorem and the fact that $w,w^{-p/p^{\prime}}\in A_{p}(\mathbb{R})$ . And as per the previous remark, $(w^{1/p}-w^{-1/p^{\prime}})\mathrm{exp}\left(i\lambda r\right)\in L^{p}(\mathbb{R})$ by Lemma 5.3.

We note (34) is equivalent to

P(r,\lambda)-\mathrm{exp}\left(i\lambda r\right)=\mathcal{P}_{[{0},{r}]}(P(r,\lambda)-\mathrm{exp}\left(i\lambda r\right))\,.

Since $P(r,\lambda)-\mathrm{exp}\left(i\lambda r\right)=-\int\limits_{0}^{r}\Gamma_{r}(r,t)\,\mathrm{exp}\left(i\lambda t\right)\,dt$ with $\Gamma_{r}(r,\cdot)\in L^{2}([0,r])$ as in the discussion in Section 2, then this clearly holds.

Meanwhile (35) is equivalent to

\mathcal{P}_{[0,r]}(wP(r,\lambda)-\mathrm{exp}\left(i\lambda r\right))=0\,.

From (32) it follows that $wP(r,\lambda)-\mathrm{exp}\left(i\lambda r\right)\in L^{1}(\mathbb{R})+L^{2}(\mathbb{R})$ . Thus by Lemma 4.2, it suffices to show

\langle wP(r,\lambda)-\mathrm{exp}\left(i\lambda r\right),\int\limits_{0}^{r}f(s)\mathrm{exp}\left(i\lambda s\right)\,ds\,\rangle_{d\lambda}=0

for each $f\in C^{\infty}_{c}(0,r)$ . This follows from Lemma 4.1.

To get (36), subtract (35) from (34) and rearrange. ∎

As it turns out, we can invert $I-Q_{w,p}$ for $p$ sufficiently close to $2$ .

Lemma 5.4.

For $w\geq 0$ , consider the formal operator

Q_{w,p}\stackrel{{\scriptstyle\rm def}}{{=}}w^{1/p}\mathcal{P}_{[{0},{r}]}w^{-1/p}-w^{-1/p^{\prime}}\mathcal{P}_{[{0},{r}]}w^{1/p^{\prime}}\,,

If $[w]_{A_{2}(\mathbb{R})}\leq\gamma$ , then there exists $\epsilon(\gamma)\in(0,\frac{1}{2})$ , independent of $r$ , with $\lim\limits_{\gamma\to 1}\epsilon(\gamma)=\frac{1}{2}$ , such that for all $p$ satisfying $|\frac{1}{p}-\frac{1}{2}|<\epsilon(\gamma)$ , $I-Q_{w,p}$ has bounded inverse on $L^{p}(\mathbb{R})$ with operator bound

\|(I-Q_{w,p})^{-1}\|_{p,p}\lesssim 1\,.

(39)

Let us briefly discuss the strategy for proving Theorem 1.2. Using Lemma 5.4 and (36), we have

X_{p}=-(I-Q_{w,p})^{-1}w^{-1/p^{\prime}}\mathcal{P}_{[0,r]}w^{1/p^{\prime}}(w^{1/p}-w^{-1/p^{\prime}})\mathrm{exp}\left(i\lambda r\right)\,\quad\text{ in }L^{p}(\mathbb{R})\,

(40)

for all $p\geq 2$ sufficiently close to $2$ . Then, we can estimate $\|X_{p}\|_{L^{p}(\mathbb{R})}$ by estimating $\|(I-Q_{w,p})^{-1}\|_{p,p}$ , $\|w^{-1/p^{\prime}}\mathcal{P}_{[0,r]}w^{1/p^{\prime}}\|_{p,p}$ , and $\|(w^{1/p}-w^{-1/p^{\prime}})\mathrm{exp}\left(i\lambda r\right)\|_{p}$ . However, such a process will only give us a bound for $\|X_{p}\|_{p}$ when $p\geq 2$ since our starting point, (36), was only valid for $p\geq 2$ . We address this is by noting that all elements involving $p$ in (40) are actually analytic in variable $\frac{1}{p}$ , and the right-side of (40) is well-defined for $p<2$ . Hence equality must hold for $p<2$ . In particular, we have the following Proposition.

Proposition 5.5.

Suppose $w-1\in L^{1}(\mathbb{R})+L^{2}(\mathbb{R})$ . If $[w]_{A_{2}(\mathbb{R})}\leq\gamma$ , then there exists $\epsilon(\gamma)\in(0,\frac{1}{2})$ , with $\lim\limits_{\gamma\to 1}\epsilon(\gamma)=\frac{1}{2}$ , such that

X_{p}=-(I-Q_{w,p})^{-1}w^{-1/p^{\prime}}\mathcal{P}_{[0,r]}w^{1/p^{\prime}}(w^{1/p}-w^{-1/p^{\prime}})\mathrm{exp}\left(i\lambda r\right)\,,

(41)

for all $p$ satisfying $\left|\frac{1}{p}-\frac{1}{2}\right|<\epsilon(\gamma)$ .

We prove Lemma 5.4 and Proposition 5.5 in Section 9 and Appendix A respectively.

Proof of Theorem 1.2.

By Proposition 5.5, we may estimate

\|X_{p}\|_{L^{p}(\mathbb{R})}\leq\|(I-Q_{w,p})^{-1}\|_{p,p}\|w^{-1/p^{\prime}}\mathcal{P}_{[0,r]}w^{1/p^{\prime}}\|_{p,p}\|w^{1/p}-w^{-1/p^{\prime}}\|_{L^{p}(\mathbb{R})}\,

for all $p$ satisfying $|\frac{1}{p}-\frac{1}{2}|<\epsilon(\gamma)$ , where $\lim\limits_{\gamma\to 1}\epsilon(\gamma)=\frac{1}{2}$ .

Since $\mathcal{P}_{[{0},{r}]}$ satisfies (28) for some $\cal{F}$ independent of $r$ , we may in fact write

\|X_{p}\|_{L^{p}(\mathbb{R})}\leq\|(I-Q_{w,p})^{-1}\|_{p,p}\cal{F}([w^{-p/p^{\prime}}]_{A_{p}(\mathbb{R})},p)\|w^{1/p}-w^{-1/p^{\prime}}\|_{L^{p}(\mathbb{R})}\,.

Since $[w^{-p/p^{\prime}}]_{A_{p}(\mathbb{R})}\lesssim_{\gamma}1$ by (31), then $\cal{F}([w^{-p/p^{\prime}}]_{A_{p}(\mathbb{R})},p)\lesssim_{\gamma,p}1$ . And by applying Lemma 5.4, we get $\|(I-Q_{w,p})^{-1}\|_{p,p}\lesssim 1$ for $\left|\frac{1}{p}-\frac{1}{2}\right|<\epsilon(\gamma)$ . Thus

\|X_{p}\|_{L^{p}(\mathbb{R})}\lesssim_{p,\gamma}\|w^{1/p}-w^{-1/p^{\prime}}\|_{L^{p}(\mathbb{R})}

for all $p$ satisfying $|\frac{1}{p}-\frac{1}{2}|<\epsilon(\gamma)$ . Note all of these estimates are uniform in $r$ .

Part (a) now follows from noting $\|w^{1/p}-w^{-1/p^{\prime}}\|_{L^{p}(\mathbb{R})}<\infty$ by Lemma 5.3 so long as $p\in[p_{2},\infty)$ .

Meanwhile part (b) follows by fixing $p$ , taking $\tau_{0}(p)$ small enough so that

•

$|\frac{1}{p}-\frac{1}{2}|<\epsilon(\gamma)$ .
•

$[w]_{A_{\infty}(\mathbb{R})}-1\leq[w]_{A_{2}(\mathbb{R})}-1=\tau$ is sufficiently small that (38) applies.

Then (38) implies (5). ∎

We now turn our attention towards Proposition 1.5. Let us first discuss its meaning: suppose $w-1\in L^{1}(\mathbb{R})+L^{p_{2}}(\mathbb{R})$ for $p_{2}\in[1,2]$ and $w-1\notin L^{1}(\mathbb{R})$ ; we think of $p_{2}$ as measuring the decay of $w-1$ , with smaller $p_{2}$ ’s indicating better decay. Recall Theorem 1.2 (a), which says that if $[w]_{A_{2}(\mathbb{R})}\leq\gamma$ , then

\|P(r,\lambda;w)-\mathrm{exp}\left(i\lambda r\right)\|_{L^{p}_{w}(\mathbb{R})}<\infty

whenever $p\in[p_{2},\infty)$ satisfies $|\frac{1}{p}-\frac{1}{2}|<\epsilon(\gamma)$ , where $\lim\limits_{\gamma\to 1}\epsilon(\gamma)=\frac{1}{2}$ . So let $\gamma-1$ be sufficiently small that $|\frac{1}{p_{2}}-\frac{1}{2}|<\epsilon(\gamma)$ . Then the requirement that $p\geq p_{2}$ in Theorem 1.2 (a) is in part saying that $P(r,\lambda;w)-\mathrm{exp}\left(i\lambda r\right)$ cannot hope to decay faster than $w-1$ decays. This seemingly makes sense a priori: since the accelerant satisfies $H(x)=\frac{1}{2\pi}\widecheck{(w-1)}\left(\frac{x}{2\pi}\right)$ , then whatever decay $w-1$ possesses gets “converted” into the regularity of $H$ . But then the resolvent kernel $\Gamma_{r}(s,t)$ , as a rule of thumb, is at most as regular as $H$ . Since by (18) we have

P(r,\lambda)-\mathrm{exp}\left(i\lambda r\right)=-\int\limits_{0}^{r}\Gamma_{r}(r,t)\mathrm{exp}\left(i\lambda t\right)\,dt\,,

i.e. $P(r,\lambda)-\mathrm{exp}\left(i\lambda r\right)$ is essentially the inverse Fourier transform of $-\Gamma_{r}(r,\cdot)$ and so whatever regularity $\Gamma_{r}(r,\cdot)$ possesses gets “converted” into the decay of $P(r,\lambda)-\mathrm{exp}\left(i\lambda r\right)$ . Thus heuristically, we expect the decay of $P(r,\lambda)-\mathrm{exp}\left(i\lambda r\right)$ to not exceed that of $w-1$ , as the former “inherits” its decay from the latter.

Proof of Proposition 1.5.

By Hölder’s inequality, we can assume without loss of generality that $p\in(1,p_{2})$ .

Without loss of generality, assume $\delta\leq\frac{1}{2}$ . Let $u$ be an even, real-valued function on $\mathbb{R}$ such that $0\leq u\leq 1$ and $u\in L^{q}(\mathbb{R})$ if and only if $q\geq p_{2}$ . Then define $w=1+\delta u$ so that $[w]_{A_{2}(\mathbb{R})}\leq 1+\delta$ and $w-1\in L^{p_{2}}(\mathbb{R})$ .

Without loss of generality, assume $\delta$ sufficiently small that $|\frac{1}{p}-\frac{1}{2}|<\epsilon(1+\delta)$ , where $\epsilon$ is as in Lemma 5.4. Then by Lemma 5.4, $\|(I-Q_{w,p})^{-1}\|_{p,p}\lesssim 1$ . Thus by Proposition 5.5, we have

X_{p}=-(I-Q_{w,p})^{-1}w^{-1/p^{\prime}}\mathcal{P}_{[{0},{r}]}w^{1/p^{\prime}}(w^{1/p}-w^{-1/p^{\prime}})\mathrm{exp}\left(i\lambda r\right)\,,

or rather

(I-Q_{w,p})X_{p}=-\delta\,\mathrm{exp}\left(i\lambda r\right)w^{-1/p^{\prime}}\mathcal{P}_{[{-r},{0}]}u\,.

Estimating $L^{p}(\mathbb{R})$ norms yields

\|I-Q_{w,p}\|_{p,p}\|X_{p}\|_{L^{p}(\mathbb{R})}\gtrsim_{p}\|\mathcal{P}_{[{-r},{0}]}u\|_{L^{p}(\mathbb{R})}\,.

Since $w\sim 1$ , then $[w]_{A_{p}(\mathbb{R})},[w^{-p/p^{\prime}}]_{A_{p}(\mathbb{R})}\sim_{p}1$ and so by Lemma 4.3, we have $\|I-Q_{w,p}\|_{p,p}\lesssim 1$ and so

\|X_{p}\|_{L^{p}(\mathbb{R})}\gtrsim_{p}\|\mathcal{P}_{[{-r},{0}]}u\|_{L^{p}(\mathbb{R})}\,.

This then means

\sup\limits_{r\geq 0}\|X_{p}\|_{L^{p}(\mathbb{R})}\gtrsim_{p}\sup\limits_{r\geq 0}\|\mathcal{P}_{[{-r},{0}]}u\|_{L^{p}(\mathbb{R})}\,.

It suffices to show

\sup\limits_{r\geq 0}\|\mathcal{P}_{[{-r},{0}]}u\|_{L^{p}(\mathbb{R})}=\infty\,.

Since $u\in L^{2}(\mathbb{R})$ , write $u(\lambda)=\int\limits_{-\infty}^{\infty}v(s)\mathrm{exp}\left(i\lambda s\right)\,ds$ , where $v\in L^{2}(\mathbb{R})$ . Since $u$ is real-valued and even, then so is $v$ . In particular this means $\mathcal{P}_{[{-r},{0}]}u=\overline{\mathcal{P}_{[{0},{r}]}u}$ , and combined with the fact that $u$ is real-valued, we get $\sup\limits_{r\geq 0}\|\mathcal{P}_{[{-r},{0}]}u\|_{L^{p}(\mathbb{R})}=\infty$ if $\sup\limits_{r\geq 0}\|\mathcal{P}_{[{-r},{r}]}u\|_{L^{p}(\mathbb{R})}=\infty$ .

In fact, $\{\mathcal{P}_{[{-n},{n}]}u\}_{n\geq 0}$ is unbounded in $L^{p}(\mathbb{R})$ . Indeed, suppose to the contrary the sequence is bounded. Then there exists some increasing sequence of integers $n_{k}$ and some $\widetilde{u}\in L^{p}(\mathbb{R})$ such that $\mathcal{P}_{[{-n_{k}},{n_{k}}]}u$ converges to $\widetilde{u}$ weakly in $L^{p}(\mathbb{R})$ . In particular, for every Schwarz function $f$ , we have

\langle\widetilde{u},f\rangle=\lim\limits_{k\to\infty}\langle\mathcal{P}_{[{-n_{k}},{n_{k}}]}u,f\rangle=\langle u,f\rangle\,,

where the last equality follows from Plancherel’s theorem. Thus $u=\widetilde{u}\in L^{p}(\mathbb{R})$ , which contradicts our requirement that $u\in L^{q}(\mathbb{R})$ if and only if $q\geq p_{2}$ . This completes the proof. ∎

6. A mixed norm remainder estimate

In this section, we prove Theorem 1.6.

Definition. If $\Gamma_{r}(\cdot,\cdot)\in C^{k}([0,r]^{2})$ , integrate (18) by parts $k$ times to yield

P(r,\lambda)-\mathrm{exp}\left(i\lambda r\right)=\sum\limits_{l=1}^{k}\frac{a_{l,r}(\lambda)}{\lambda^{l}}+\frac{R_{k,r}(\lambda)}{\lambda^{k}}\,,

where

a_{l,r}(\lambda)\stackrel{{\scriptstyle\rm def}}{{=}}\begin{cases}(i)^{l}\left(\mathrm{exp}\left(i\lambda r\right)(\partial_{t})^{l-1}\Gamma_{r}(r,t)\Big{|}_{t=r}-(\partial_{t})^{l-1}\Gamma_{r}(r,t)\Big{|}_{t=0}\right)\quad&l\geq 1\\ 0&l=0\end{cases}

(42)

and the remainder term $R_{k,r}(\lambda)$ is defined by

R_{k,r}(\lambda)\stackrel{{\scriptstyle\rm def}}{{=}}-(i)^{k}\int\limits_{0}^{r}\left((\partial_{t})^{k}\Gamma_{r}(r,t)\right)\mathrm{exp}\left(i\lambda t\right)\,dt\,.

(43)

We can also express the remainder in terms of the solution $P(r,\lambda)$ to the Krein system and the “coefficients” $a_{l,r}(\lambda)$ , i.e.

R_{k,r}(\lambda)=\lambda^{k}(P(r,\lambda)-\mathrm{exp}\left(i\lambda r\right))-\sum\limits_{l=1}^{k}\lambda^{k-l}a_{l,r}(\lambda)\,.

Note that $R_{0,r}=P(r,\lambda;w)-\mathrm{exp}\left(i\lambda r\right)$ .

The Steklov problem can be reformulated in terms of mixed norms: it asks when does one have the bound

\|P(r,\lambda)\|_{L^{\infty}_{dr}(\mathbb{R}^{+};L^{p}_{w}(\Delta))}<\infty

(44)

for every compact $\Delta\subset\mathbb{R}$ ? In Theorem 1.2, we estimated

\|R_{0,r}\|_{L^{\infty}_{dr}(\mathbb{R}^{+};L^{p}_{w}(\mathbb{R}))}

for some $p$ close to $2$ , which implied (44). However, it also makes sense to consider other mixed norms, as we do in Theorem 1.6. To estimate $R_{1,r}$ in $L^{p}_{w}(\mathbb{R})$ , it suffices to estimate $Y_{p}\stackrel{{\scriptstyle\rm def}}{{=}}w^{1/p}R_{1,r}(\lambda)$ in $L^{p}(\mathbb{R})$ .

We begin with two lemmas.

Lemma 6.1.

Suppose $w\in A_{2}(\mathbb{R})$ and $w-1\in L^{1}(\mathbb{R})$ . If Condition (6) holds, then $A\in L^{2}(\mathbb{R}^{+})$ , where $A(r)$ is as given by (21).

Proof.

By [3, Theorem 12.14], $A(r)\in L^{2}(\mathbb{R}^{+})$ follows if both Condition (6) holds and $\log w\in L^{1}(\mathbb{R})$ . Thus it suffices to check our assumptions imply $\log w\in L^{1}(\mathbb{R})$ .

Note that when $x\geq\frac{1}{2}$ , then $|\log(x)|\lesssim|x-1|$ . Whence

\int\limits_{\{w\geq\frac{1}{2}\}}|\log w|\lesssim\int\limits_{\{w\geq\frac{1}{2}\}}|w-1|\leq\|w-1\|_{L^{1}(\mathbb{R})}<\infty\,.

Since $\int\limits_{\mathbb{R}}|w-1|d\lambda<\infty$ , then $\{w\leq\frac{1}{2}\}$ has finite Lebesgue measure. By taking $\widetilde{p},p^{\prime}=2$ and $q=0$ in Lemma 5.3, we get $w^{1/2}-w^{-1/2}\in L^{2}(\mathbb{R})$ and in particular $w^{1/2}-w^{-1/2}$ is square-integrable on $\{w\leq\frac{1}{2}\}$ . Meanwhile $w$ is integrable on $\{w\leq\frac{1}{2}\}$ since $w-1\in L^{1}(\mathbb{R})$ . Thus the $L^{2}(\mathbb{R})$ -triangle inequality implies $w^{-1}$ is integrable on $\{w\leq\frac{1}{2}\}$ : indeed, $\int\limits_{\{w\leq\frac{1}{2}\}}w^{-1}$ equals

\int\limits_{\{w\leq\frac{1}{2}\}}|w^{-\frac{1}{2}}|^{2}\lesssim\int\limits_{\{w\leq\frac{1}{2}\}}|w^{\frac{1}{2}}-w^{-\frac{1}{2}}|^{2}+\int\limits_{\{w\leq\frac{1}{2}\}}|w^{\frac{1}{2}}|^{2}=\int\limits_{\{w\leq\frac{1}{2}\}}|w^{\frac{1}{2}}-w^{-\frac{1}{2}}|^{2}+\int\limits_{\{w\leq\frac{1}{2}\}}w-1+\int\limits_{\{w\leq\frac{1}{2}\}}1<\infty\,.

And hence $\log w$ must be integrable on $\{w\leq\frac{1}{2}\}$ as well. ∎

Lemma 6.2.

Suppose $w\in A_{2}(\mathbb{R})$ , $\langle\lambda\rangle(w-1)\in L^{1}(\mathbb{R})$ and Condition (6) holds. Then

a_{1,r}(\lambda)=\mathrm{exp}\left(i\lambda r\right)\alpha_{\infty}(r)+\alpha_{2}(r)\,,

(45)

where $\alpha_{2}\in L^{2}(\mathbb{R}^{+})$ , $\alpha_{\infty}\in L^{\infty}(\mathbb{R}^{+})$ and $\lim\limits_{r\to\infty}\alpha_{\infty}(r)$ exists. In particular,

|a_{1,r}(\lambda)|\leq|\alpha_{\infty}(r)|+|\alpha_{2}(r)|\in L^{\infty}_{dr}(\mathbb{R}^{+})+L^{2}_{dr}(\mathbb{R}^{+})

for all $\lambda\in\mathbb{R}$ .

Proof.

Since $\langle\lambda\rangle(w-1)\in L^{1}(\mathbb{R})$ , then by Lemma 3.3 it follows that $\Gamma_{r}(\cdot,\cdot)\in C^{1}([0,r]^{2})$ . Hence $a_{1,r}$ is well-defined and is given by

a_{1,r}(\lambda)=i[\mathrm{exp}\left(i\lambda r\right)\Gamma_{r}(r,r)-\Gamma_{r}(r,0)]=i[\mathrm{exp}\left(i\lambda r\right)\Gamma_{r}(0,0)-\overline{A(r)}]\,,

where we used (15) and (21) in the last equality.

Then use (16) and the fundamental theorem of Calculus to write

\Gamma_{r}(0,0)=\Gamma_{0}(0,0)+\int\limits_{0}^{r}\partial_{x}\Gamma_{x}(0,0)\,dx=\Gamma_{0}(0,0)-\int\limits_{0}^{r}\Gamma_{x}(0,x)\Gamma_{x}(x,0)\,dx\,.

Use (21) and the resolvent symmetries (15) again, to get

a_{1,r}(\lambda)=-i\left(-\mathrm{exp}\left(i\lambda r\right)\Gamma_{0}(0,0)+\mathrm{exp}\left(i\lambda r\right)\int\limits_{0}^{r}|A(s)|^{2}\,ds+\overline{A(r)}\right)\,.

Define

\alpha_{\infty}(r)\stackrel{{\scriptstyle\rm def}}{{=}}-i\left(-\Gamma_{0}(0,0)+\int\limits_{0}^{r}|A(s)|^{2}\,ds\right)\,,\quad\alpha_{2}(r)\stackrel{{\scriptstyle\rm def}}{{=}}-i\overline{A(r)}\,,

so that (45) holds. Since $A(r)\in L^{2}(\mathbb{R}^{+})$ by Lemma 6.1, then $\alpha_{2}(r)\in L^{2}(\mathbb{R}^{+})$ and

|\alpha_{\infty}(r)|\leq\left|\Gamma_{0}(0,0)\right|+\int\limits_{0}^{\infty}|A(s)|^{2}\,ds<\infty\,,\quad\lim\limits_{r\to\infty}\alpha_{\infty}(r)=-i\left(-\Gamma_{0}(0,0)+\int\limits_{0}^{\infty}|A(s)|^{2}\,ds\right)\,.

∎

Now we can do functional analysis like in the proof of Theorem 1.2.

Lemma 6.3.

Suppose $\langle\lambda\rangle^{q}(w-1)\in L^{1}(\mathbb{R})$ for some $q>2$ . If $[w]_{A_{2}(\mathbb{R})}\leq\gamma$ , then there exists $\epsilon(\gamma)\in(0,\frac{1}{2})$ , with $\lim\limits_{\gamma\to 1}\epsilon(\gamma)=\frac{1}{2}$ , such that for all $p\in(1,q]$ satisfying $|\frac{1}{p}-\frac{1}{2}|<\epsilon(\gamma)$ , we have

	$\displaystyle Y_{p}$	$\displaystyle=w^{1/p}\mathcal{P}_{[0,r]}w^{-1/p}Y_{p}\,,$		(46)
	$\displaystyle w^{-1/p^{\prime}}\mathcal{P}_{[0,r]}w^{1/p^{\prime}}Y_{p}$	$\displaystyle=-w^{-1/p^{\prime}}\mathcal{P}_{[0,r]}w^{1/p^{\prime}}(w^{1/p}-w^{-1/p^{\prime}})\left(\lambda\mathrm{exp}\left(i\lambda r\right)+a_{1,r}(\lambda)\right)$		(47)

in $L^{p}(\mathbb{R})$ . In particular, for all such values of $p$ , we have

(I-Q_{w,p})Y_{p}=-w^{-1/p^{\prime}}\mathcal{P}_{[0,r]}w^{1/p^{\prime}}(w^{1/p}-w^{-1/p^{\prime}})\left(\lambda\mathrm{exp}\left(i\lambda r\right)+a_{1,r}(\lambda)\right)\,,

(48)

where $Q_{w,p}\stackrel{{\scriptstyle\rm def}}{{=}}w^{1/p}\mathcal{P}_{[{0},{r}]}w^{-1/p}-w^{-1/p^{\prime}}\mathcal{P}_{[{0},{r}]}w^{1/p^{\prime}}$ .

Remark. Since $\langle\lambda\rangle^{2}(w-1)\in L^{1}(\mathbb{R})$ , then $\Gamma_{r}(\cdot,\cdot)\in C^{2}([0,r]^{2})$ by Lemma 3.3. Integrating (43) by parts once yields $R_{1,r}(\lambda)\in L^{p}(\mathbb{R})$ for $1<p\leq\infty$ , and so $Y_{p}\in L^{p}(\mathbb{R})$ for $1<p\leq\infty$ .

We also remark that the right-side of (47) is well-defined, i.e.

(w^{1/p}-w^{-1/p^{\prime}})(\lambda\mathrm{exp}\left(i\lambda r\right)+a_{1,r}(\lambda))\in L^{p}(\mathbb{R})

for $1<p\leq q$ , for each $r>0$ . Indeed this follows from Lemma 5.3, (45), and that $\langle\lambda\rangle^{q}(w-1)\in L^{1}(\mathbb{R})$ . We require $q>2$ just so that we may consider $p>$ , which we will do later.

Proof.

Take $\epsilon(\gamma)$ as in (30), which in particular implies (31) holds for all $p$ of concern, i.e. for all $p$ such that $\left|\frac{1}{p}-\frac{1}{2}\right|<\epsilon$ . Then note all relevant quantities are well-defined as elements of, or operators on, $L^{p}(\mathbb{R})$ : $w^{-1/p^{\prime}}\mathcal{P}_{[0,r]}w^{1/p^{\prime}}$ and $w^{-1/p^{\prime}}\mathcal{P}_{[0,r]}w^{1/p^{\prime}}$ are bounded operators on $L^{p}(\mathbb{R})$ by the Hunt-Muckenhoupt-Wheeden theorem and that $w,w^{-p/p^{\prime}}\in A_{p}(\mathbb{R})$ . And as per the remark above, $Y_{p}\in L^{p}(\mathbb{R})$ , and $(w^{1/p}-w^{-1/p^{\prime}})(\lambda\mathrm{exp}\left(i\lambda r\right)+a_{1,r}(\lambda))\in L^{p}(\mathbb{R})$ by Lemma 5.3.

Notice (43) implies (46) is equivalent to

R_{1,r}=\mathcal{P}_{[{0},{r}]}R_{1,r}\,.

Since $R_{1,r}=-i\int\limits_{0}^{r}\partial_{t}\Gamma_{r}(r,t)\,\mathrm{exp}\left(i\lambda t\right)\,dt$ with $\partial_{t}\Gamma_{r}(r,\cdot)\in C([0,r])$ , it clearly holds.

Concerning (47), it is equivalent to

\mathcal{P}_{[0,r]}[w(R_{1,r}(\lambda)+\lambda\mathrm{exp}\left(i\lambda r\right)+a_{1,r}(\lambda))-(\lambda\mathrm{exp}\left(i\lambda r\right)+a_{1,r}(\lambda))]=0\,.

We note that $\mathcal{P}_{[{0},{r}]}$ is acting on

w(R_{1,r}(\lambda)+\lambda\mathrm{exp}\left(i\lambda r\right)+a_{1,r}(\lambda))-(\lambda\mathrm{exp}\left(i\lambda r\right)+a_{1,r}(\lambda))\in L^{1}(\mathbb{R})+L^{2}(\mathbb{R})\,;

to see this, rewrite it as

(w-1)R_{1,r}+R_{1,r}+(w-1)(\lambda\mathrm{exp}\left(i\lambda r\right)+a_{1,r})\,.

By the previous remark, $R_{1,r}\in L^{2}(\mathbb{R})$ and also $R_{1,r}\in L^{\infty}(\mathbb{R})$ . If we combine these two estimates with the assumption that $\langle\lambda\rangle(w-1)\in L^{1}(\mathbb{R})$ , then we get $(w-1)R_{1,r}\in L^{1}(\mathbb{R})$ . Finally, combine assumption $\langle\lambda\rangle(w-1)\in L^{1}(\mathbb{R})$ with the estimate $a_{1,r}\in L^{\infty}(\mathbb{R})$ , which follows from (42), to get $(w-1)(\lambda\mathrm{exp}\left(i\lambda r\right)+a_{1,r})\in L^{1}(\mathbb{R})$ .

Thus by Lemma 4.2, it suffices to show

\langle w(R_{1,r}+a_{1,r}+\lambda\mathrm{exp}\left(i\lambda r\right))-(\lambda\mathrm{exp}\left(i\lambda r\right)+a_{1,r}),\int\limits_{0}^{r}f(s)\mathrm{exp}\left(i\lambda s\right)\,ds\rangle_{d\lambda}=0\,,

or equivalently

\langle w\lambda P(r,\lambda)-(\lambda\mathrm{exp}\left(i\lambda r\right)+a_{1,r}),\int\limits_{0}^{r}f(s)\mathrm{exp}\left(i\lambda s\right)\,ds\rangle_{d\lambda}=0

for each $f\in C_{c}^{\infty}(0,r)$ . This now follows from Lemma 4.1 and the identity

a_{1,r}(\lambda)=i(\mathrm{exp}\left(i\lambda r\right)\Gamma_{r}(r,r)-\Gamma_{r}(r,0))\,,

as given by (42).

Add (47) and (46) and rearrange to obtain (48). ∎

Proof of Theorem 1.6.

It suffices to estimate $\|Y_{p}\|_{L^{\infty}_{dr}(\mathbb{R};L^{p}(\mathbb{R}))+L^{2}_{dr}(\mathbb{R};L^{p}(\mathbb{R}))}$ ; we first estimate $\|Y_{p}\|_{L^{p}(\mathbb{R})}$ in terms of $r$ .

From (48) we have

\displaystyle\|Y_{p}\|_{L^{p}(\mathbb{R})}\leq\|(I-Q_{w,p})^{-1}\|_{p,p}\,\cdot\,\|w^{-1/p^{\prime}}\mathcal{P}_{[0,r]}w^{1/p^{\prime}}\|_{p,p}\,\cdot\,\|(w^{1/p}-w^{-1/p^{\prime}})(\lambda\mathrm{exp}\left(i\lambda r\right)+a_{1,r}(\lambda))\|_{L^{p}(\mathbb{R})}\,.

By Lemma 5.4, we have $\|(I-Q_{w,p})^{-1}\|_{p,p}\lesssim 1$ whenever $|\frac{1}{p}-\frac{1}{2}|<\epsilon(\gamma)$ , where $\lim\limits_{\gamma\to 1}\epsilon(\gamma)=\frac{1}{2}$ . Since $\mathcal{P}_{[{0},{r}]}$ satisfies (28) for some $\cal{F}$ independent of $r$ , we can also estimate

\|w^{-1/p^{\prime}}\mathcal{P}_{[0,r]}w^{1/p^{\prime}}\|_{p,p}\leq\cal{F}([w^{-p/p^{\prime}}]_{A_{p}(\mathbb{R})},p)\lesssim_{\gamma,p}1\,.

If additionally $p\in(1,q]$ , then

\|(w^{1/p}-w^{-1/p^{\prime}})(\lambda\mathrm{exp}\left(i\lambda r\right)+a_{1,r}(\lambda))\|_{L^{p}_{d\lambda}(\mathbb{R})}\lesssim\|w^{1/p}-w^{-1/p^{\prime}}\|_{L^{p}_{\langle\lambda\rangle^{q}d\lambda}(\mathbb{R})}(1+|\alpha_{2}(r)|+|\alpha_{\infty}(r)|)

where $\alpha_{2},\alpha_{\infty}$ arise from Lemma 6.2. By Lemma 5.3 with $\widetilde{p}=p$ , we have

\|w^{1/p}-w^{-1/p^{\prime}}\|_{L^{p}_{\langle\lambda\rangle^{q}d\lambda}(\mathbb{R})}\lesssim_{p,w,q}1\,.

All together, we get

\|Y_{p}\|_{L^{p}_{w}(\mathbb{R})}\lesssim_{p,w,q}(1+\alpha_{2}(r)+\alpha_{\infty}(r))\,.

Since $1+\alpha_{2}(r)+\alpha_{\infty}(r)\in L^{\infty}_{dr}(\mathbb{R}^{+})+L^{2}_{dr}(\mathbb{R}^{+})$ by Lemma 6.2, then

\|Y_{p}\|_{L^{\infty}_{dr}(\mathbb{R}^{+};L^{p}_{w}(\mathbb{R}))+L^{2}_{dr}(\mathbb{R}^{+};L^{p}_{w}(\mathbb{R}))}<\infty\,.

∎

7. Higher order remainder estimates: proof of Theorem 1.7

We will spend the rest of this subsection proving Theorem 1.7. Suppose

(w-1)\langle\lambda\rangle^{k}\in L^{1}(\mathbb{R})\,.

Then $\Gamma\in C^{k}(\mathbb{R}^{+})$ by Lemma 3.3, and so $R_{k,r}$ is well-defined. The following Lemma shows $\{a_{l,r}\}$ are uniformly bounded in $(\lambda,r)$ .

Lemma 7.1.

Suppose $L\stackrel{{\scriptstyle\rm def}}{{=}}\|(w-1)\langle\lambda\rangle^{k}\|_{L^{1}(\mathbb{R})}<\infty$ . If $\|\langle\lambda\rangle^{k}(w-1)\|_{L^{\infty}(\mathbb{R})}\leq\delta<1$ , then

\|\partial_{t}^{j}\Gamma_{r}(r,t)\|_{L^{\infty}((0,r))}\lesssim\frac{L}{1-\delta}\,,\,\,j=0,\ldots,k\,.

In particular $|a_{l,r}(\lambda)|\lesssim\frac{L}{1-\delta}$ .

Proof.

The bound on $|a_{l,r}(\lambda)|$ follows from its definition in (42) and the bound on $\|\partial_{t}^{j}\Gamma_{r}(r,t)\|_{L^{\infty}((0,r))}$ . To get this latter bound, by the resolvent symmetries (15) it suffices to estimate $\Gamma_{r}(u,0)$ and all its derivatives in $[0,r]$ . The resolvent identity (14) is equivalent to

(I+\mathcal{H}_{r})\Gamma_{r}(\cdot,s)=H(\cdot-s)\,,\quad s\in[0,r]

and thus

\Gamma_{r}(u,0)=(I+\mathcal{H}_{r})^{-1}H(u)\,.

By Lemma 2.1 we have $H(x)=\mathcal{F}^{-1}(w(2\pi\cdot)-1)$ ; we in fact claim $\|\mathcal{H}_{r}\|_{L^{2}([0,r]),L^{2}([0,r])}\leq\delta<1$ . Indeed:

	$\displaystyle\\|\mathcal{H}_{r}f\\|_{L^{2}(\mathbb{R})}=\\|\int\limits_{0}^{r}\mathcal{F}^{-1}(w(2\pi\cdot)-1)(x-y)f(y)dy\\|_{L^{2}(\mathbb{R})}$	$\displaystyle=\\|(w(2\pi\cdot)-1)\mathcal{F}(f\chi_{[0,r]})\\|_{L^{2}(\mathbb{R})}$
		$\displaystyle\leq\\|w-1\\|_{L^{\infty}(\mathbb{R})}\\|\mathcal{F}(f\chi_{[0,r]})\\|_{L^{2}(\mathbb{R})}$
		$\displaystyle\leq\delta\\|f\\|_{L^{2}([0,r])}\,.$

Thus by geometric sum

\Gamma_{r}(u,0)=(I+\mathcal{H}_{r})^{-1}H(u)=\sum\limits_{l=0}^{\infty}(-\mathcal{H}_{r})^{l}H(u)\,.

We estimate $\sum\limits_{l=0}^{\infty}(-\mathcal{H}_{r})^{l}H(u)$ and all its derivatives. Differentiate (10) to get

\left|\left(\frac{d}{du}\right)^{j}\mathcal{H}_{r}(f)(u)\right|=\left|\int\limits_{0}^{r}H^{(j)}(u-v)f(v)\,dv\right|\leq\|H^{(j)}\|_{L^{2}(\mathbb{R})}\|f\|_{L^{2}([0,r])}\,

for all $0\leq j\leq k$ . Take $f=(-\mathcal{H}_{r})^{l-1}H$ to get

	$\displaystyle\left\|\left(\frac{d}{du}\right)^{j}(-\mathcal{H}_{r})^{l}H(u)\right\|$	$\displaystyle\leq\\|H^{(j)}\\|_{L^{2}(\mathbb{R})}\\|\mathcal{H}_{r}^{l-1}H\\|_{L^{2}([0,r])}$
		$\displaystyle=\\|H^{(j)}\\|_{L^{2}([0,r])}\\|\mathcal{H}_{r}\\|_{L^{2}([0,r]),L^{2}(\mathbb{R})}^{l-1}\\|H\\|_{L^{2}([0,r])}$
		$\displaystyle\lesssim L\delta^{l}\,,$

where in the last inequality we estimated

\|H^{(j)}\|_{L^{2}(\mathbb{R})}\lesssim\|(w-1)\lambda^{j}\|_{L^{2}(\mathbb{R})}\leq\|(w-1)\langle\lambda\rangle^{k}\|_{L^{2}(\mathbb{R})}\leq\delta^{1/2}L^{1/2}\,,

and similarly for $\|H\|_{L^{2}([0,r])}$ .

Thus the sum $\sum\limits_{l=0}^{\infty}\left(\frac{d}{du}\right)^{j}(-\mathcal{H}_{r})^{l}H(u)$ converges absolutely and so one can use the dominated convergence theorem to show we get the desired estimate on

\partial_{u}^{j}\Gamma_{r}(u,0)=\partial_{u}^{j}\sum\limits_{l=0}^{\infty}(-\mathcal{H}_{r})^{l}H(u)\,.

∎

Now we proceed with our usual functional-analytic approach.

Lemma 7.2.

Suppose that $\langle\lambda\rangle^{k}(w-1)\in L^{1}(\mathbb{R})$ . Then

R_{k,r}=\mathcal{P}_{[0,r]}R_{k,r}\,

(49)

and

\mathcal{P}_{[0,r]}(w-1)R_{k,r}=-\mathcal{P}_{[0,r]}R_{k,r}-\mathcal{P}_{[0,r]}(w-1)(\lambda^{k}\mathrm{exp}\left(i\lambda r\right)+\sum\limits_{l=1}^{k}a_{l,r}\lambda^{k-l})\,

(50)

in $L^{p}(\mathbb{R})$ for $2\leq p<\infty$ . If in addition $|\langle\lambda\rangle^{k}(w-1)|\leq\delta<1$ , then

R_{k,r}(\lambda)=-(I-\mathcal{P}_{[0,r]}(1-w))^{-1}\mathcal{P}_{[0,r]}(w-1)(\lambda^{k}\mathrm{exp}\left(i\lambda r\right)+\sum\limits_{l=1}^{k}a_{l,r}\lambda^{k-l})\,,

(51)

where both sides are well-defined in, e.g., $L^{2}(\mathbb{R})$ .

Proof.

First note that $R_{k,r}\in L^{p}(\mathbb{R})$ for $2\leq p\leq\infty$ . Indeed, $p=2$ follows by (43) and Plancherel, and $p=\infty$ from (43) and that $\Gamma_{r}\in C^{k}([0,r])$ . From this and the fact that $\langle\lambda\rangle^{k}(w-1)\in L^{1}(\mathbb{R})$ we have both sides of (49) and (50) are sensical. Trivially, (49) holds. Meanwhile (50) is equivalent to

\mathcal{P}_{[0,r]}\left((w-1)R_{k,r}+R_{k,r}+(w-1)(\lambda^{k}\mathrm{exp}\left(i\lambda r\right)+\sum\limits_{l=1}^{k}a_{l,r}\lambda^{k-l})\right)=0\,

or rather

\mathcal{P}_{[0,r]}(w\lambda^{k}P(r,\lambda)-\lambda^{k}\mathrm{exp}\left(i\lambda r\right)-\sum\limits_{l=1}^{k}a_{l,r}\lambda^{k-l})=0\,,

which follows by applying Lemma 4.1 and (42).

Add (50) and (49) to yield

(I-\mathcal{P}_{[0,r]}(1-w))R_{k,r}(\lambda)=-\mathcal{P}_{[0,r]}(w-1)(\lambda^{k}\mathrm{exp}\left(i\lambda r\right)+\sum\limits_{l=1}^{k}a_{l,r}\lambda^{k-l})\,.

Then on $L^{2}(\mathbb{R})$ , we have

\|\mathcal{P}_{[{0},{r}]}(1-w)\|_{2,2}\leq\left\|\mathcal{P}_{[{0},{r}]}\right\|_{2,2}\|1-w\|_{L^{\infty}(\mathbb{R})}\leq 1\cdot\delta<1\,.

Thus the operator $(I-\mathcal{P}_{[0,r]}(1-w))$ has bounded inverse $\sum\limits_{k=0}^{\infty}(\mathcal{P}_{[0,r]}(1-w))^{k}$ on $L^{2}(\mathbb{R})$ , and so (51) follows. ∎

We will also need the following Lemma to estimate the Fourier projections.

Lemma 7.3.

If $\delta\in(0,1)$ , then there exists $\epsilon(\delta)\in(0,\frac{1}{2})$ , with $\lim\limits_{\delta\to 0}\epsilon(\delta)=\frac{1}{2}$ , such that for all $p$ satisfying $|\frac{1}{p}-\frac{1}{2}|<\epsilon(\delta)$ we have

\|\mathcal{P}_{[{0},{r}]}\|_{p,p}\leq\frac{1}{\sqrt{\delta}}\,.

Proof.

By (29) and self-adjointness of $\mathcal{P}_{[{0},{r}]}$ , it follows that $\|\mathcal{P}_{[{0},{r}]}\|_{p^{\prime},p^{\prime}}=\|\mathcal{P}_{[{0},{r}]}\|_{p,p}\leq\|\cal{H}\|_{p,p}$ , where $\cal{H}$ is the Hilbert Transform. By [10], $1\leq\|\cal{H}\|_{p,p}=\tan(\frac{\pi}{2p})$ for $p\in(1,2]$ . Let $p_{0}=p_{0}(\delta)\in(1,2]$ be the unique element of $(1,2]$ that satisfies $\tan(\frac{\pi}{2p_{0}})=\delta^{-1/2}$ , and let $\epsilon(\delta)\stackrel{{\scriptstyle\rm def}}{{=}}\frac{1}{p_{0}}-\frac{1}{2}$ .

By duality and (29), we have $\|\mathcal{P}_{[{0},{r}]}\|_{p_{0}^{\prime},p_{0}^{\prime}},\|\mathcal{P}_{[{0},{r}]}\|_{p_{0},p_{0}}\leq\delta^{-1/2}$ . Interpolate between both estimates to get

\|\mathcal{P}_{[{0},{r}]}\|_{p,p}\leq\frac{1}{\sqrt{\delta}}\,

for all $p$ satisfying $|\frac{1}{p}-\frac{1}{2}|<\epsilon(\delta)$ .

To see that $\lim\limits_{\delta\to 0}\epsilon(\delta)=\frac{1}{2}$ , it suffices to note that $\tan(\frac{\pi}{2p})$ is increasing in $\frac{1}{p}$ for $p\in(1,2]$ and has singularity at $\frac{1}{p}=1$ . Thus as $\delta\to 0$ , we have $\delta^{-1/2}\to\infty$ , meaning that for our definition of $p_{0}$ , we have $\frac{1}{p_{0}}\to 1$ and so $\epsilon(\delta)\to\frac{1}{2}$ . ∎

We are now in a position to prove Theorem 1.7.

Proof of Theorem 1.7.

By (51) and Lemma 7.1, it follows that

\|R_{k,r}\|_{L^{p}(\mathbb{R})}\lesssim_{k}\|(I-\mathcal{P}_{[0,r]}(1-w))^{-1}\|_{p,p}\|\mathcal{P}_{[{0},{r}]}\|_{p,p}\|\langle\lambda\rangle^{k}(w-1)\|_{L^{p}(\mathbb{R})}\left(1+\frac{\|\langle\lambda\rangle^{k}(w-1)\|_{L^{1}(\mathbb{R})}}{1-\delta}\right)\,.

First note that by (29), we have $\|\mathcal{P}_{[{0},{r}]}\|_{p,p}\lesssim\|\cal{H}\|_{p,p}\lesssim_{p}1$ .

Next, by Lemma 7.3 we can choose $\epsilon(\delta)\in(0,\frac{1}{2})$ , with $\lim\limits_{\delta\to 0}\epsilon(\delta)=\frac{1}{2}$ , such that for all $p$ satisfying $|\frac{1}{p}-\frac{1}{2}|<\epsilon(\delta)$ , we have

\|\mathcal{P}_{[0,r]}(1-w)\|_{p,p}\leq\|\mathcal{P}_{[0,r]}\|_{p,p}\|1-w\|_{\infty}\leq\sqrt{\delta}<1\,.

Now by geometric sum, $I-\mathcal{P}_{[0,r]}(1-w)$ has inverse $\sum\limits_{j=0}^{\infty}(\mathcal{P}_{[{0},{r}]}(1-w))^{k}$ on $L^{p}(\mathbb{R})$ , with the bound

\|(I-\mathcal{P}_{[0,r]}(1-w))^{-1}\|_{p,p}\leq\sum\limits_{k=0}^{\infty}\|(\mathcal{P}_{[0,r]}(1-w))^{k}\|_{p,p}\leq\frac{1}{1-\delta^{1/2}}\sim\frac{1}{1-\delta}

whenever $|\frac{1}{p}-\frac{1}{2}|<\epsilon(\delta)$ .

Combine these estimates with the fact that $\frac{1}{1-\delta}\geq 1$ , we get

\|R_{k,r}\|_{L^{p}(\mathbb{R})}\lesssim_{p,k}\frac{1}{(1-\delta)^{2}}\|\langle\lambda\rangle^{k}(w-1)\|_{L^{p}(\mathbb{R})}(1+\|\langle\lambda\rangle^{k}(w-1)\|_{L^{1}(\mathbb{R})})\,

(52)

for all $p$ satisfying $|\frac{1}{p}-\frac{1}{2}|<\epsilon(\delta)$ , from which part (a) follows. As for part (b), given a $p\in(1,\infty)$ choose $\delta_{0}(p)$ small enough so that $|\frac{1}{p}-\frac{1}{2}|<\epsilon(\delta)$ for all $\delta\in(0,\delta_{0})$ . Then the estimate

\|\langle\lambda\rangle^{k}(w-1)\|_{L^{p}(\mathbb{R})}\leq\delta^{1/p^{\prime}}\|\langle\lambda\rangle^{k}(w-1)\|_{L^{1}(\mathbb{R})}^{1/p}\,,

combined with (52), yields part (b). ∎

8. Proof of Lemma 5.3

Before proving Lemma 5.3, we will need terminology and lemmas for a Calderón-Zygmund decomposition.

Definition. An interval $[a,a+r]$ has left neighbor $[a-r,a]$ and right neighbor $[a+r,a+2r]$ .

Two intervals are almost disjoint if their intersection is empty or a single point.

Given an interval $I$ , let $D_{I}\stackrel{{\scriptstyle\rm def}}{{=}}\mathrm{dist}(I,0)$ denote its distance from the origin.

An interval $I$ in $\mathbb{R}$ is dyadic if it is of the form $\{[j2^{-n},(j+1)2^{-n}]\}_{j,n\in\mathbb{Z}}$ .

Two dyadic intervals $I$ and $J$ are siblings if their union is a dyadic interval $K$ of strictly larger size, which we call the parent.

So define a partial ordering $\preceq$ on the dyadic intervals: if $I\subset K$ , we write $I\preceq K$ , and say $K$ is an ancestor of $I$ . Note that two dyadic intervals are either almost disjoint, or comparable via $\preceq$ .

See e.g. [13, Chapter 1, Section 3] or [14, Chapter 1, Theorem 4] for other variants of the Calderón-Zygmund decomposition below.

Lemma 8.1 (Calderón-Zygmund decomposition).

Let $d\mu=\langle\lambda\rangle^{q}d\lambda$ for some $q\geq 0$ , and suppose $u\in L^{1}_{\mu}(\mathbb{R})$ . If $\beta>0$ and $E_{\beta}\stackrel{{\scriptstyle\rm def}}{{=}}\{\lambda\in\mathbb{R}~{}:~{}|u|>\beta\}$ , then there exists a collection of dyadic intervals $\{I_{j}\}$ with the following properties:

(i)

$E_{\beta}\stackrel{{\scriptstyle\rm def}}{{=}}\{\lambda\in\mathbb{R}~{}:~{}|u|>\beta\}\subset\bigcup\limits_{j}I_{j}$ .
(ii)

Each $I_{j}$ is maximal with respect to $\preceq$ among those dyadic intervals $I$ satisfying $\beta<\frac{1}{\mu(I)}\int\limits_{I}|u|d\mu$ .
(iii)

The $\{I_{j}\}$ are pairwise almost disjoint.
(iv)

We have the estimate

$\mu(E_{\beta})\leq\sum\limits_{j}\mu(I_{j})\leq\frac{1}{\beta}\|u\|_{L^{1}_{\mu}(\mathbb{R})}<\infty\,.$ (53)

This is known as the Calderón-Zygmund decomposition of $u$ at level $\beta$ .

Proof.

Fix $\beta\in(0,1)$ , and by substituting $u$ with $|u|$ , assume without loss of generality that $u\geq 0$ . Since $u\in L^{1}_{\mu}(\mathbb{R})$ , then for any interval $I$ of size $|I|\geq\frac{1}{\beta}\int\limits_{\mathbb{R}}u\,d\mu$ , we have $\langle u\rangle_{I,\mu}\leq\beta$ , since $\mu(I)\geq|I|$ . Thus any dyadic interval $J$ has an ancestor $I_{0}\succeq J$ such that for all dyadics $I\succeq I_{0}$ , we have $\langle u\rangle_{I,\mu}\leq\beta$ . Meaning if we consider the set of dyadic intervals

\cal{I}\stackrel{{\scriptstyle\rm def}}{{=}}\{I\text{ dyadic}~{}:~{}\beta<\langle u\rangle_{I,\mu}\}\,,

then for each $I\in\cal{I}$ , there exists $I_{\max}\in\cal{I}$ such that $I\preceq I_{\max}$ , and $I_{\max}$ is maximal in $\cal{I}$ with respect to $\preceq$ . Define $\{I_{j}\}$ to be the set of maximal dyadic intervals in $(\cal{I},\preceq)$ . By definition, $\{I_{j}\}$ satisfies Property (ii), which then immediately implies Property (iii).

Meanwhile (53) follows from Property (i) and that $\beta<\frac{1}{\mu(I_{j})}\int\limits_{I_{j}}u\,d\mu$ :

\mu(E_{\beta})\leq\sum\limits_{j}\mu(I_{j})\leq\sum\limits_{j}\frac{1}{\beta}\int\limits_{I_{j}}|u|d\mu\leq\frac{1}{\beta}\|u\|_{L^{1}_{\mu}(\mathbb{R})}<\infty\,.

We are left with showing Property (i), which will follow if we can show $u\leq\beta$ for almost every $x$ in $(\bigcup\limits_{j}I_{j})^{c}$ . Consider the $L^{1}(\mathbb{R})$ function $v\stackrel{{\scriptstyle\rm def}}{{=}}u\langle\lambda\rangle^{q}$ . By the dyadic Lebesgue differentiation theorem, for almost every $x\in\mathbb{R}$ , we have $\lim\limits_{n\to\infty}\langle v\rangle_{J_{n}}=v(x)$ , where $J_{n}$ is any sequence of dyadic intervals shrinking to $x$ . Fix such a point $x\in(\bigcup\limits_{j}I_{j})^{c}$ . Then by construction of $\{I_{j}\}$ , for every dyadic interval $J$ containing $x$ , it follows that

\beta\geq\langle u\rangle_{J,\mu}=\frac{|J|}{\mu(J)}\langle v\rangle_{J}\,.

Letting $J$ shrink down to $x$ , the above yields

\beta\geq\frac{1}{\langle x\rangle^{q}}u(x)\langle x\rangle^{q}\,,

i.e. $\beta\geq u(x)$ . Since $x$ arbitrary in $(\bigcup\limits_{j}I_{j})^{c}\setminus N$ where $N$ is a set of measure $0$ , we get Property (i). ∎

Lemma 8.2 (Calderón-Zygmund decomposition for $L^{p_{1}}(\mathbb{R})+L^{p_{2}}(\mathbb{R})$ ).

Let $d\mu=\langle\lambda\rangle^{q}d\lambda$ for some $q\geq 0$ , and suppose $u=u_{1}+u_{2}$ with $u_{1}\in L^{p_{1}}_{\mu}(\mathbb{R})$ , $u_{2}\in L^{p_{2}}_{\mu}(\mathbb{R})$ , where $1\leq p_{1}\leq p_{2}<\infty$ . For each $\beta>0$ , define $E_{\beta}\stackrel{{\scriptstyle\rm def}}{{=}}\{\lambda\in\mathbb{R}~{}:~{}|u|>\beta\}$ and $E^{i}_{\beta/2}\stackrel{{\scriptstyle\rm def}}{{=}}\{\lambda~{}:~{}|u_{i}|>\frac{\beta}{2}\}$ , $i=1,2$ . Then $E_{\beta}\subset E_{\beta/2}^{1}\cup E_{\beta/2}^{2}$ , all of which are of finite $\mu$ -measure, and there exists a collection of dyadic intervals $\{I_{j}\}$ with the following properties:

(i)

$E_{\beta/2}^{1}\cup E_{\beta/2}^{2}\subset\bigcup\limits_{j}I_{j}$ .
(ii)

The $\{I_{j}\}$ are pairwise almost disjoint.

(iii)

We have the estimate

\sum\limits_{j}\mu(I_{j})\lesssim_{p_{1},p_{2}}\frac{1}{\beta^{p_{2}}}(\beta^{p_{2}-p_{1}}\|u_{1}\|_{L^{p_{1}}_{\mu}(\mathbb{R})}^{p_{1}}+\|u_{2}\|_{L^{p_{2}}_{\mu}(\mathbb{R})}^{p_{2}})\,.

(54)

(iv)

We have

$\langle|u|\rangle_{K,\mu}\leq\beta\,,\quad K\text{ any ancestor of }I_{j}\,.$ (55)

Proof.

By the triangle inequality, $E_{\beta}\subset E_{\beta/2}^{1}\cup E_{\beta/2}^{2}$ . Thus $\mu(E_{\beta})\leq\mu(E_{\beta/2}^{1})+(E_{\beta/2}^{2})<\infty$ , since $u_{i}\in L^{p_{i}}_{\mu}(\mathbb{R})$ .

For $i=1,2$ , we apply Lemma 8.1 to $|u_{i}|^{p_{i}}$ at level $\left(\frac{\beta}{2}\right)^{p_{i}}$ and get a collection of pairwise disjoint dyadic intervals $\{I_{j}^{i}\}_{j\geq 0}$ such that

E^{i}_{\beta/2}\subset\bigcup\limits_{j}I_{j}^{i}\,.

By Lemma 8.1 (ii),

\langle|u_{i}|^{p_{i}}\rangle_{I_{j}^{i},\mu}^{1/p_{i}}>\frac{\beta}{2}\,.

Let $\{\dot{I_{j}}\}_{j}=\{I_{j}^{1}\}_{j}\cup\{I_{j}^{2}\}_{j}$ . Note by maximality of the $I_{j}^{i}$ ’s specified by Lemma 8.1 (ii), each $\dot{I_{j}}$ is contained in at most one other $\dot{I_{k}}$ . So now let $\{I_{j}\}$ be the maximal intervals among $\{\dot{I_{j}}\}$ with respect to $\preceq$ ; maximality ensures the $\{I_{j}\}$ are pairwise almost disjoint and so (ii) holds. Note maximality also yields

E_{\beta}\subset E_{\beta/2}^{1}\cup E_{\beta/2}^{2}\subset\left(\bigcup\limits_{j}I_{j}^{1}\right)\cup\left(\bigcup\limits_{j}I_{j}^{2}\right)\subset\bigcup\limits_{j}I_{j}\,,

i.e. (i) holds. And (53) yields (54), as

\sum\limits\mu(I_{j})\leq\sum\limits_{j}\mu(I_{j}^{1})+\sum\limits_{j}\mu(I_{j}^{2})\lesssim_{p_{1},p_{2}}\frac{1}{\beta^{p_{1}}}\|u_{1}\|_{L^{p_{1}}_{\mu}(\mathbb{R})}^{p_{1}}+\frac{1}{\beta^{p_{2}}}\|u_{2}\|_{L^{p_{2}}_{\mu}(\mathbb{R})}^{p_{2}}=\frac{1}{\beta^{p_{2}}}(\beta^{p_{2}-p_{1}}\|u_{1}\|_{L^{p_{1}}_{\mu}(\mathbb{R})}^{p_{1}}+\|u_{2}\|_{L^{p_{2}}_{\mu}(\mathbb{R})}^{p_{2}})\,.

Also note by our choice of $I_{j}$ as the maximal intervals among $\{I_{j}^{1}\}_{j}\cup\{I_{j}^{2}\}_{j}$ , then for any ancestor $K$ of $I_{j}$ , it follows that

\langle|u_{1}|^{p_{1}}\rangle_{K,\mu}^{1/p_{1}}\,,\,\langle|u_{2}|^{p_{2}}\rangle_{K,\mu}^{1/p_{2}}\leq\frac{\beta}{2}\,.

The triangle inequality followed by Hölder’s inequality then imply (55), i.e.

\langle|u|\rangle_{K,\mu}\leq\langle|u_{1}|\rangle_{K,\mu}+\langle|u_{2}|\rangle_{K,\mu}\leq\langle|u_{1}|^{p_{1}}\rangle_{K,\mu}^{1/p_{1}}+\langle|u_{2}|^{p_{2}}\rangle_{K,\mu}^{1/p_{2}}\leq\beta\,.

∎

We also need the following additional lemma. Recall that for an interval $I$ , we defined $D_{I}=\mathrm{dist}\left(I,0\right)$ .

Lemma 8.3 ( $\mu$ is flat away from $0$ ).

Let $d\mu=\langle\lambda\rangle^{q}d\lambda$ for some $q\geq 0$ , and suppose $I$ is an interval such that $\mu(I)\leq L$ , where $L$ is some constant. Then there exists $D=D(L,q)\geq 0$ sufficiently large so that if $D_{I}\geq D$ , then

\langle\lambda\rangle^{q}\sim_{q}\langle D_{I}\rangle^{q}\,,\quad\lambda\text{ in }I\text{ or either of its neighbors}\,,

(56)

In particular,

\langle f\rangle_{J,\mu}\sim_{q}\langle f\rangle_{J}

(57)

for all $f\geq 0$ , where $J$ may equal $I$ , either of its neighbors, or its parent in the case that $I$ is dyadic.

Proof.

If $q=0$ , the lemma is trivial.

For $q>0$ , choose $D$ so large that $\langle D\rangle^{q}\geq 100L$ . Thus if $D_{I}\geq D$ , then

100L|I|\leq|I|\langle D\rangle^{q}\leq|I|\langle D_{I}\rangle^{q}\leq\mu(I)\leq L

and so $|I|<\frac{1}{100}$ . Thus on $I$ or either of its neighbors, we have $\langle\lambda\rangle^{q}\sim_{q}\langle D_{I}\rangle^{q}$ . Hence for $J=I$ or either of its neighbors, (57) holds as

\langle f\rangle_{J,\mu}=\frac{1}{\mu(J)}\int\limits_{J}fd\mu=\frac{1}{\mu(J)}\int\limits_{J}f(\lambda)\langle\lambda\rangle^{q}d\lambda\sim_{q}\frac{1}{\langle D_{I}\rangle^{q}|J|}\int\limits_{J}f(\lambda)\langle D_{I}\rangle^{q}d\lambda=\langle f\rangle_{J}\,.

If $I$ is dyadic, then since the parent of $I$ is the union of $I$ and one of its neighbors, it follows $\langle\lambda\rangle^{q}\sim_{q}\langle D_{I}\rangle^{q}$ on the parent of $I$ . Then the same computations as done previously yield (57) when $J$ is the parent of $I$ . ∎

Proof that $\|w^{1/\widetilde{p}}-w^{-1/\widetilde{p}^{\prime}}\|_{L^{p}_{\mu}(\mathbb{R})}<\infty$ in Lemma 5.3.

Let $\widetilde{p}\in(1,\infty)$ and $p\geq p_{2}$ . Let’s apply Lemma 8.2 with $u=w-1$ and $\beta=\alpha$ , where $\alpha\in(0,\frac{1}{2})$ , so that $E_{\alpha}\stackrel{{\scriptstyle\rm def}}{{=}}\{\lambda~{}:~{}|w(\lambda)-1|>\alpha\}$ and $E^{i}_{\alpha/2}\stackrel{{\scriptstyle\rm def}}{{=}}\{\lambda~{}:~{}|u_{i}(\lambda)|>\alpha/2\}$ . Then $E_{\alpha}\subset E_{\alpha/2}^{1}\cup E_{\alpha/2}^{2}$ .

We begin by splitting

\|w^{1/\widetilde{p}}-w^{-1/\widetilde{p}^{\prime}}\|_{L^{p}_{\mu}(\mathbb{R})}^{p}=\|w^{1/\widetilde{p}}-w^{-1/\widetilde{p}^{\prime}}\|_{L^{p}_{\mu}((E_{\alpha/2}^{1}\cup E_{\alpha/2}^{2})^{c})}^{p}+\|w^{1/\widetilde{p}}-w^{-1/\widetilde{p}^{\prime}}\|_{L^{p}_{\mu}(E_{\alpha/2}^{1}\cup E_{\alpha/2}^{2})}^{p}\stackrel{{\scriptstyle\rm def}}{{=}}A+B\,.

(58)

Since $(E_{\alpha/2}^{1}\cup E_{\alpha/2}^{2})^{c}\subset E_{\alpha}^{c}=\{|w-1|\leq\alpha\}$ , we may estimate

A=\int\limits_{(E_{\alpha/2}^{1}\cup E_{\alpha/2}^{2})^{c}}w^{-p/\widetilde{p}^{\prime}}|w-1|^{p}\,d\mu\leq(1-\alpha)^{-p/\widetilde{p}^{\prime}}\int\limits_{(E_{\alpha/2}^{1})^{c}\cap(E_{\alpha/2}^{2})^{c}}|w-1|^{p}\,d\mu\,.

By the triangle inequality, $A$ is

\lesssim_{p,\widetilde{p}}\int\limits_{(E_{\alpha/2}^{1})^{c}\cap(E_{\alpha/2}^{2})^{c}}|u_{1}|^{p}\,d\mu+\int\limits_{(E_{\alpha/2}^{1})^{c}\cap(E_{\alpha/2}^{2})^{c}}|u_{2}|^{p}\,d\mu\,\lesssim_{p_{1},p_{2}}\alpha^{p-p_{1}}\int\limits_{(E_{\alpha/2}^{1})^{c}}|u_{1}|^{p_{1}}\,d\mu+\alpha^{p-p_{2}}\int\limits_{(E_{\alpha/2}^{2})^{c}}|u_{2}|^{p_{2}}\,d\mu

Thus,

A\lesssim_{p,\widetilde{p},p_{1},p_{2}}\alpha^{p-p_{2}}(\alpha^{p_{2}-p_{1}}\|u_{1}\|_{L^{p_{1}}_{\mu}(\mathbb{R})}^{p_{1}}+\|u_{2}\|_{L^{p_{2}}_{\mu}(\mathbb{R})}^{p_{2}})\,.

(59)

By the triangle inequality followed by Lemma 8.2 (i),

B\lesssim_{p}\int\limits_{E_{\alpha/2}^{1}\cup E_{\alpha/2}^{2}}w^{p/\widetilde{p}}d\mu+\int\limits_{E_{\alpha/2}^{1}\cup E_{\alpha/2}^{2}}w^{-p/\widetilde{p}^{\prime}}d\mu\leq\left(\sum\limits_{j}\int\limits_{I_{j}}w^{p/\widetilde{p}}d\mu\right)+\left(\sum\limits_{j}\int\limits_{I_{j}}w^{-p/\widetilde{p^{\prime}}}d\mu\right)\stackrel{{\scriptstyle\rm def}}{{=}}B_{1}+B_{2}\,.

(60)

Recall $w\in A_{p^{\prime}}(\mathbb{R})$ and so $w^{-p/p^{\prime}}\in A_{p}(\mathbb{R})$ . By applying Lemma 5.1 to both of these weights, we can choose $\epsilon(\gamma,p)\in(0,\frac{1}{2})$ , with $\lim\limits_{\gamma\to 1}\epsilon(\gamma,p)=\min\{\frac{1}{p},\frac{1}{p^{\prime}}\}$ , so that whenever $\widetilde{p}$ is chosen so that $|\frac{1}{\widetilde{p}}-\frac{1}{p}|<\epsilon(\gamma,p)$ , we have

\langle w^{p/\widetilde{p}}\rangle_{I}\lesssim_{\gamma,p}\langle w\rangle_{I}^{p/\widetilde{p}}\,,\quad\langle w^{-p/\widetilde{p^{\prime}}}\rangle_{I}\lesssim_{\gamma,p}\langle w^{-p/p^{\prime}}\rangle_{I}^{p^{\prime}/\widetilde{p^{\prime}}}\,,\quad\text{for all intervals }I\,.

(61)

In particular, this means $w^{p/\widetilde{p}}$ and $w^{-p/\widetilde{p}^{\prime}}$ are locally integrable.

Let us focus on $B_{2}$ first, as $B_{1}$ will be similar. Set $D_{j}\stackrel{{\scriptstyle\rm def}}{{=}}\mathrm{dist}(I_{j},0)$ , then write

\sum\limits_{j}\int\limits_{I_{j}}w^{-p/\widetilde{p^{\prime}}}d\mu=\sum\limits_{j\,:\,D_{j}<D}\int\limits_{I_{j}}w^{-p/\widetilde{p^{\prime}}}d\mu+\sum\limits_{j\,:\,D_{j}\geq D}\int\limits_{I_{j}}w^{-p/\widetilde{p^{\prime}}}d\mu\,,

where $D>0$ is a constant which will be chosen later.

Since $|I_{j}|\leq\mu(I_{j})\lesssim_{p_{1},p_{2}}\frac{1}{\alpha^{p_{2}}}(\beta^{p_{2}-p_{1}}\|u_{1}\|_{L^{p_{1}}_{\mu}(\mathbb{R})}^{p_{1}}+\|u_{2}\|_{L^{p_{2}}_{\mu}(\mathbb{R})}^{p_{2}})$ by (54), then the length of each interval $I_{j}$ is bounded. Since $w^{-p/\widetilde{p}^{\prime}}$ is locally integrable, then

\sum\limits_{j\,:\,D_{j}<D}\int\limits_{I_{j}}w^{-p/\widetilde{p}^{\prime}}d\mu\leq\int\limits_{\widetilde{I}}w^{-p/\widetilde{p}^{\prime}}\langle\lambda\rangle^{q}\,d\lambda<\infty\,,

where $\widetilde{I}$ is some sufficiently large interval centered at $0$ . And so it suffices to show

\sum\limits_{j\,:\,D_{j}\geq D}\int\limits_{I_{j}}w^{-p/\widetilde{p}^{\prime}}d\mu=\sum\limits_{j\,:\,D_{j}\geq D}\mu(I_{j})\langle w^{-p/\widetilde{p}^{\prime}}\rangle_{I_{j},\mu}<\infty\,.

Since $\sum_{j}\mu(I_{j})<\infty$ , to show $B_{2}<\infty$ , it suffices to show $\langle w^{-p/\widetilde{p}^{\prime}}\rangle_{I_{j},\mu}\lesssim_{p,\gamma,q}1$ ; similarly $B_{1}<\infty$ will follow from showing $\langle w^{p/\widetilde{p}}\rangle_{I_{j},\mu}\lesssim_{p,\gamma}1$ for a suitable choice of $D$ .

Since there exists some constant $C=C(p_{1},p_{2})$ such that

\mu(I_{j})\leq C\frac{1}{\alpha^{p_{2}}}\left(\alpha^{p_{2}-p_{1}}\|u_{1}\|_{L^{p_{1}}_{\mu}(\mathbb{R})}^{p_{1}}+\|u_{2}\|_{L^{p_{2}}_{\mu}(\mathbb{R})}^{p_{2}}\right)\,,

we can take $D=D(C\frac{1}{\alpha^{p_{2}}}(\alpha^{p_{2}-p_{1}}\|u_{1}\|_{L^{p_{1}}_{\mu}(\mathbb{R})}^{p_{1}}+\|u_{2}\|_{L^{p_{2}}_{\mu}(\mathbb{R})}^{p_{2}}),q)$ as in Lemma 8.3 so that when $D_{j}\geq D$ , we have $\langle\lambda\rangle^{q}\sim_{q}\langle D_{j}\rangle^{q}$ on $I_{j}$ , its neighbors and its parent $K_{j}$ .

For such an interval $I_{j}$ , recall (55) implies

|\langle w-1\rangle_{K_{j},\mu}|\leq\langle|w-1|\rangle_{K_{j},\mu}\leq\alpha\leq\frac{1}{2}\,,

where $K_{j}$ is the parent of $I_{j}$ . And so $\langle\lambda\rangle^{q}\sim_{q}\langle D_{j}\rangle^{q}$ on $K_{j}$ and (57) together imply

\langle w\rangle_{K_{j}}\sim\langle w\rangle_{K_{j},\mu}=1+\langle w-1\rangle_{K_{j},\mu}\sim 1\,.

In particular,

\langle w\rangle_{I_{j},\mu}\sim_{q}\langle w\rangle_{I_{j}}\lesssim\langle w\rangle_{K_{j}}\sim 1\,

and so (61) implies

\langle w^{p/\widetilde{p}}\rangle_{I_{j},\mu}\sim_{q}\langle w^{p/\widetilde{p}}\rangle_{I_{j}}\lesssim_{\gamma,p}\langle w\rangle_{I_{j}}^{p/\widetilde{p}}\sim_{q,p,\gamma}\langle w\rangle_{I_{j},\mu}^{p/\widetilde{p}}\lesssim_{p,\gamma,q}1\,.

Similarly, since $w^{-p/p^{\prime}}\in A_{p}(\mathbb{R})$ , then

\langle w^{-p/p^{\prime}}\rangle_{I_{j},\mu}\sim_{q}\langle w^{-p/p^{\prime}}\rangle_{I_{j}}\lesssim\langle w^{-p/p^{\prime}}\rangle_{K_{j}}\leq[w^{-p/p^{\prime}}]_{A_{p}(\mathbb{R})}\langle w\rangle_{K_{j}}^{-p^{\prime}/p}\lesssim_{p,\gamma}1\,

and therefore (61) implies

\langle w^{-p/\widetilde{p}^{\prime}}\rangle_{I_{j},\mu}\,,\quad\langle w^{p/\widetilde{p}}\rangle_{I_{j},\mu}\lesssim_{p,\gamma}1\,.

∎

For the proof of (38), we need to understand how $[w]_{A_{\infty}(\mathbb{R})}$ affects $\log w$ .

Lemma 8.4.

Suppose $w\in A_{\infty}(\mathbb{R})$ and let $J$ be an interval. If $\langle|w-1|\rangle_{J}\leq\frac{1}{2}$ , then

|\langle\log w\rangle_{J}|\lesssim\langle|w-1|\rangle_{J}+\log[w]_{A_{\infty}(\mathbb{R})}\,.

Proof.

By Jensen’s inequality,

\langle\log w\rangle_{J}\leq\log\,\langle w\rangle_{J}=\log(\langle w-1\rangle_{J}+1)\leq\log(\langle|w-1|\rangle_{J}+1)\lesssim\langle|w-1|\rangle_{J}\,.

But also since $[w]_{A_{\infty}(\mathbb{R})}<\infty$ , then

\mathrm{exp}\left(-\langle\log w\rangle_{J}\right)\leq\frac{[w]_{A_{\infty}(\mathbb{R})}}{\langle w\rangle_{J}}\leq\frac{[w]_{A_{\infty}(\mathbb{R})}}{1-\langle|w-1|\rangle_{J}}\,,

which implies

\langle\log w\rangle_{J}\geq\log(1-\langle|w-1|\rangle_{J})-\log[w]_{A_{\infty}(\mathbb{R})}\,.

Combine the estimates of $\langle\log w\rangle_{J}$ from above and below to get

|\langle\log w\rangle_{J}|\lesssim\langle|w-1|\rangle_{J}+|\log(1-\langle|w-1|\rangle_{J})|+\log[w]_{A_{\infty}(\mathbb{R})}\lesssim\langle|w-1|\rangle_{J}+\log[w]_{A_{\infty}(\mathbb{R})}\,.

∎

Next, we recall a few facts about $\rm{BMO}$ , the space of functions with bounded mean oscillation, and how it relates to $A_{p}(\mathbb{R})$ weights. Recall that $f\in{\rm BMO}(\mathbb{R}^{d})$ if

\|f\|_{{\rm BMO}(\mathbb{R}^{d})}\stackrel{{\scriptstyle\rm def}}{{=}}\sup_{B}\,\langle|f-\langle f\rangle_{B}|\rangle_{B}<\infty\,,

where $B$ denotes a ball in $\mathbb{R}^{d}$ (see, e.g., p.140 in [13]). Functions in $\rm BMO(\mathbb{R}^{d})$ all satisfy the John-Nirenberg estimates below.

Theorem 8.5 (John-Nirenberg, [13, p.144-146]).

Suppose $f\in\mathrm{BMO}(\mathbb{R}^{d})$ . Then

(a)

There exist positive absolute constants $c_{1},c_{2}$ such that for each $\alpha>0$ and every ball $B$ ,

\frac{1}{|B|}|\{x\in B~{}:~{}|f(x)-\langle f\rangle_{B}|>\alpha\}|\leq c_{1}\mathrm{exp}\left(-c_{2}\alpha/\|f\|_{\mathrm{BMO}}\right)\,.

(b)

For any $p<\infty$ , $f\in L^{p}_{loc}(\mathbb{R}^{d})$ and

$\langle|f-\langle f\rangle_{B}|^{p}\rangle_{B}\lesssim_{p}\|f\|_{\mathrm{BMO}(\mathbb{R}^{d})}^{p}\,$

for all balls $B$ .
(c)

If $0\leq\mu\|f\|_{\mathrm{BMO}(\mathbb{R}^{d})}\lesssim 1$ , then

$\langle\mathrm{exp}\left(\mu|f-\langle f\rangle_{B}|\right)\rangle_{B}\lesssim 1\,.$

It is well known that if $w\in A_{p}(\mathbb{R}^{d})$ , then $\log w\in\mathrm{BMO}(\mathbb{R}^{d})$ . We also have the following well-known quantification (see, e.g., [7, Corollary 6]).

Lemma 8.6.

If $w\in A_{p}(\mathbb{R})$ for some $p\in(1,\infty]$ , then $\|\log w\|_{\rm BMO}\leq\log(2[w]_{A_{p}(\mathbb{R}^{d})})$ . If in addition $[w]_{A_{p}(\mathbb{R}^{d})}=1+\tau,\tau\in[0,1]$ , then

\|\log w\|_{\rm BMO}\lesssim\sqrt{\tau}\,.

Proof of (38).

By Lemma 8.6, when $\tau_{0}(p)\leq\frac{1}{2}$ we have $\|\log w\|_{\mathrm{BMO}(\mathbb{R})}\lesssim\tau^{1/2}$ . Take $\tau_{0}(p)$ small enough so that additionally $\|\log w\|_{\mathrm{BMO}(\mathbb{R})}\leq\frac{1}{10}$ .

Set $\alpha\stackrel{{\scriptstyle\rm def}}{{=}}\tau^{1/2}$ and define $E_{\alpha}$ , $E_{\alpha/2}^{i}$ and $\{I_{j}\}$ as in the proof that $\|w^{1/\widetilde{p}}-w^{-1/\widetilde{p}^{\prime}}\|_{L^{p}_{\mu}(\mathbb{R})}<\infty$ , i.e. by applying Lemma 8.2 with $u=|w-1|$ and $\beta=\alpha$ . We begin by splitting $\|w^{1/p}-w^{-1/p^{\prime}}\|_{L^{p}(\mathbb{R})}^{p}$ as in (58), with the same definitions of $A$ and $B$ . By (59) and our choice of $\alpha$ , it follows that

A\lesssim_{p,p_{1},p_{2}}\tau^{(p-p_{2})/2}(\tau^{\frac{p_{2}-p_{1}}{2}}\|u_{1}\|_{L^{p_{1}}(\mathbb{R})}^{p_{1}}+\|u_{2}\|_{L^{p_{2}}(\mathbb{R})}^{p_{2}})\,.

As for estimating $B$ , we split slightly differently than in (60):

B\lesssim_{p}\|w^{1/p}-1\|_{L^{p}(E_{\alpha/2}^{1}\cup E_{\alpha/2}^{2})}^{p}+\|w^{-1/p^{\prime}}-1\|_{L^{p}(E_{\alpha/2}^{1}\cup E_{\alpha/2}^{2})}^{p}\stackrel{{\scriptstyle\rm def}}{{=}}B_{1}+B_{2}\,.

It suffices to show

B_{1},B_{2}\lesssim_{p,p_{1},p_{2}}\tau^{(p-p_{2})/2}(\tau^{\frac{p_{2}-p_{1}}{2}}\|u_{1}\|_{L^{p_{1}}(\mathbb{R})}^{p_{1}}+\|u_{2}\|_{L^{p_{2}}(\mathbb{R})}^{p_{2}})\,.

(62)

We prove this for $B_{1}$ ; the estimate for $B_{2}$ will follow similarly.

By Lemma 8.2 (i),

B_{1}\leq\sum\limits_{j}\|w^{1/p}-1\|_{L^{p}(I_{j})}^{p}\,.

Note that (62) will follow if we can show

\|w^{1/p}-1\|_{L^{p}(I)}^{p}\lesssim_{p}|I|\tau^{p/2}

(63)

for all $I\in\{I_{j}\}$ : indeed, sum over $I=I_{j}$ and use (54) to get (62). As such, fix $I\in\{I_{j}\}$ ; our goal is to show (63).

Now note

|\langle\log w\rangle_{I}|\lesssim\tau^{1/2}\,.

(64)

Indeed, if $K$ is the dyadic parent of $I$ , then (55) guarantees $\langle|w-1|\rangle_{I}\lesssim\langle|w-1|\rangle_{K}\leq\alpha$ . By Lemma 8.4, we get (64) for $\tau_{0}(p)$ sufficiently small, which we will make use of repeatedly.

Split the left side of (63) into

\|w^{1/p}-1\|_{L^{p}(I)}^{p}\leq\|w^{1/p}-\mathrm{exp}\left(\langle\log w\rangle_{I}/p\right)\|_{L^{p}(I)}^{p}+\|\mathrm{exp}\left(\langle\log w\rangle_{I}/p\right)-1\|_{L^{p}(I)}^{p}\stackrel{{\scriptstyle\rm def}}{{=}}B_{11}+B_{12}\,.

By (64) it follows that

B_{12}=|I||\mathrm{exp}\left(\langle\log w\rangle_{I}/p\right)-1|^{p}\lesssim_{p}|I|\tau^{p/2}\,.

Thus it suffices to show $B_{11}\lesssim_{p}|I|\tau^{p/2}$ .

But $B_{11}$ equals

\mathrm{exp}\left(\langle\log w\rangle_{I}/p\right)\|\mathrm{exp}\left((\log w-\langle\log w\rangle_{I})/p\right)-1\|_{L^{p}(I)}^{p}\lesssim\|\mathrm{exp}\left(f/p\right)-1\|_{L^{p}(I)}^{p}\,,

where $f\stackrel{{\scriptstyle\rm def}}{{=}}\log w-\langle\log w\rangle_{I}$ and we applied (64) in the inequality. Write

\mathrm{exp}\left(f/p\right)-1=\int\limits_{0}^{1/p}\mathrm{exp}\left(sf\right)f\,ds\,

and apply Minkowski’s inequality, followed by the Cauchy-Schwarz inequality, to get

\|\mathrm{exp}\left(f/p\right)-1\|_{L^{p}(I)}\lesssim\|f\mathrm{exp}\left(|f|/p\right)\|_{L^{p}(I)}\leq\|f\|_{L^{2p}(I)}\|\mathrm{exp}\left(|f|/p\right)\|_{L^{2p}(I)}\,.

The John-Nirenberg Theorem 8.5 yields

\|f\|_{L^{2p}(I)}\lesssim_{p}\|\log w\|_{\mathrm{BMO}(\mathbb{R})}|I|^{\frac{1}{2p}}\,,\quad\|\mathrm{exp}\left(|f|/p\right)\|_{L^{2p}(I)}\lesssim_{p}|I|^{\frac{1}{2p}}

for $\|\log w\|_{\mathrm{BMO}(\mathbb{R})}$ small enough, which we can arrange by taking $\tau_{0}(p)$ as small as necessary and applying Lemma 8.6. Thus

\displaystyle\|\mathrm{exp}\left(f/p\right)-1\|_{L^{p}(I)}

\displaystyle\lesssim_{p}|I|^{1/p}\|\log w\|_{\mathrm{BMO}(\mathbb{R})}\lesssim|I|^{1/p}\tau^{1/2}\,.

Hence $B_{11}\lesssim_{p}|I|^{1/p}\tau^{1/2}$ , which completes the proof. ∎

9. Inverting $I-Q_{w,p}$ : proof of Lemma 5.4

Let us introduce some notation and definitions.

Notation. Given $z=\mathbb{C}$ , we will generally write $z=t+iy$ , where $t,y$ are real-valued.

Given $K\subset\mathbb{C}$ , we let $N(K)$ denote an open set containing $K$ , although we allow for the particular set to change line to line.

Definition. Given some $p_{*}\in[1,\infty]$ , define

\frac{1}{p(z)}\stackrel{{\scriptstyle\rm def}}{{=}}\frac{z}{p_{*}}+\frac{1-z}{2},\quad\frac{1}{p^{\prime}(z)}\stackrel{{\scriptstyle\rm def}}{{=}}1-\frac{1}{p(z)}=\frac{1+z}{2}-\frac{z}{p_{*}},\quad z=t+iy\,:-1\leq t\leq 1\,.

(65)

Given $\epsilon\geq 0$ , define the open strip

\Omega_{<\epsilon}\stackrel{{\scriptstyle\rm def}}{{=}}\left\{z~{}:~{}\left|\frac{1}{p(t)}-\frac{1}{2}\right|<\epsilon\right\}\,.

We define the closed strip $\Omega_{\leq\epsilon}$ similarly.

Given $I$ an interval, let

\Omega_{I}\stackrel{{\scriptstyle\rm def}}{{=}}\{z\in\mathbb{C}~{}:~{}t\in I\}\,.

Lemma 9.1.

Let ${\bf\Lambda}>1$ be an arbitrary absolute constant, and let $p(z)$ is as in (65) for some $p_{*}\in[1,\infty]$ . For $[w]_{A_{2}(\mathbb{R})}\leq\gamma$ , consider the operator

Q_{w,p(z),T}\stackrel{{\scriptstyle\rm def}}{{=}}w^{1/p(z)}Tw^{-1/p(z)}-w^{-1/p^{\prime}(z)}Tw^{1/p^{\prime}(z)}\,,

where $T$ is an operator satisfying (28) for some $\cal{F}$ and is also self-adjoint as an operator on $L^{2}(\mathbb{R})$ .

Then there exists $\epsilon=\epsilon(\gamma,\cal{F},{\bf\Lambda})\in(0,\frac{1}{2})$ , with $\lim\limits_{\gamma\to 1}\epsilon(\gamma,\cal{F},{\bf\Lambda})=\frac{1}{2}$ , such that on $\Omega_{<\epsilon}$ , $I-Q_{w,p(z)}$ has bounded inverse on $L^{p(t)}(\mathbb{R})$ with operator bound

\|(I-Q_{w,p(z),T})^{-1}\|_{p(t),p(t)}\leq 2\bf{\Lambda}\,,

(66)

and $(I-Q_{w,p(z),T})^{-1}$ is analytic as a map $\Omega_{<\epsilon}\to\cal{L}(L^{2}(\mathbb{R}))$ .

Lemma 5.4 follows from Lemma 9.1 by taking $T=\mathcal{P}_{[{0},{r}]}$ , $\mathbf{\Lambda}=10$ and applying Lemma 4.3. As such, this section is dedicated to the proof of Lemma 9.1. The proof strategy will proceed more or less as follows:

(1)

Show $\|Q_{w,p(z),T}\|_{p(t),p(t)}$ uniformly bounded on the strip.
(2)

Show that if we have uniform bounds on $\|(I-\kappa Q_{w,p(z),T})^{-1}\|_{p(t),p(t)}$ is uniformly bounded on the strip, then $(I-\kappa Q_{w,p(z),T})^{-1}$ is weakly analytic.
(3)

Chop $(I-\kappa Q_{w,p(z),T})^{-1}-I$ into small pieces and show we have uniform bounds for $I$ plus the small piece; using the two previous parts of the strategy, we have our function is weakly analytic. We maintain boundedness while adding the small pieces by applying analytic interpolation.

Before we begin, we will need a few preliminary lemmas.

Proposition 9.2.

Suppose $X$ is an Banach space and $H,V$ are linear bounded operators from $X$ to $X$ . Then,

	$\displaystyle(I+H+V)^{-1}=(I+H)^{-1}-(I+H+V)^{-1}V(I+H)^{-1},$
	$\displaystyle(I+H+V)^{-1}=(I+H)^{-1}(I+V(I+H)^{-1})^{-1}\,,$

provided the operators involved are well-defined and bounded in $X$ . Moreover, assuming $\|V\|\cdot\|(I+H)^{-1}\|<1$ , we get

\|(I+H+V)^{-1}\|\leqslant\frac{\|(I+H)^{-1}\|}{1-\|V\|\cdot\|(I+H)^{-1}\|}\,.

(67)

Finally, if $\|V\|<1$ , then

\|(I+V)^{-1}\|\leqslant\frac{1}{1-\|V\|}\,.

(68)

The proof of this proposition is a straightforward calculation.

We will also need continuity of weighted operators as proved in [1, Theorem 1.2].

Theorem 9.3 ([1, Theorem 1.2]).

Suppose $p\in(1,\infty)$ , $[w]_{A_{p}(\mathbb{R}^{d})}<\infty$ , $\|f\|_{\rm BMO}<\infty$ , and $T$ satisfies (28). Consider $w_{\delta}=we^{\delta f}$ . Then, there is $\delta_{0}(p,[w]_{A_{p}},\|f\|_{\rm BMO})>0$ such that

\|w_{\delta}^{1/p}Tw_{\delta}^{-1/p}-w^{1/p}Tw^{-1/p}\|_{p,p}<|\delta|C(p,[w]_{A_{p}},\|f\|_{\rm BMO},\cal{F})

for all $\delta:|\delta|<\delta_{0}$ .

Definition. Suppose $\cal{D}$ is a set of linear functionals acting on a vector space $V$ . If $U\subset\mathbb{C}$ is an open set, and $\{a(z)\}_{z\in U}$ is contained within $V$ , then $a(z)$ is weakly analytic with respect to $\cal{D}$ if $\ell(a(z))$ is analytic for all $\ell\in\cal{D}$ .

We will generally be concerned with two cases:

(1)

$V=\bigcup\limits_{p\geq 1}L^{p}(\mathbb{R})$ for $p\in(1,\infty)$ , and $\cal{D}=\cal{S}_{c}(\mathbb{R})$ , the space of simple functions with compact support, where $g\in\cal{S}_{c}(\mathbb{R})$ acts on $f\in\bigcup\limits_{p\geq 1}L^{p}(\mathbb{R})$ by $\langle f,g\rangle$ .
(2)

$V=\bigcup\limits_{p\geq 1}\cal{L}(L^{p}(\mathbb{R}))$ and $\cal{D}=\{\langle\cdot f,g\rangle\}_{f,g\,\in\cal{S}_{c}(\mathbb{R})}$ .

Lemma 9.4.

Let $T$ be an operator satisfying (28) that is self-adjoint on $L^{2}(\mathbb{R})$ . If $[w]_{A_{2}(\mathbb{R})}\leq\gamma$ , then there exists $\epsilon_{0}(\gamma)\in(0,\frac{1}{2})$ , with $\lim\limits_{\gamma\to 1}\epsilon_{0}(\gamma)=\frac{1}{2}$ , such that for all $z=t+iy\in N(\Omega_{\leq\epsilon_{0}(\gamma)})$ , we have

[w]_{A_{p(t)}(\mathbb{R})},[w]_{A_{p^{\prime}(t)}(\mathbb{R})}\lesssim_{\gamma}1,\,

(69)

and

\|Q_{w,p(z),T}\|_{p,p}\leq C_{\gamma,\cal{F}}\,

(70)

and $Q_{w,p(z),T}$ is weakly analytic with respect to $\{\langle\cdot f,g\rangle_{f,g\in\cal{S}_{c}(\mathbb{R})}\}$ on $N(\Omega_{\leq\epsilon_{0}(\gamma)})$ .

Proof.

Initially define $\epsilon_{0}(\gamma)\stackrel{{\scriptstyle\rm def}}{{=}}\frac{1}{s(2,\gamma)}-\frac{1}{2}$ where $s$ is as in Lemma 5.1. Then $\epsilon_{0}(\gamma)\in(0,\frac{1}{2})$ , $\lim\limits_{\gamma\to 1}\epsilon_{0}(\gamma)=\frac{1}{2}$ , and (69) holds for $z\in\Omega_{\leq\epsilon_{0}}$ ; by taking a slightly smaller $\epsilon_{0}(\gamma)\in(0,\frac{1}{2})$ , still with $\lim\limits_{\gamma\to 1}\epsilon_{0}(\gamma)=\frac{1}{2}$ , then (69) holds for $z\in N(\Omega_{\leq\epsilon_{0}(\gamma)})$ .

If $T$ satisfies (28), then

\|w^{1/p}Tw^{-1/p}\|_{p,p}\leq C_{\gamma,\cal{F},p}\,,

for all $p$ satisfying $\left|\frac{1}{2}-\frac{1}{p}\right|\leq\epsilon_{0}(\gamma)$ . Since $\|w^{1/p}Tw^{-1/p}\|_{p,p}=\|T\|_{L^{p}_{w}(\mathbb{R}),L^{p}_{w}(\mathbb{R})}$ , we can interpolate between the extremal values of $p$ given by $\left|\frac{1}{2}-\frac{1}{p}\right|=\epsilon_{0}(\gamma)$ to in fact get

\|w^{1/p}Tw^{-1/p}\|_{p,p}\leq C_{\gamma,\cal{F}}\,

for all $p$ satisfying $\left|\frac{1}{2}-\frac{1}{p}\right|\leq\epsilon_{0}(\gamma)$ . In particular,

\|w^{1/p(z)}Tw^{-1/p(z)}\|_{p(t),p(t)}=\|w^{1/p(t)}Tw^{-1/p(t)}\|_{p(t),p(t)}\leq C_{\gamma,\cal{F}}\,

for all $z\in\Omega_{\leq\epsilon_{0}(\gamma)}$ . If we swap the role of $p_{*}$ with $p_{*}^{\prime}$ , this has the net effect of swapping the role $p(z)$ with $p^{\prime}(z)$ . Thus if $T$ is self-adjoint on $L^{2}(\mathbb{R})$ , then swapping as described and taking adjoints yields

\|w^{-1/p^{\prime}(z)}Tw^{1/p^{\prime}(z)}\|_{p(t),p(t)}=\|w^{1/p^{\prime}(z)}Tw^{-1/p^{\prime}(z)}\|_{p^{\prime}(t),p^{\prime}(t)}\leq C_{\gamma,\cal{F}}\,

for $z\in\Omega_{\leq\epsilon_{0}(\gamma)}$ . Combining all of this together yields (70) for $z\in\Omega_{\epsilon_{0}(\gamma)}$ ; by slightly shrinking $\epsilon_{0}(\gamma)\in(0,1/2)$ , while still requiring $\lim\limits_{\gamma\to 1}\epsilon_{0}(\gamma)=\frac{1}{2}$ , we get (70) for $z\in N(\Omega_{\leq\epsilon_{0}(\gamma)})$ .

By Proposition A.4, if we again shrinking $\epsilon_{0}(\gamma)\in(0,\frac{1}{2})$ we may assume without loss of generality that $Q_{w,p(z),T}$ is analytic as a map $N(\Omega_{\leq\epsilon_{0}(\gamma)})\to\cal{L}(L^{2}(\mathbb{R}))$ , and hence weakly analytic with respect to $\{\langle\cdot f,g\rangle_{f,g\,\in\cal{S}_{c}(\mathbb{R})}\}$ on $N(\Omega_{\leq\epsilon_{0}(\gamma)})$ . ∎

Lemma 9.5.

Let $[w]_{A_{2}(\mathbb{R})}\leq\gamma$ , $\epsilon_{0}(\gamma)$ is as Lemma 9.4 and let $p(z)$ as in (65) be given for some $p_{*}\in[1,\infty]$ . If

\sup\limits_{z\in\Omega_{\leq\epsilon_{0}(\gamma)}}\|(I-\kappa Q_{w,p(z),T})^{-1}\|_{p(t),p(t)}<\infty\,,

then $(I-\kappa Q_{w,p(z),T})^{-1}$ is weakly analytic with respect to $\{\langle\cdot f,g\rangle\}_{f,g\in\cal{S}_{c}(\mathbb{R})}$ on $N(\Omega_{\leq\epsilon_{0}(\gamma)})$ .

Proof.

Let $F(z)\stackrel{{\scriptstyle\rm def}}{{=}}(I-\kappa Q_{w,p(z),T})^{-1}$ . We show that for each $z_{0}\in\Omega_{\leq\epsilon_{0}(\gamma)}$ , there exists a ball $B_{\delta}(z_{0})$ on which $F(z)$ is analytic as a map $B_{\delta}(z_{0})\to\cal{L}(L^{p(t_{0})}(\mathbb{R}))$ ; this will clearly imply weak analyticity on $N(\Omega_{\leq\epsilon_{0}(\gamma)})$ . Fix $z_{0}$ , define

V(z,z_{0})\stackrel{{\scriptstyle\rm def}}{{=}}\kappa(Q_{w,p(z),T}-Q_{w,p(z_{0}),T})

and write

F(z)=F(z_{0})+(F(z_{0})^{-1}-V(z,z_{0}))^{-1}-F(z_{0})=F(z_{0})+((I-F(z_{0})V(z,z_{0}))^{-1}-I)F(z_{0})\\ =(I-F(z_{0})V(z,z_{0}))^{-1}F(z_{0})\,.

(71)

We claim that there exists a ball $B_{\delta}(z_{0})$ on which $Q_{w,p(z),T}$ , and hence $V(z,z_{0})$ , is analytic as a map $B_{\delta}(z_{0})\to\cal{L}(L^{p(t_{0})}(\mathbb{R}))$ . Given the claim, we can then take $\delta$ small enough so that $\|V(z,z_{0})\|_{p(t_{0}),p(t_{0})}<\frac{1}{2\|F(z_{0})\|_{p(t_{0}),p(t_{0})}}$ . Then (71) yields

F(z)=(I-F(z_{0})V(z,z_{0}))^{-1}F(z_{0})=\sum\limits_{k=0}^{\infty}(F(z_{0})V(z,z_{0}))^{k}F(z_{0})\,,

where the sum converges uniformly in $\cal{L}(L^{p(t_{0})(\mathbb{R})})$ . Since each partial sum is analytic in $z$ and the sum converges uniformly, we have $F(z)$ is analytic as a map $B_{\delta}(z_{0})\to\cal{L}(L^{p(t_{0})}(\mathbb{R}))$ .

We must now show there exists a ball $B_{\delta}(z_{0})$ on which $Q_{w,p(z),T}$ is analytic as a map $B_{\delta}(z_{0})\to\cal{L}(L^{p(t_{0})}(\mathbb{R}))$ . As per Lemma 9.4, $Q_{w,p(z),T}$ is weakly analytic with respect to $\{\langle\cdot f,g\rangle\}_{f,g\in\cal{S}_{c}(\mathbb{R})}$ on $N(\Omega_{\leq\epsilon_{0}(\gamma)})$ , and so by Proposition A.1 it suffices to show there exists $\delta>0$ such that for all $z\in B_{\delta}(z_{0})$ , we have $w^{1/p(z)}Tw^{-1/p(z)},w^{-1/p^{\prime}(z)}Tw^{1/p(z)}\in\cal{L}(L^{p(t_{0})}(\mathbb{R}))$ . We show this for $w^{1/p(z)}Tw^{-1/p(z)}$ , as the proof for $w^{-1/p^{\prime}(z)}Tw^{1/p^{\prime}(z)}$ will follow similarly. Write

\|w^{1/p(z)}Tw^{-1/p(z)}\|_{p(t_{0}),p(t_{0})}=\|w^{1/p(t)}Tw^{-1/p(t)}\|_{p(t_{0}),p(t_{0})}\\ \leq\|w^{1/p(t)}Tw^{-1/p(t)}-w^{1/p(t_{0})}Tw^{-1/p(t_{0})}\|_{p(t_{0}),p(t_{0})}+\|w^{1/p(t_{0})}Tw^{-1/p(t_{0})}\|_{p(t_{0}),p(t_{0})}\,.

The last term is clearly finite. As for $\|w^{1/p(t)}Tw^{-1/p(t)}-w^{1/p(t_{0})}Tw^{-1/p(t_{0})}\|_{p(t_{0}),p(t_{0})}$ , by (69) we have $w\in A_{p(t)}$ . Thus we can apply the continuity of weighted operators Theorem 9.3 with $f=\log w$ , to argue that for $|t-t_{0}|$ sufficiently small, we have $\|w^{1/p(t)}Tw^{-1/p(t)}-w^{1/p(t_{0})}Tw^{-1/p(t_{0})}\|_{p(t_{0}),p(t_{0})}\leq 1$ . This completes the proof. ∎

Proposition 9.6 ([1]).

Let $[w]_{A_{2}(\mathbb{R})}\leq\gamma$ and let $p(z)$ as in (65) be given for some $p_{*}\in[1,\infty]$ such that $\Omega_{[-1,1]}\subset\Omega_{\leq\epsilon_{0}(\gamma)}$ , where $\epsilon_{0}(\gamma)$ is as in Lemma 9.4. Suppose

\sup_{-1\leqslant\mathop{\rm Re}z\leqslant 1}\|Q_{w,p(z),T}\|_{p(t),p(t)}<\infty\,,

where $t\stackrel{{\scriptstyle\rm def}}{{=}}\mathop{\rm Re}z$ . If there is a positive number ${\bf\Lambda}>1$ such that

\|(I-\kappa Q_{w,p(z),T})^{-1}\|_{p(t),p(t)}\leqslant 2{\bf\Lambda}

for all $z\in\Omega_{[-1,1]}$ , then there is a $t_{*}({\bf\Lambda})\in(0,1]$ , so that

\|(I-\kappa Q_{w,p(z),T})^{-1}\|_{p(t),p(t)}\leqslant{\bf\Lambda}

for all $z\in\Omega_{[-t_{*},t_{*}]}$ .

Proof.

We notice that $Q_{w,p(iy),T}$ is bounded and antisymmetric operator in Hilbert space $L^{2}(\mathbb{R})$ . Therefore, $\|(I-\kappa Q_{w,p(iy),T})^{-1}\|_{2,2}\leqslant 1$ . By Lemma 9.5, $(I-\kappa Q_{w,p(z),T})^{-1}$ is weakly analytic and continuous in the sense of Stein (p.209, [2]). Applying Stein’s interpolation theorem on the strips $\Omega_{[0,1]}$ and $\Omega_{[-1,0]}$ , we get

\|(I-\kappa Q_{w,p(z),T})^{-1}\|_{p(t),p(t)}\leqslant\exp\left(\frac{\sin(\pi|t|)}{2}\int_{\mathbb{R}}\frac{\log(2{\bf\Lambda)}}{\cosh(\pi y)+\cos(\pi|t|)}dy\right)=1+O(|t|),\quad t\to 0\,.

∎

Remark. We emphasize here that positive $t_{*}$ does not depend on $w$ .

Proof of Lemma 9.1.

The reader may also consult a similar, but simpler proof in [1, Proof of Lemma 3.6].

We just need to find $\epsilon(\gamma,\cal{F},\mathbf{\Lambda})$ such that on $\Omega_{<\epsilon(\gamma,\cal{F},\mathbf{\Lambda})}$ , we have $(I-Q_{w,p(z),T})^{-1}$ is bounded on $L^{p(t)}(\mathbb{R})$ and analytic as an $\cal{L}(L^{2}(\mathbb{R}))$ -valued function.

In (65), we take parameter $p_{*}$ as follows: define $p_{*}^{(1)}$ by $\frac{1}{p_{*}^{(1)}}=\frac{1}{2}-\epsilon_{0}(\gamma)$ , where $\epsilon_{0}(\gamma)$ is as in Lemma 9.4, and set $p_{1}(z)\stackrel{{\scriptstyle\rm def}}{{=}}p(z)$ ; note then $\Omega_{[-1,1]}\subset\Omega_{\leq\epsilon_{0}(\gamma)}$ . Consider $Q^{(j)}_{w,p(z),T}\stackrel{{\scriptstyle\rm def}}{{=}}jQ_{w,p(z),T}/N$ , $j=1,\ldots,N$ , where $N$ is large and will be fixed later (it will depend on $\gamma$ , $\cal{F}$ , $\bf{\Lambda}$ only).

We take $N$ to satisfy

1-C_{\gamma,\cal{F}}{\bf\Lambda}/N>1/2\,,

(72)

where $C_{\gamma,\cal{F}}$ is as in (70). Next, by (68) and (70),

\|(I-Q^{(1)}_{w,p(t+iy),T})^{-1}\|_{p(t),p(t)}\leqslant\frac{1}{1-C_{\gamma,\mathcal{F}}/N}\leqslant\frac{1}{1-C_{\gamma,\mathcal{F}}{\bf\Lambda}/N}\leqslant 2\leqslant 2{\bf\Lambda}\,,

since ${\bf\Lambda}>1$ .

We continue with an inductive argument in which the bound for $\{Q^{(j)}_{w,p(z)}\}$ provides the bound for $\{Q^{(j+1)}_{w,p(z),T}\}$ when $j=1,\ldots,N-1$ .

$\bullet$ Base of induction: handling $Q^{(1)}_{w,p(z)}$ . Apply Proposition 9.6 with $\kappa=1/N$ to get an absolute constant $t_{*}$ so that

\|(I-Q^{(1)}_{w,p(t+iy),T})^{-1}\|_{p(t),p(t)}\leqslant{\bf\Lambda}

for $t\in[-t_{*},t_{*}]$ and $y\in\mathbb{R}$ . Next, we use (67) with $H=-Q^{(1)}_{w,p(t+iy)}$ and $V=-N^{-1}Q_{w,p(t+iy)}$ . This gives

\|(I-Q^{(2)}_{w,p(t+iy),T})^{-1}\|_{p(t),p(t)}\leqslant\frac{{\bf\Lambda}}{1-C_{\gamma,\mathcal{F}}{\bf\Lambda}/N}\leqslant 2{\bf\Lambda},\quad t\in[-t_{*},t_{*}]

(73)

by (72).

That finishes the first step. Next, we will explain how estimates on $Q^{(2)}_{w,p(z)}$ give bounds for $Q^{(3)}_{w,p(z)}$ .

$\bullet$ Handling $Q^{(2)}_{w,p(z),T}$ . In Proposition 9.6, we now take $\kappa=\kappa_{2}\stackrel{{\scriptstyle\rm def}}{{=}}2/N,p^{(2)}_{*}\stackrel{{\scriptstyle\rm def}}{{=}}p_{1}(t_{*})=p(t_{*})$ (here $p(t_{*})$ is obtained at the previous step) and compute new $p_{2}(z),p_{2}^{\prime}(z)$ by (65):

\frac{1}{p_{2}(z)}\stackrel{{\scriptstyle\rm def}}{{=}}\frac{z}{p^{(2)}_{*}}+\frac{1-z}{2}=\frac{zt_{*}}{p_{*}}+\frac{1-zt_{*}}{2}=\frac{1}{p_{1}(zt_{*})}=\frac{1}{p(zt_{*})}\,.

(74)

Therefore, when $z$ belongs to $-1\leq\mathop{\rm Re}z\leq 1$ , $zt^{*}$ belongs to $-t_{*}\leq\mathop{\rm Re}z\leq t_{*}$ , $p_{2}(z)=p(zt_{*})$ and $p_{2}(z)$ still satisfies $\Omega_{[-1,1]}\subset\Omega_{\leq\epsilon_{0}(\gamma)}$ . In this domain, the estimate (73) can be rewritten as

\|(I-Q^{(2)}_{w,p_{2}(t+iy),T})^{-1}\|_{p_{2}(t),p_{2}(t)}\leqslant 2{\bf\Lambda},\quad t\in[-1,1],\quad y\in\mathbb{R}\,,

where $p_{2}(z)$ is different from $p_{1}(z)=p(z)$ only by the choice of parameter $p_{*}$ in (65) and is in fact a rescaling of the original $p(z)$ as follows from (74). From Proposition 9.6, we have

\|(I-Q^{(2)}_{w,p_{2}(t+iy),T})^{-1}\|_{p_{2}(t),p_{2}(t)}\leqslant{\bf\Lambda}

for $t\in[-t_{*},t_{*}],y\in\mathbb{R}$ . We use the perturbative bound (67) one more time with $H=-Q^{(2)}_{w,p_{2}(t+iy),T}$ and $V=-N^{-1}Q_{w,p_{2}(t+iy),T}$ to get

\|(I-Q^{(3)}_{w,p_{2}(t+iy),T})^{-1}\|_{p_{2}(t),p_{2}(t)}\leqslant 2{\bf\Lambda}

for $t\in[-t_{*},t_{*}],y\in\mathbb{R}$ .

$\bullet$ Induction in $j$ and the bound for $Q^{(N)}_{w,p(z),T}$ . Next, we take $p_{*}^{(3)}\stackrel{{\scriptstyle\rm def}}{{=}}p^{(2)}(t_{*})$ and repeat the process in which the bound

\|(I-Q^{(j)}_{w,p_{j}(t+iy),T})^{-1}\|_{p_{j}(t),p_{j}(t)}\leqslant 2{\bf\Lambda},\quad z=t+iy\in\Omega_{[-1,1]}\,,

implies

\|(I-Q^{(j+1)}_{w,p_{j+1}(t+iy),T})^{-1}\|_{p_{j+1}(t),p_{j+1}(t)}\leqslant 2{\bf\Lambda},\quad z=t+iy\in\Omega_{[-1,1]}\,.

Notice that each time the new $p_{j}(z)$ is in fact a rescaling of the original $p(z)$ by $t_{*}^{j-1}$ as can be seen from a calculation analogous to (74). In $N-1$ steps, we get

\displaystyle\|(I-Q^{(N)}_{w,p_{N-1}(t+iy),T})^{-1}\|_{p_{N-1}(t),p_{N-1}(t)}\leqslant 2{\bf\Lambda}\,,\quad z=t+iy\in\Omega_{[-t_{*},t_{*}]}\,.

Thus recalling that $p_{N-1}(z)=p(t_{*}^{N-2}z)$ , one has

\|(I-Q^{(N)}_{w,p(t_{*}^{N-1}z),T})^{-1}\|_{p(t_{*}^{N-1}t),p(t_{*}^{N-1}t)}\leqslant 2{\bf\Lambda}\,.

Since $Q^{(N)}_{w,p(t_{*}^{N}z)}=Q_{w,p(t_{*}^{N}z),T}$ , we get (66) with

\epsilon(\gamma,\mathbf{\Lambda},\cal{F})\stackrel{{\scriptstyle\rm def}}{{=}}t_{*}^{N-1}\epsilon_{0}(\gamma).

The estimates (72) implies that we can take $N\sim C_{\gamma,\cal{F},\mathbf{\Lambda}}\,.$

Thus, we showed there exists $\epsilon(\gamma,\cal{F},\mathbf{\Lambda})>0$ such that $\|(I-Q_{w,p,T})^{-1}\|_{p,p}\leq 2\mathbf{\Lambda}$ for all $|\frac{1}{p}-\frac{1}{2}|<\epsilon(\gamma,\cal{F},\mathbf{\Lambda})$ .

In fact, we may assume without loss of generality that $\lim\limits_{\gamma\to 1}\epsilon(\gamma,\cal{F},\mathbf{\Lambda})=\frac{1}{2}$ : indeed, it suffices to show that for any $\widetilde{\epsilon}\in(0,\frac{1}{2})$ , we can choose $\gamma-1$ sufficiently small so that $\|(I-Q_{w,p(z),T})^{-1}\|_{p(t),p(t)}\leq 2\mathbf{\Lambda}$ for all $z$ satisfying $|\frac{1}{p(t)}-\frac{1}{2}|<\widetilde{\epsilon}$ . Note that if $\|Q_{w,p(z),T}\|_{p(t),p(t)}\leq\frac{1}{2}$ , then by geometric sum

\|(I-Q_{w,p(z),T})^{-1}\|_{p(t),p(t)}\leq\sum\limits_{k=0}^{\infty}\|Q_{w,p(z)}\|_{p(t),p(t)}^{k}\leq 2\leq 2\mathbf{\Lambda}\,.

Thus it suffices to show we can take $\gamma$ sufficiently small so that

\|Q_{w,p(z),T}\|_{p(t),p(t)}\leq\frac{1}{2}\,

for $|\frac{1}{p(t)}-\frac{1}{2}|<\widetilde{\epsilon}$ . In fact, by Stein’s analytic interpolation it suffices to check this at the vertical lines such that $|\frac{1}{p(t)}-\frac{1}{2}|=\widetilde{\epsilon}$ . By duality and the triangle inequality this in turn will follow from showing

\|w^{1/p(z)}Tw^{-1/p(z)}-T\|_{p(t),p(t)}\leq\frac{1}{4}\,

for $z=t+iy$ such that $|\frac{1}{p(t)}-\frac{1}{2}|=\epsilon(\gamma)$ . Fix one the values $t$ satisfying the last equality: if $\gamma-1$ is sufficiently small, then by Lemma 8.6 we have $\|\log w\|_{\rm BMO}\lesssim\sqrt{\gamma-1}$ , which can be made arbitrarily small. Apply Theorem 9.3 with $f=\log w$ to get

\|w^{1/p}Tw^{-1/p}-T\|_{p,p}\lesssim_{p,\cal{F}}\sqrt{\gamma-1}\,.

We can now choose $\gamma$ sufficiently small that this is at most $\frac{1}{4}$ . Thus, we can choose $\epsilon(\gamma,\cal{F},\mathbf{\Lambda})$ so that as $\gamma\to 1$ , we get $\epsilon(\gamma,\cal{F},\mathbf{\Lambda})\to\frac{1}{2}$ .

As for showing $(I-Q_{w,p(z),T})^{-1}$ is analytic as an $\cal{L}(L^{2}(\mathbb{R}))$ -valued function, by Lemma 9.5 and Proposition A.1, it suffices to show $(I-Q_{w,p(z),T})^{-1}$ is bounded on $L^{2}(\mathbb{R})$ for $z\in\Omega_{\leq\epsilon(\gamma,\cal{F},\mathbf{\Lambda})}$ . Fix one such $z$ ; we just showed previously that

\|(I-Q_{w,p(z)})^{-1}\|_{p(t),p(t)}\leq 2\mathbf{\Lambda}\,.

However the proof just as easily applies to weight $w^{-1}$ , operator $-T$ and Hölder index $p^{\prime}(z)$ (as opposed to $p(z)$ ) and so we get

\|(I-Q_{w,p(z),T})^{-1}\|_{p^{\prime}(t),p^{\prime}(t)}=\|(I-Q_{w^{-1},p^{\prime}(z),-T})^{-1}\|_{p^{\prime}(t),p^{\prime}(t)}\leq 2\mathbf{\Lambda}\,\quad\text{for }z\in\Omega_{<\epsilon(\gamma,\cal{F},\mathbf{\Lambda})}.

Interpolate between both estimates to get

\|(I-Q_{w,p(z),T})^{-1}\|_{2,2}\leq 2\mathbf{\Lambda}\,.

This completes the proof. ∎

Appendix A A detour through complex analysis: proof of Proposition 5.5

We recall the following well-known lemma, which we prove for the sake of completeness.

Proposition A.1.

Let $B$ be a Banach space, let $\cal{D}\subset B^{*}$ be a dense subset of the dual space and $U\subset\mathbb{C}$ an open set. If $a:U\to B$ is weakly analytic with respect to $\cal{D}$ , then $a:U\to B$ is analytic.

Proof.

We adapt the proof of [6, Theorem 8.20].

Let $\ell\in\cal{D}$ so that $\ell\circ a(z)$ is analytic.

Fix $\zeta\in U$ and let $|h|<\epsilon$ for $\epsilon$ sufficiently small. By the Cauchy integral formula,

\ell\circ a(\zeta+h)=\frac{1}{2\pi i}\oint\limits_{|z-\zeta|=\epsilon}\frac{\ell\circ a(z)}{z-(\zeta+h)}\,dz\,.

Then

\ell\circ a(\zeta+h)-\ell\circ a(\zeta)=\frac{h}{2\pi i}\oint\limits_{|z-\zeta|=\epsilon}\frac{\ell\circ a(z)}{(z-(\zeta+h))(z-\zeta)}\,dz\,.

(75)

For $h\neq h^{\prime}$ with $0<|h|,|h^{\prime}|<\epsilon$ , define the second order difference quotient

x(h,h^{\prime})\stackrel{{\scriptstyle\rm def}}{{=}}\frac{1}{h-h^{\prime}}\left(\frac{a(\zeta+h)-a(\zeta)}{h}-\frac{a(\zeta+h^{\prime})-a(\zeta)}{h^{\prime}}\right)\,.

It suffices to show $x(h,h^{\prime})$ is uniformly bounded in $h,h^{\prime}$ , for both sufficiently small. Indeed, uniform boundedness in $B$ would then yield

h\mapsto\frac{a(\zeta+h)-a(\zeta)}{h}

is Lipschitz for $h$ near $0$ , and so has a limit as $h\to 0$ .

By (75),

\displaystyle\ell(x(h,h^{\prime}))

\displaystyle=\frac{1}{2\pi i}\oint\limits_{|z-\zeta|=\epsilon}\frac{\ell\circ a(z)}{(z-\zeta)(z-(\zeta+h))(z-(\zeta+h^{\prime}))}\,dz\,.

For $|h|,|h^{\prime}|<\epsilon/2$ , the denominator in the integral is bounded uniformly away from $0$ , and the numerator is bounded above by $\|\ell\|_{B^{*}}\sup\limits_{|z-\zeta|=\epsilon}\|a(z)\|_{B}$ , which is finite by the uniform boundedness principle. Whence

\ell(x(h,h^{\prime}))\leq M\|\ell\|_{B^{*}}

where $M$ does not depend on $\ell$ . By duality

\|x(h,h^{\prime})\|_{B}\leq M\,.

∎

Fix $\chi$ a characteristic function of some finite interval $I$ , and let $X_{p}\stackrel{{\scriptstyle\rm def}}{{=}}w^{1/p}(P(r,\lambda;w)-\mathrm{exp}\left(i\lambda r\right))$ , like in Section 5.

Proposition A.2.

Suppose $[w]_{A_{2}(\mathbb{R})}\leq\gamma$ , $p_{*}\in[1,\infty]$ and $p(z)$ is as in (65). Then $z\mapsto\chi w^{1/p(z)-1/2}$ is analytic as a map

\Omega_{(-1,1)}\to L^{2}(\mathbb{R})\,.

Proof.

Without loss of generality, assume $p_{*}=\infty$ ; the general case follows by a rescaling argument.

We first compute

\|\chi w^{1/p(z)-1/2}\|_{L^{q}(\mathbb{R})}^{q}=\int\limits_{I}w^{-qt/2}\,d\lambda\,.

Since $[w]_{A_{2}(\mathbb{R})},\,[w^{-1}]_{A_{2}(\mathbb{R})}\leq\gamma$ , then by Lemma 5.1 there exists $q(\gamma)>2$ such that for all $|t|<1$ , $w^{-qt/2}\in A_{2}(\mathbb{R})$ . Since $A_{2}(\mathbb{R})\subset L^{1}_{loc}(\mathbb{R})$ , then $\int\limits_{I}w^{-qt/2}\,d\lambda<\infty$ and so $\chi w^{1/p(z)-1/2}\in L^{q}(\mathbb{R})$ for some $q>2$ by (65).

Now let us show analyticity: write

	$\displaystyle\chi w^{1/p(z)-1/2}$	$\displaystyle=(\chi w^{1/p(z)-1/p(z_{0})})(\chi w^{1/p(z_{0})-1/2})$
		$\displaystyle=\sum\limits_{k=0}^{\infty}\frac{(-1)^{k}}{k!\,2^{k}}(\chi\log w)^{k}(z-z_{0})^{k}(\chi w^{1/p(z_{0})-1/2})\,$

by Taylor expansion of $w^{1/p(z)-1/p(z_{0})}=\mathrm{exp}\left(-\frac{(z-z_{0})}{2}\log w\right)$ . We simply need to check the series converges uniformly in $L^{2}(\mathbb{R})$ norm for $|z-z_{0}|$ sufficiently small; by Stirling’s approximation, it suffices to show

\|(\chi\log w)^{k}(\chi w^{1/p(z_{0})-1/2})\|_{L^{2}(\mathbb{R})}\lesssim_{\log w,I,\gamma}C(\log w,I,\gamma)^{k}k^{k}\,.

Apply Hölder’s inequality to bound

\|(\chi\log w)^{k}(\chi w^{1/p(z_{0})-1/2})\|_{L^{2}(\mathbb{R})}\lesssim\|\chi w^{1/p(z_{0})-1/2}\|_{L^{q}(\mathbb{R})}\|(\chi\log w)^{k}\|_{L^{2(q/2)^{\prime}}(\mathbb{R})}^{1/2-1/q}\,,

where $(q/2)^{\prime}$ is the dual exponent to $q/2$ .

Since we already showed $\chi w^{1/p(z)-1/2}\in L^{q}(\mathbb{R})$ , then we will be done once we show $\|(\chi\log w)^{k}\|_{L^{2(q/2)^{\prime}}(\mathbb{R})}\lesssim_{\log w,I,\gamma}C(\log w,I,\gamma)^{k}k^{k}$ . If $p=2(q/2)^{\prime}$ , write

\|(\chi\log w)^{k}\|_{L^{p}(\mathbb{R})}=\|(\log w)^{k}\|_{L^{p}(I)}=\|\log w\|_{L^{pk}(I)}^{k}\,.

It suffices to show $\|\log w\|_{L^{pk}(I)}\lesssim_{\log w,I,\gamma}k$ .

Write

\|\log w\|_{L^{pk}(I)}\leq\|\log w-\langle\log w\rangle_{I}\|_{L^{pk}(I)}+\langle\log w\rangle_{I}|I|^{1/(pk)}

The last term is $\lesssim_{\log w,I,p}k$ . For the first term, the John-Nirenberg Theorem 8.5 yields

\|\log w-\langle\log w\rangle_{I}\|_{L^{pk}(I)}\leq(c|I|pk\|\log w\|_{\rm BMO})^{1/(pk)}\Gamma(pk)^{1/(pk)}\lesssim_{\log w,I,\gamma}\Gamma(pk)^{1/(pk)}

where $\Gamma$ function is the usual analytic extension of factorial. Finally, Stirling’s formula yields

\|\log w-\langle\log w\rangle_{I}\|_{L^{pk}(I)}\lesssim_{\log w,I,\gamma}\Gamma(pk)^{1/pk}\lesssim pk\lesssim_{p}k\,,

thereby completing the proof. ∎

Remark. To avoid repeating similar arguments, from now onwards if we write that a function or operator is analytic (or weakly analytic) thanks to a “John-Nirenberg argument,” the reader should understand this as meaning that analyticity follows from an argument similar to the one above where we showed $\chi w^{1/p(z)-1/2}$ was analytic as a map $\{|\mathop{\rm Re}z|<1\}\to L^{2}(\mathbb{R})$ .

Lemma A.3.

Suppose $[w]_{A_{2}(\mathbb{R})}\leq\gamma$ , $p_{*}\in[1,\infty]$ and $p(z)$ is as in (65). Then there exists $\epsilon(\gamma)\in(0,\frac{1}{2})$ , with $\lim\limits_{\gamma\to 1}\epsilon(\gamma)=\frac{1}{2}$ , such that $z\mapsto w^{1/p(z)}-w^{-1/p^{\prime}(z)}$ is analytic as a map $\Omega_{<\epsilon(\gamma)}\to L^{2}(\mathbb{R})$ .

Proof.

We first note that $w^{1/p(z)}-w^{-1/p^{\prime}(z)}\in L^{2}(\mathbb{R})$ for $z\in\Omega_{<\epsilon}$ for some $\epsilon>0$ . Indeed,

\left|w^{1/p(z)}-w^{-1/p^{\prime}(z)}\right|=\left|w^{-1/p^{\prime}(z)}\right|\left|w-1\right|=w^{-1/p(t)}\left|w-1\right|=\left|w^{1/p(t)}-w^{-1/p^{\prime}(t)}\right|\,.

By Lemma 5.3, this belongs to $L^{2}(\mathbb{R})$ so long as $z\in\Omega_{<\epsilon(\gamma)}$ , where $\epsilon(\gamma)\in(0,\frac{1}{2})$ with $\lim\limits_{\gamma\to 1}\epsilon(\gamma)=\frac{1}{2}$ .

By a John-Nirenberg argument, $w^{1/p(z)}$ and $w^{-1/p^{\prime}(z)}$ are weakly analytic with respect to $\cal{S}_{c}(\mathbb{R})$ on $\Omega_{<\epsilon(\gamma)}$ . By Proposition A.1, this means $w^{1/p(z)}-w^{-1/p^{\prime}(z)}$ is analytic as a map $\Omega_{<\epsilon(\gamma)}\to L^{2}(\mathbb{R})$ . ∎

Proposition A.4.

Suppose $[w]_{A_{2}(\mathbb{R})}\leq\gamma$ , $T$ is an operator satisfying (28) for some $\cal{F}$ , and $p(z)$ is as in (65) for some $p_{*}\in[1,\infty]$ . Then there exists $\epsilon(\gamma,\cal{F})\in(0,\frac{1}{2}]$ , with $\lim\limits_{\gamma\to 1}\epsilon(\gamma,\cal{F})=\frac{1}{2}$ , such that $w^{1/p(z)}Tw^{-1/p(z)}$ and $w^{-1/p^{\prime}(z)}Tw^{1/p^{\prime}(z)}$ are analytic as maps $\Omega_{<\epsilon(\gamma,\cal{F})}\to\cal{L}(L^{2}(\mathbb{R}))$ .

Proof.

We will prove the portion of the proposition involving $w^{1/p(z)}Tw^{-1/p(z)}$ ; the result for $w^{-1/p^{\prime}(z)}Tw^{1/p^{\prime}(z)}$ follows by then replacing $w$ by $w^{-1}$ and $p_{*}$ by $p_{*}^{\prime}$ .

A John-Nirenberg argument shows that $w^{1/p(z)}Tw^{-1/p(z)}$ is weakly analytic with respect to $\{\langle\cdot f,g\rangle\}_{f,g\,\in\cal{S}_{c}(\mathbb{R})}$ for all $z=t+iy$ where $w\in A_{p(t)}$ . By Proposition A.1, $w^{1/p(z)}Tw^{-1/p(z)}$ is then analytic as a map $\Omega_{<\epsilon}\to\cal{L}(L^{2}(\mathbb{R}))$ if we can show $w^{1/p(z)}Tw^{-1/p(z)}\in\cal{L}(L^{2}(\mathbb{R}))$ on $\Omega_{<\epsilon}$ for some $\epsilon>0$ . To see this latter part, notice that

\|w^{1/p(z)}Tw^{-1/p(z)}\|_{2,2}=\|w^{1/p(t)}Tw^{-1/p(t)}\|_{2,2}=\|(w^{2/p(t)})^{1/2}T(w^{2/p(t)})^{-1/2}\|_{2,2}\,.

By Lemma 5.1, there exists $\epsilon(\gamma,\cal{F})$ such that when $t+0i\in\Omega_{<\epsilon(\gamma,\cal{F})}$ , we have $w^{2/p(t)}\in A_{2}(\mathbb{R})$ . By (28), the operator is then indeed bounded on $L^{2}(\mathbb{R})$ . ∎

The above Proposition A.4, combined with Lemma 4.3, yield the following.

Corollary A.5.

Suppose $[w]_{A_{2}(\mathbb{R})}\leq\gamma$ and $p(z)$ is as in (65) for some $p_{*}\in[1,\infty]$ . Then there exists $\epsilon(\gamma)\in(0,\frac{1}{2})$ , with $\lim\limits_{\gamma\to 1}\epsilon(\gamma)=\frac{1}{2}$ , such that $w^{1/p(z)}\mathcal{P}_{[{0},{r}]}w^{-1/p(z)},w^{-1/p^{\prime}(z)}\mathcal{P}_{[{0},{r}]}w^{1/p^{\prime}(z)}$ are analytic as maps $\Omega_{<\epsilon(\gamma)}\to\cal{L}(L^{2}(\mathbb{R}))$ .

Remark. In what follows, we will need to show some $\epsilon(\gamma)\in(0,\frac{1}{2})$ exists, such that $\lim\limits_{\gamma\to 1}\epsilon(\gamma)=\frac{1}{2}$ . In our reasoning, we will consider various other such functions $\epsilon(\gamma)$ . We consider only finitely many such functions, so by taking the minimum of all of them we will assume without loss of generality that all the $\epsilon(\gamma)$ ’s are the same.

Corollary A.6.

Suppose $[w]_{A_{2}(\mathbb{R})}\leq\gamma$ and $p(z)$ is as in (65) for some $p_{*}\in[1,\infty]$ . Then there exists $\epsilon(\gamma)\in(0,\frac{1}{2})$ with $\lim\limits_{\gamma\to 1}\epsilon(\gamma)=\frac{1}{2}$ such that

\chi(I-Q_{w,p(z)})^{-1}w^{-1/p^{\prime}(z)}\mathcal{P}_{[0,r]}w^{1/p^{\prime}(z)}(w^{1/p(z)}-w^{-1/p^{\prime}(z)})\mathrm{exp}\left(i\lambda r\right)

is analytic as a map $\Omega_{<\epsilon(\gamma)}\to L^{2}(\mathbb{R})$ .

Proof.

By Corollary A.5, Lemma A.3 and Lemma 9.1 with $T=\mathcal{P}_{[{0},{r}]}$ and e.g. $\mathbf{\Lambda}=10$ , we can choose $\epsilon(\gamma)\in(0,\frac{1}{2})$ , with $\lim\limits_{\gamma\to 1}\epsilon(\gamma)=\frac{1}{2}$ , such that $w^{-1/p^{\prime}(z)}\mathcal{P}_{[0,r]}w^{1/p^{\prime}(z)}$ , $w^{1/p(z)}-w^{-1/p^{\prime}(z)}$ and $(I-Q_{w,p(z)})^{-1}$ are analytic as elements or operators on $L^{2}(\mathbb{R})$ , for $\Omega_{<\epsilon(\gamma)}$ . A difference quotient computation then yields the Corollary statement. ∎

Proof of Proposition 5.5.

We begin with (36). By applying Lemma 4.3 and Lemma 9.1 with $T=\mathcal{P}_{[{0},{r}]}$ and e.g. $\ \mathbf{\Lambda}=10$ , we can invert $I-Q_{w,p}$ to get (41) for all $p\in[2,\infty)$ satisfying $|\frac{1}{2}-\frac{1}{p}|<\epsilon(\gamma)$ , where $\epsilon(\gamma)\in(0,\frac{1}{2})$ and $\lim\limits_{\gamma\to 1}\epsilon(\gamma)=\frac{1}{2}$ . So for all $p\geq 2$ satisfying $|\frac{1}{2}-\frac{1}{p}|<\epsilon(\gamma)$ , we have

\langle\chi X_{p},f\rangle=-\langle\chi(I-Q_{w,p})^{-1}w^{-1/p^{\prime}}\mathcal{P}_{[0,r]}w^{1/p^{\prime}}(w^{1/p}-w^{-1/p^{\prime}})\mathrm{exp}\left(i\lambda r\right),f\rangle

(76)

for all $f\in\cal{S}_{c}(\mathbb{R})$ .

Now replace $p$ by $p(z)$ with $p_{*}=\infty$ . Then by Corollary A.6, we have

\langle\chi(I-Q_{w,p(z)})^{-1}w^{-1/p^{\prime}(z)}\mathcal{P}_{[0,r]}w^{1/p^{\prime}(z)}(w^{1/p(z)}-w^{-1/p^{\prime}(z)})\mathrm{exp}\left(i\lambda r\right),f\rangle

is analytic on $\Omega_{<\epsilon(\gamma)}$ . Recall $X_{2}\in L^{2}(\mathbb{R})$ by Proposition 3.1. Thus, if we write $\langle\chi X_{p(z)},f\rangle=\langle\chi w^{1/p(z)-1/2}X_{2},f\rangle$ , then by Proposition A.2 and that $X_{2}\in L^{2}(\mathbb{R})$ , we may assume $\langle\chi X_{p(z)},f\rangle$ is analytic on the same region.

Thus (76) shows two analytic functions are equals when $z$ is in the interval $\Omega_{<\epsilon(\gamma)}\cap\mathbb{R}$ . Thus we must conclude that they are in fact equal on the strip $\Omega_{<\epsilon(\gamma)}$ , i.e.

\langle\chi X_{p(z)},f\rangle=-\langle\chi(I-Q_{w,p(z)})^{-1}w^{-1/p^{\prime}(z)}\mathcal{P}_{[0,r]}w^{1/p^{\prime}(z)}(w^{1/p(z)}-w^{-1/p^{\prime}(z)})\mathrm{exp}\left(i\lambda r\right),f\rangle

on $\Omega_{<\epsilon(\gamma)}$ for all $f\in\cal{S}_{c}(\mathbb{R})$ . Since $\chi$ is the indicator of an arbitrary finite interval, then by duality and density of $\cal{S}_{c}(\mathbb{R})$ in $L^{p}(\mathbb{R})$ ,

X_{p(z)}=-(I-Q_{w,p(z)})^{-1}w^{-1/p^{\prime}(z)}\mathcal{P}_{[0,r]}w^{1/p^{\prime}(z)}(w^{1/p(z)}-w^{-1/p^{\prime}(z)})\mathrm{exp}\left(i\lambda r\right)\,

for $z\in\Omega_{<\epsilon(\gamma)}$ . Taking $z=t$ yields (41) for $|\frac{1}{p}-\frac{1}{2}|<\epsilon(\gamma)$ . ∎

Remark. Both sides of (41) always makes sense as elements of $L^{2}(\mathbb{R})$ for $|\frac{1}{p}-\frac{1}{2}|<\epsilon(\gamma)$ .

References

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	$\displaystyle\\|\mathcal{H}_{r}f\\|_{L^{2}(\mathbb{R})}=\\|\int\limits_{0}^{r}\mathcal{F}^{-1}(w(2\pi\cdot)-1)(x-y)f(y)dy\\|_{L^{2}(\mathbb{R})}$	$\displaystyle=\\|(w(2\pi\cdot)-1)\mathcal{F}(f\chi_{[0,r]})\\|_{L^{2}(\mathbb{R})}$
		$\displaystyle\leq\\|w-1\\|_{L^{\infty}(\mathbb{R})}\\|\mathcal{F}(f\chi_{[0,r]})\\|_{L^{2}(\mathbb{R})}$
		$\displaystyle\leq\delta\\|f\\|_{L^{2}([0,r])}\,.$

	$\displaystyle\left\|\left(\frac{d}{du}\right)^{j}(-\mathcal{H}_{r})^{l}H(u)\right\|$	$\displaystyle\leq\\|H^{(j)}\\|_{L^{2}(\mathbb{R})}\\|\mathcal{H}_{r}^{l-1}H\\|_{L^{2}([0,r])}$
		$\displaystyle=\\|H^{(j)}\\|_{L^{2}([0,r])}\\|\mathcal{H}_{r}\\|_{L^{2}([0,r]),L^{2}(\mathbb{R})}^{l-1}\\|H\\|_{L^{2}([0,r])}$
		$\displaystyle\lesssim L\delta^{l}\,,$

The Steklov problem and Remainder Estimates for Krein Systems generated by a Muckenhoupt weight

Abstract.

1. Introduction

Problem 1.1 (The Steklov problem for Krein systems).

Theorem 1.2.

Corollary 1.3.

Proof.

Corollary 1.4.

Proof.

Proposition 1.5.

Theorem 1.6.

Theorem 1.7.

Corollary 1.8.

1.1. Lingering questions: some potentially open problems

1.2. Organization of this Paper

1.3. Notation

2. Krein Systems Basics

Lemma 2.1.

Proof.

3. A priori estimates: when does P​(r,λ)−exp​(i​λ​r)∈Lp​(ℝ)P(r,\lambda)-\mathrm{exp}\left(i\lambda r\right)\in L^{p}(\mathbb{R})?

Proposition 3.1.

Lemma 3.2.

Proof.

Lemma 3.3.

Proof.

Proof of Proposition 3.1.

4. Krein system solutions are orthogonal to lower Fourier frequencies

Lemma 4.1.

Proof.

Lemma 4.2.

Proof.

Lemma 4.3.

Proof.

5. The Steklov problem for an A2​(ℝ)A_{2}(\mathbb{R}) weight: proof of Theorem 1.2 and Proposition 1.5

Lemma 5.1 (Reverse Hölder inequality, open inclusion of ApA_{p} weights).

Lemma 5.2.

Lemma 5.3 (Integrability of ApA_{p} weights).

Proof of Lemma 5.2.

Lemma 5.4.

Proposition 5.5.

Proof of Theorem 1.2.

Proof of Proposition 1.5.

6. A mixed norm remainder estimate

Lemma 6.1.

Proof.

Lemma 6.2.

Proof.

Lemma 6.3.

Proof.

Proof of Theorem 1.6.

7. Higher order remainder estimates: proof of Theorem 1.7

Lemma 7.1.

Proof.

Lemma 7.2.

Proof.

Lemma 7.3.

Proof.

Proof of Theorem 1.7.

8. Proof of Lemma 5.3

Lemma 8.1 (Calderón-Zygmund decomposition).

Proof.

Lemma 8.2 (Calderón-Zygmund decomposition for Lp1​(ℝ)+Lp2​(ℝ)L^{p_{1}}(\mathbb{R})+L^{p_{2}}(\mathbb{R})).

Proof.

Lemma 8.3 (μ\mu is flat away from 0).

Proof.

Proof that ‖w1/p~−w−1/p~′‖Lμp​(ℝ)<∞\|w^{1/\widetilde{p}}-w^{-1/\widetilde{p}^{\prime}}\|_{L^{p}_{\mu}(\mathbb{R})}<\infty in Lemma 5.3.

Lemma 8.4.

Proof.

Theorem 8.5 (John-Nirenberg, [13, p.144-146]).

Lemma 8.6.

Proof of (38).

9. Inverting I−Qw,pI-Q_{w,p}: proof of Lemma 5.4

Lemma 9.1.

Proposition 9.2.

Theorem 9.3 ([1, Theorem 1.2]).

Lemma 9.4.

Proof.

Lemma 9.5.

Proof.

Proposition 9.6 ([1]).

3. A priori estimates: when does $P(r,\lambda)-\mathrm{exp}\left(i\lambda r\right)\in L^{p}(\mathbb{R})$ ?

5. The Steklov problem for an $A_{2}(\mathbb{R})$ weight: proof of Theorem 1.2 and Proposition 1.5

Lemma 5.1 (Reverse Hölder inequality, open inclusion of $A_{p}$ weights).

Lemma 5.3 (Integrability of $A_{p}$ weights).

Lemma 8.2 (Calderón-Zygmund decomposition for $L^{p_{1}}(\mathbb{R})+L^{p_{2}}(\mathbb{R})$ ).

Lemma 8.3 ( $\mu$ is flat away from $0$ ).

Proof that $\|w^{1/\widetilde{p}}-w^{-1/\widetilde{p}^{\prime}}\|_{L^{p}_{\mu}(\mathbb{R})}<\infty$ in Lemma 5.3.

9. Inverting $I-Q_{w,p}$ : proof of Lemma 5.4