Necessity of a logarithmic estimate for hypoellipticity of some degenerately elliptic operators

Timur Akhunov Department of Mathematics and Computer Science
Wabash College
301 W Wabash Ave, Crawfordsville, IN 47933, USA and Lyudmila Korobenko Mathematics Department
Reed College
3203 Southeast Woodstock Boulevard
Portland, OR 97202-8199, USA

Abstract.

This paper extends a class of degenerate elliptic operators for which hypoellipticity requires more than a logarithmic gain of derivatives of a solution in every direction. Work of Hoshiro and Morimoto in late 80s characterized a necessity of a super-logarithmic gain of derivatives for hypoellipticity of a sum of a degenerate operator and some non-degenerate operators like Laplacian. The operators we consider are similar, but more general. We examine operators of the form $L_{1}(x)+g(x)L_{2}(y)$ , where $L_{1}(x)$ is one-dimensional and $g(x)$ may itself vanish. The argument of the paper is based on spectral projections, analysis of a spectral differential equation, and interpolation between standard and operator-adapted derivatives. Unlike prior results in the literature, our methods do not require explicit analytic construction in the non-degenerate direction. In fact, our result allows non-analytic and even non-smooth coefficients for the non-degenerate part.

Key words and phrases:

hypoellipticity, infinite vanishing, loss of derivatives

1991 Mathematics Subject Classification:

35H10, 35H20, 35S05, 35G05, 35B65, 35A18

The authors thank Cristian Rios, who introduced us to the problem of hypoellipticity and worked with us on [AKR19] that motivated results of this paper. We also thank the anonymous referee for numerous insightful suggestions

1. Introduction

Consider a degenerate elliptic operator

(1)

\displaystyle L(x,y)=L_{1}(x)+g(x)L_{2}(y)

where

\displaystyle L_{1}=-\sum_{j,k=1}^{n}a_{jk}(x)\partial_{x_{j}}\partial_{x_{k}}+\sum_{j=1}^{n}a_{j}(x)\partial_{x_{j}}+a_{0}(x)

with $x\in\operatorname{\mathbb{R}}^{n}$ , $a_{jk}$ , $a_{j}$ smooth real valued coefficients with $a_{jk}$ a non-negative matrix. Similarly, let $L_{2}$ be denoted as

(2)

\displaystyle L_{2}=-\sum_{j,k=1}^{m}b_{jk}(y)\partial_{y_{j}}\partial_{y_{k}}+\sum_{j=1}^{m}b_{j}(y)\partial_{y_{j}}+b_{0}(y)

where $y\in\operatorname{\mathbb{R}}^{m}$ , $b_{jk}$ , $b_{j}$ are smooth real functions and $b_{jk}$ is a non-negative matrix:

(3)

\displaystyle\sum_{j,k=1}^{m}b_{j,k}(y)\eta_{j}\eta_{k}\geq 0

In this paper we investigate conditions under which local hypoellipticity of $L$ , or smoothness of solutions of $Lu=f$ whenever $f$ is locally smooth (see Definition 2), requires the following superlogarithmic estimate introduced by Morimoto [Mor87]:

Definition 1 (Superlogarithmic estimate).

We say that an operator $L$ satisfies a superlogarithmic estimate near $y_{0}\in R^{m}$ if for any small enough compact set $K\Subset R^{m}$ containing $y_{0}$ and for all $\varepsilon>0$ , there exists a family of constants $C_{\varepsilon,K}$ , such that the following estimate holds.

(4)

||\log\!\left<\xi\right>\hat{u}(\xi)||^{2}\leq\varepsilon\,\mathrm{Re}(Lu,u)+C_{\varepsilon,K}||u||^{2},\quad u\in C_{0}^{\infty}(K).

where

(5)

\displaystyle\!\left<\xi\right>:=\sqrt{e^{2}+|\xi|^{2}}

This estimate is satisfied by certain operators $L_{2}$ with degeneracy in (3) so that the vanishing is faster than any polynomial in $y$ in some directions. For example,

(6)

\displaystyle L_{2}=\partial_{y_{1}}^{2}+e^{-\frac{1}{|y_{1}|^{1-\kappa}}}\partial_{y_{2}}^{2}

considered by [KS84] satisfies (4) for $\kappa>0$ , see [Chr01] Lemma 5.2. The Hörmander bracket condition for $L_{2}$ corresponds to a polynomial degeneracy for $L_{2}$ and leads to subelliptic estimate, defined as

(7)

\displaystyle||\!\left<\xi\right>^{\delta}\hat{u}(\xi)||^{2}\leq C(\delta,K)\left(\mathrm{Re}(L_{2}u,u)+||u||^{2}\right),\text{ for }\quad u\in C_{0}^{\infty}(K).

for some $\delta>0$ and $\!\left<\xi\right>$ defined in (5) as before. Note that (7) implies (4). Superlogarithmic estimate (4) limits the degree of vanishing that an operator $L_{2}$ can exhibit. If a stronger vanishing is considered, e.g. (6) for $\kappa\leq 0$ , then (4) fails and a weaker gain function should replace the logarithm in (4). See [Chr01] for a further discussion of degeneracy and gain function.

This is a companion paper to [AKR19], where we proved hypoellipticity of $L$ from (1) under the assumption that operators $L_{1}$ and $L_{2}$ satisfy Definition 1, see [AKR19, Theorem 1] and [Mor87]. It was also demonstrated that, at least for particular classes of operators, the superlogarithmic estimate (4) for $L_{1}$ is necessary, see [AKR19, Example 1]. Here we address in greater generality the question of necessity of these superlogarithmic estimates.

We first set up relevant notation. In what follows, for any subsets $E$ , $F$ in $\operatorname{\mathbb{R}}^{n}$ we write $E\Subset F$ to indicate that the closure of $E$ , $\overline{E}$ , is compact and $\overline{E}\subset F$ .

For completeness, we first define local smooth hypoellipticity here.

Definition 2.

The operator $L(x,y)$ is locally ( $C^{\infty}$ ) hypoelliptic at $(x_{0},y_{0})$ if there exists a neighborhood $\Omega$ of $(x_{0},y_{0})$ , such that for every $f(x,y)$ smooth in $\Omega$ and a distributional solution $u$ satisfying $Lu=f$ in the weak sense in $\Omega$ , then $u\in C^{\infty}(\Omega^{\prime})$ for all neighbohoods $\Omega^{\prime}$ of $(x_{0},y_{0})$ such that $\Omega^{\prime}\Subset\Omega$ .

In particular, the region of smoothness of a weak solution depends on the operator $L$ and the region $\Omega$ , but not on the specific function $f$ , so long as $f$ is smooth. Our method allows us to work with a limited regularity form of hypoellipticity as well. We model this form of hypoellipticity on the Definition 2.

Definition 3.

We say that $L$ is (locally) $H^{s}$ -hypoelliptic at $(x_{0},y_{0})$ , if there exists a neighborhood $\Omega$ of $(x_{0},y_{0})$ , such that for every $f\in H^{s}(\Omega)$ and $u\in L^{2}(\Omega)$ satisfying $Lu=f$ in the weak sense in $\Omega$ , then $u\in H^{s}(\Omega^{\prime})$ for all neighbohoods $\Omega^{\prime}$ of $(x_{0},y_{0})$ such that $\Omega^{\prime}\Subset\Omega$ .

Morimoto in [Mor87], motivated by the results in [KS84], provided a non-probabilistic proof that in order for a symmetric operator $L(x,y)=-\partial_{x}^{2}+L_{2}(y)$ , $x\in\operatorname{\mathbb{R}},\ y\in\operatorname{\mathbb{R}}^{m}$ to be hypoelliptic, $L_{2}$ has to satisfy Definition 1. Independently at the same time, Hoshiro [Hos87] obtained a similar result as a corollary of an a priori estimate for a general self-adjoint partial differential operator. Our main theorem is a generalization of the Morimoto’s and Hoshiro’s necessity estimates to the operator of the form $L=L_{1}(x)+g(x)L_{2}(y)$ , where the function $g$ is allowed to vanish, allowing the operator $L$ itself to be more degenerate than $L_{2}$ and hence not satisfy (4) at some $(x_{0},y_{0})$ .

Theorem 4.

Consider a possibly degenerate elliptic operator

(8)

\displaystyle L=-a_{2}(x)\partial_{x}^{2}+a_{1}(x)\partial_{x}+a_{0}(x)+g(x)L_{2}(y,\partial_{y}),

where $L_{2}$ , defined in (2), is a self-adjoint operator, i.e.

(9)

\displaystyle(L_{2}u,v)=(u,L_{2}v)\text{, for }u\in C^{\infty}_{0}

Assume further $a_{2}(x_{0})>0$ , $g(x)\geq 0$ and the coefficients $g$ and $a_{i}$ for $i=0,1,2$ are smooth.

Suppose that $L$ from (8) is $H^{s}$ -hypoelliptic near $(x_{0},y_{0})$ for $s>\frac{1}{2}$ . Then $L_{2}$ must satisfy (4) near $y_{0}$ .

Remark 5.

Proof of Theorem 4 for $\frac{1}{2}<s\leq 1$ is valid with $a_{2}(x)\in C^{1}_{x}$ and other coefficients merely continuous.

Corollary 6.

Suppose $L$ from (8) satisfies all of the assumptions of Theorem 4, with the exception of $H^{s}$ hypoellipticity. We replace that assumption with the $C^{\infty}$ hypoellipticity of $L$ at $(x_{0},y_{0})$ . Then $L_{2}$ satisfies (4) near $y_{0}$ . Note that to establish $C^{\infty}$ hypoellipticity, more smoothness of the coefficients than continuity in Theorem 4 may be needed.

Remark 7.

Note, that the focus in Theorem 4 is on the implications of hypoellipticity. In particular, if $g(x)\equiv 0$ , the operator $L$ fails to be hypoelliptic and the Theorem 4 does not apply.

When $g(x)$ has an isolated zero at $x_{0}$ , our earlier work with Cristian Rios in [AKR19] proved $C^{\infty}$ hypoellipticity of $L$ if it satisfies (4) extending the non-degenerate case of $g(x)$ studied by Morimoto in [Mor87]. Thus, in the context of an isolated zero (or non-vanishing) $g$ , Theorem 4 is sharp for $C^{\infty}$ hypoellipticity of (8).

Finally, [MM97a] proved hypoellipticity of $L$ from (8), when $L_{2}=\partial_{y}^{2}$ assuming only that the stochastic average

\displaystyle g_{I}:=\int_{I}g(x)dx>0\text{, for any interval }I\subset\operatorname{\mathbb{R}},

including the case of non-isolated zeros. Morimoto and Morioka further connected quantitative estimates for $g_{I}$ with the logarithmic estimate (4) for some operators in [MM97b]. Thus Theorem 4 is likely non-trivial in some cases of $g(x)$ with non-isolated zeros.

The strength of Theorem 4 lies both in the fact that we replace $-\partial^{2}_{x}$ by a general one-dimensional elliptic operator, and, more importantly, allow for broader conditions on the degeneracy of the non-negative weight function $g(x)$ in (8). This is in line with the results by Fediĭ [Fed71], who considered $L=\partial^{2}_{x}+g(x)\partial^{2}_{y}$ , and Kohn [Koh98] who generalized Fediĭ’s result to $L=L_{1}(x)+g(x)L_{2}(y)$ where $L_{1}$ and $L_{2}$ are subelliptic in their variables. Note that the subelliptic estimate (7) is stronger that (4), so Theorem 4 together with [AKR19, Theorem 1] characterizes hypoellipticity in the case of $g(x)$ with isolated vanishing. Furthermore, Theorem 4 extends a necessary condition for hypoellipticity to a wide class of operators, many of them not previously considered in the literature.

Theorem 4 can be generalized in several ways. First, one can ask if the result still holds for a more general operator $L_{1}(x)$ . We believe that the ultimate goal would be the following conjecture.

Conjecture 8.

Suppose that the operator $L$ from (1) is symmetric, i.e. satisfies (9) and $g(x)\geq 0$ . The operator $L$ is locally $C^{\infty}$ -hypoelliptic, if and only if both $L_{1}$ and $L_{2}$ satisfy (4).

As discussed in Remark 7, results in [AKR19] imply the sufficiency part of this conjecture in the case of $g(x)$ that has at most isolated zeros. Theorem 4 confirms the necessity direction in the case that $L_{1}$ is one dimensional and elliptic.

The second direction to generalize Theorem 4 is to allow for a more general dependence on the variables, ultimately considering operators of the form $L_{1}(x,y)+L_{2}(x,y)$ with either $L_{1}$ or $L_{2}$ being degenerate elliptic. It is a nontrivial question, what happens to hypoellipticity when two (or more) operators are added. For example, Fediĭ’s operator $\partial^{2}_{x}+g(x)\partial^{2}_{y}$ is hypoelliptic whenever $g$ is positive away from zero [Fed71], while a very similar three-dimensional operator $\partial^{2}_{x}+\partial^{2}_{y}+g(x)\partial^{2}_{z}$ considered by Kusuoka and Strook [KS84] is hypoelliptic if and only if $g$ vanishes slower than $e^{-1/x^{2}}$ . See also [Chr01], [Mor87], [Mor92], and references within for more discussion of these “intertwined cases”. It is possible that methods used in our paper may also extend to some complex valued elliptic operators satisfying Hörmander bracket condition [Koh05], similar to the note of Christ [Chr05].

To summarize, Theorem 4 together with [AKR19, Theorem 1], contribute to our understanding of how regularity properties of operators in lower dimensions play into the regularity of their sum in higher dimensions. On the technical level, prior works of [Hos87], [Mor86] and [Mor87] used an explicit construction of an analytic function of $x$ motivated by techniques of analytic hypoellipticity [M7́8] or complex analysis. In contrast, the argument of this paper is grounded in analysis of ODEs, which allows for non-analytic and even nonsmooth coefficients. Additionally, we have adapted Littlewood-Paley projections to the degenerate operator to interpolate between standard derivatives and functional calculus, what we hope improves exposition.

This paper is organized as follows. The proof of Theorem 4 is split into the next three sections. Section 2 uses an initial value problem for an ODE and spectral projections to construct a special solution to $\tilde{L}u=0$ . In Section 3 we use the Closed Graph Theorem and a version of Sobolev Embedding to obtain qualitative estimates for the solution of a hypoelliptic operator. In section 4 we apply the results of the previous two section to spectral projections of any $L^{2}$ function. Section 5 concludes the proof by using interpolation to combine the estimates for the spectral projections into (4). Finally, the Appendix A, contains some technical results about spectral projections used throughout the paper.

2. Spectral solutions via an ODE

We begin the proof of Theorem 4 with a simple change of operator. It is convenient to reduce (8) to a case of a constant coefficient: $a_{2}\equiv 1$ first.

Lemma 9.

Consider the operator $\tilde{P}=\frac{1}{a_{2}(x)}L$ in a neighborhood small enough, where $a_{2}(x)$ does not vanish. I.e.

\displaystyle\tilde{P}=-\partial_{x}^{2}+\frac{a_{1}}{a_{2}}(x)\partial_{x}+\frac{a_{0}}{a_{2}}(x)+\frac{g}{a_{2}}(x)L_{2}(y,\partial_{y}).

Then $\tilde{P}$ is hypoelliptic if and only if $L$ from (8) is hypoelliptic. By relabeling $a_{1}$ , $a_{0}$ and $g(x)$ with analogues in $\tilde{P}$ , we can assume $a_{2}(x)=1$ in $L$ without loss of generality.

Proof.

Suppose $L$ is hypoelliptic. Note that $\tilde{P}$ has the same structure as $L$ , except for a coefficient of $\partial_{x}^{2}$ .

If $F=\tilde{P}u$ is smooth near $(x_{0},y_{0})$ , so is $f(x,y)=a_{2}(x)F(x,y)=Lu$ . By hypoellipticity of $L$ , $u$ must be smooth near $(x_{0},y_{0})$ . Thus $\tilde{P}$ is hypoelliptic. A similar argument works for the converse. ∎

The strategy of the proof of Theorem 4 is to convert the problem $Lu=0$ into an ODE using spectral projections, formally discussed in appendix A. Informally, the operator $L_{2}$ is replaced by a spectral parameter $\lambda$ . The following ODE becomes relevant to our analysis in light of Lemma 9.

(10)

\displaystyle\begin{cases}-\partial_{x}^{2}v(x,\lambda)+a_{1}(x)\partial_{x}v(x,\lambda)+[a_{0}(x)+g(x)\lambda]v(x,\lambda)=0\\ v(x_{0},\lambda)=1,\,\dot{v}(x_{0},\lambda)=0\end{cases}

We analyze this ODE first before returning to the original operator $L$ from (8) in Lemma 13.

Proposition 10 (Proposition 1.2.4, [Hör97]).

Let $x_{0}\in\operatorname{\mathbb{R}}$ , $I$ a connected neighborhood of $x_{0}$ and let $\Phi:I\to R^{m\times m}$ be a continuous function with values in $m\times m$ matrices. Then the initial value problem

(11)

\displaystyle\begin{cases}y^{\prime}(x)=\Phi(x)y&\,\,\text{ for }|x-x_{0}|\leq r_{x_{0}}\\ y(x_{0})=y_{0}&\end{cases}

has a unique solution for all $y_{0}\in\operatorname{\mathbb{R}}^{m}$ . Moreover, if $\lVert\Phi\rVert_{L^{\infty}_{I}}\leq M$ for some $M>1$ , then $|y(x)|\leq|y_{0}|e^{M|x-x_{0}|}$ for all $x\in I$ .

We need estimates for derivatives of the solutions to (11), primarily for $y^{\prime}(x)$ .

Lemma 11.

If in addition $\lVert\Phi^{(j)}(x)\rVert\leq M$ for $0\leq j\leq k-1$ , $k\geq 1$ , then

\displaystyle|\partial_{x}^{k}y(x)|\leq C(k)(1+M)^{k}e^{M|x-x_{0}|}|y_{0}|

Moreover, estimates for the derivatives of $\Phi$ are not needed for $k=1$ .

Proof.

Note, that

\displaystyle|y^{\prime}(x)|\leq\lvert\Phi(x)\rvert\lvert y(x)\rvert\leq Me^{M|x-x_{0}|}|y_{0}|

Differentiating (11) we get

\displaystyle y^{\prime\prime}=\Phi y^{\prime}+\Phi^{\prime}y

Hence

\displaystyle\lvert y^{\prime\prime}(x)\rvert\leq(Me^{M|x-x_{0}|}+M\cdot Me^{M|x-x_{0}|})|y_{0}|

Proceeding inductively, we get the estimates for higher derivatives of $y$ up to $k$ . ∎

We now use these results to solve (10).

Lemma 12.

Let $x_{0}$ be given. Then (10) has a unique solution $v(x,\lambda)$ of (10) on $[x_{0}-1,x_{0}+1]$ . Furthermore, the following estimate holds:

\displaystyle|\partial_{x}^{k}v(x,\lambda)|\leq C({k,x_{0}})\!\left<\lambda\right>^{\max\{0,\frac{k-1}{2}\}}\exp(C_{x_{0}}\!\left<\lambda\right>^{\frac{1}{2}}|x-x_{0}|)

where $C_{x_{0}}$ depends on $x_{0}$ through the coefficients of (10). Moreover, the constant for $k=0$ and $k=1$ , $C({0,x_{0}})=1$ . For $k=0,1,2$ no derivatives of the coefficients $a_{0}$ , $a_{1}$ and $g$ are needed. Note, that we use (5) for $\!\left<\lambda\right>$ .

Proof.

To apply Proposition 10 to (10), define $a(x)=[a_{0}(x)+g(x)\lambda]$ and $y(x)=\begin{pmatrix}v\\ v^{\prime}\end{pmatrix}$ . Then $y(x_{0})=\begin{pmatrix}1\\ 0\end{pmatrix}$ and $y^{\prime}(x)=\Phi(x)y(x)$ with

(12)

\displaystyle\Phi(x)=\begin{pmatrix}0&1\\ a(x)&a_{1}(x)\end{pmatrix}

Looking at the eigenvalues of $\Phi(x)$ , we obtain

\lVert\Phi(x)\rVert\leq C(|a_{1}(x)|+\sqrt{|a(x)|}),

where $C$ is a universal constant. Therefore, from the definition of $a(x)$ , we estimate

(13)

\displaystyle\lVert\Phi(\cdot)\rVert_{L^{\infty}_{loc}}\leq C(\sqrt{\lVert g\rVert_{L^{\infty}_{loc}}\lambda+\lVert a_{0}\rVert_{L^{\infty}_{loc}}}+\lVert a_{1}\rVert_{L^{\infty}_{loc}})\leq C(x_{0})\!\left<\lambda\right>^{\frac{1}{2}}

Proposition 10 completes the proof for $k=0$ and $k=1$ .

For higher derivatives of $v$ , we observe that differentiating (12), will give an estimate similar to (13) for $\lVert\partial_{x}^{j}\Phi(\cdot)\rVert_{L^{\infty}_{loc}}$ . That is, $\partial_{x}a(x)$ and $\partial_{x}a_{1}(x)$ have similar to dependence on $\lambda$ , i.e.

\lVert\partial_{x}^{j}\Phi(\cdot)\rVert_{L^{\infty}_{loc}}\leq C(j,x_{0})\!\left<\lambda\right>^{\frac{1}{2}}.

As control of $v^{(k)}(x)$ requires estimates of $y^{(k-1)}(x)$ for $k\geq 1$ , Lemma 11 completes the proof. ∎

We are now ready to construct solutions to (8) via Proposition 23 and other results from the Appendix. We define an operator $v(x,B)$ on a subset of $L^{2}$ via $v(x,\lambda)$ . The following Lemma summarizes the connection between the spectral construction and (8). We will later extend the domain of the $v$ operator, by pre-applying Littlewood-Paley projections.

Lemma 13.

Let $u(y)$ be such that, $e^{\sqrt{B}\varepsilon}u\in L^{2}$ for some $\varepsilon>0$ . Define $w$ by

(14)

\displaystyle w(x,y)=v(x,B)u(y)=\int_{1}^{\infty}v(x,\lambda)dE_{\lambda}u(y).

where $v(x,\lambda)$ is the solution of (10). Let $0<\varepsilon^{\prime}<\varepsilon$ . Then for $|x-x_{0}|\leq\frac{\varepsilon^{\prime}}{2C_{x_{0}}}$ for $C_{x_{0}}$ from Lemma 12, $w$ is well-defined. Likewise, $Lw(x,y)\in L^{2}$ for $L$ from (8). Moreover, using Lemma 9 to replace $a_{2}$ with $1$ , we establish that

Lw=0.

Proof.

By Proposition 23

Bw=\int_{1}^{\infty}\lambda v(x,\lambda)dE_{\lambda}u(y)

Hence by Lemma 12 for $|x-x_{0}|\leq\frac{\varepsilon^{\prime}}{2C_{x_{0}}}$

	$\displaystyle\lVert Bw\rVert_{L^{2}}^{2}$	$\displaystyle\leq\int_{1}^{\infty}\lambda^{2}(\exp(C_{x_{0}}\!\left<\lambda\right>^{\frac{1}{2}}\|x-x_{0}\|))^{2}d\lVert E_{\lambda}u(y)\rVert^{2}$
		$\displaystyle\leq C\int_{1}^{\infty}\lambda^{2}e^{\varepsilon^{\prime}\sqrt{\lambda}}d\lVert E_{\lambda}u(y)\rVert^{2}\leq C\int_{1}^{\infty}e^{\varepsilon\sqrt{\lambda}}d\lVert E_{\lambda}u(y)\rVert<\infty$

for $\varepsilon^{\prime}<\varepsilon$ and $x$ within the interval specified above.
We now justify that $\partial_{x}w\in L^{2}$ and can be expressed as

(15)

\displaystyle\partial_{x}w=\int_{1}^{\infty}\partial_{x}v(x,\lambda)dE_{\lambda}u(y).

To do that we use difference quotients to interchange derivatives and integration in $\lambda$ . Indeed,

\displaystyle D_{h}w(x):=\frac{w(x+h,y)-w(x,y)}{h}=\int_{1}^{\infty}\frac{1}{h}[v(x+h,\lambda)-v(x,\lambda)]dE_{\lambda}u(y)

Hence

\displaystyle D_{h}w(x)=\int_{1}^{\infty}\int_{0}^{1}\partial_{x}v(x+sh,\lambda)dsdE_{\lambda}u(y)

Since $|x-x_{0}|\leq\frac{\varepsilon^{\prime}}{2C_{x_{0}}}$ and for $|h|<\frac{\varepsilon-\varepsilon^{\prime}}{2sC_{x_{0}}}$ we use Lemma 12 to obtain

\displaystyle|\partial_{x}v(x+sh,\lambda)|\leq\!\left<\lambda\right>^{\frac{1}{2}}e^{C_{x_{0}}\varepsilon\sqrt{\lambda}|x-x_{0}+sh|}\leq e^{\varepsilon\sqrt{\lambda}}

Therefore, $D_{h}w\in L^{2}$ for $h$ sufficiently small. A similar argument shows that for any $h_{n}\to 0$ , $D_{h_{n}}w$ is Cauchy and thus has a limit, that we call $\partial_{x}v$ . By Lebesgue Dominated Convergence $\int_{0}^{1}\partial_{x}v(x+sh,\lambda)ds\to v(x,\lambda)$ as $h\to 0$ . This proves (15).

Analysis of $\partial_{x}^{2}w\in L^{2}$ and an analogue of (15) is similar. Combining the terms implies $Lw\in L^{2}$ .

We now demonstrate that under the hypothesis of the Lemma $Lw=0$ . Using spectral projections, e.g. Proposition 23

\displaystyle Lw(x,y)=-\partial_{x}^{2}w+a_{1}(x)\partial_{x}w+\int_{1}^{\infty}[a_{0}(x)+g(x)\lambda]v(x,\lambda)dE_{\lambda}u(y)

We have shown above that we can pass $x$ derivatives into the projection for $w$ . Hence

Lw(x,y)=\int_{1}^{\infty}[-\partial_{x}^{2}v(x,\lambda)+a_{1}(x)\partial_{x}v(x,\lambda)+[a_{0}(x)+g(x)\lambda]v(x,\lambda)dE_{\lambda}u(y)

As $v(x,\lambda)$ solves (10), $Lw=0$ . ∎

3. Hypoellipticity and closed graph theorem

For generic $u\in L^{2}$ or even $C^{\infty}_{0}$ , we may not be able to guarantee the hypothesis of Lemma 13. To go around this concern, we first use the Closed Graph Theorem to control smoothness of solutions to $Lw=0$ using hypoellipticity. We start with a bit of notation.

Definition 14.

For $r>0$ define $I_{r}:=(x_{0}-r,x_{0}+r)$

Lemma 15.

Let $y_{0}\in R^{n}_{y}$ with $y_{0}\in\Omega\Subset\operatorname{\mathbb{R}}^{n}_{y}$ and let $r>0$ . For $I_{r}$ from Definition 14, define

(16)

\displaystyle X:=\{w\in L^{2}({I_{r}}\times\Omega)\mid Lw=0\text{ in the weak sense on }I_{r}\times\Omega\}

Then $X$ is a closed subspace of $L^{2}_{loc}({I_{r}}\times\Omega)$ .

Proof.

We invoke duality. Let $w_{n}\to w\in L^{2}_{loc}$ with $Lw_{n}=0$ weakly. Then for $\phi\in C^{\infty}_{0}({I_{r}}\times\Omega)$

\displaystyle(Lw,\phi)=(w,L^{*}\phi)=\lim_{n}(w_{n},L^{*}\phi)=0

Thus $Lw=0$ weakly and $w\in X$ . ∎

Lemma 16.

[Closed Graph] Let $\Omega^{\prime}\Subset\Omega\subset\operatorname{\mathbb{R}}^{n}_{y}$ and $0<r^{\prime}<r$ . Let $s>0$ . Define $I_{r}$ and $I_{r^{\prime}}$ by Definition 14.
Suppose the operator $L$ from (8) is $H^{s}$ hypoelliptic at $(x_{0},y_{0})$ in the sense of Definition 3 and $X$ is as in (16). Then the operator

\displaystyle T:X\to H^{s}(I_{r^{\prime}}\times\Omega^{\prime})\text{ defined by }Tw=w

has a closed graph. In particular, if $w\in X$ , then

(17)

\displaystyle\lVert w\rVert_{H^{s}(I_{r^{\prime}}\times\Omega^{\prime})}\leq\tilde{C}(\Omega,\Omega^{\prime},r,r^{\prime})\lVert w\rVert_{L^{2}({I_{r}}\times\Omega)}

Remark 17.

As $r^{\prime}\to r$ or $d(\partial\Omega^{\prime},\partial\Omega)\to 0$ the constant in (17) may go to infinity.

Proof.

First, we demonstrate $T$ is well defined. Let $\psi\in C^{\infty}_{0}(I_{r}\times\Omega)$ be a bump function $0\leq\psi\leq 1$ with $\psi(x,y)\equiv 1$ on $I_{r^{\prime}}\times\Omega^{\prime}$ . Then, for all $u\in X$ , $L(\psi u)(x,y)=0$ for $(x,y)\in I_{r^{\prime}}\times\Omega^{\prime}$ . By Definition 3 $\psi u\in H^{s}$ . Hence $u\in H^{s}(I_{r^{\prime}}\times\Omega^{\prime})$ .

To demonstrate that $T$ has a closed graph, suppose $w_{n}\in X$ ,

w_{n}\to w\in X,\text{ and }Tw_{n}\to\tilde{w}\in H^{s}(I_{r^{\prime}}\times\Omega^{\prime}),

Passing to subsequences if necessary, we may assume that $w_{n}\to\tilde{w}$ and $w_{n}\to w$ for a.e. $(x,y)\in I_{r^{\prime}}\times\Omega^{\prime}$ . Since $H^{s}(I_{r^{\prime}}\times\Omega^{\prime})\subset L^{2}(I_{r^{\prime}}\times\Omega^{\prime})$ , it is then clear that $w=\tilde{w}$ and the graph of $T$ is closed. By the closed graph theorem, c.f. [Yos95] p.80, $T$ is continuous, and hence satisfies (17). ∎

We now need a version of Sobolev Embedding to apply Lemma 16 in $H^{s}_{y}$ for $\frac{1}{2}<s\leq\frac{n+1}{2}$ , where $y\in\operatorname{\mathbb{R}}^{n}$ . If we use $C^{\infty}$ hypoellipticity or $H^{s}$ hypoellipticity for $s$ bigger than this range, the standard Sobolev Embedding $H^{\frac{n+1}{2}^{+}}(\operatorname{\mathbb{R}}^{n+1})\subset C^{0}(\operatorname{\mathbb{R}}^{n+1})$ suffices.

Lemma 18.

[Sobolev Embedding] Let

(18)

\displaystyle s=s_{1}+s_{2}\text{ for }s_{1}>\frac{1}{2}\text{ and }s_{2}\geq 0

Suppose $u(x,y)\in H^{s}_{x,y}(\operatorname{\mathbb{R}}_{x}\times\operatorname{\mathbb{R}}^{n}_{y})$ . Then $x\to u(x,\cdot)$ is a continuous uniformly bounded function from $\operatorname{\mathbb{R}}\to H^{s_{2}}_{y}$ with

(19)

\displaystyle\lVert u\rVert_{L^{\infty}_{x}H^{s_{2}}_{y}}\leq C_{s_{1},s_{2}}\lVert u\rVert_{H^{s}_{x,y}}

Proof.

Define

(20)

\displaystyle u_{s}:=\!\left<\partial_{y}\right>^{s_{2}}\!\left<\partial_{x}\right>^{s_{1}}u,

where $\!\left<\partial_{x}\right>^{s_{1}}$ is a pseudodifferential operator with symbol $\!\left<\xi\right>^{s_{1}}$ and similarly for $\!\left<\partial_{y}\right>^{s_{2}}$ . We can re-write this formula as

\displaystyle\hat{u}_{s}(\xi,\eta)=\frac{(1+|\xi|^{2})^{s_{1}}(1+|\eta|^{2})^{s_{2}}}{(1+|\xi|^{2}+|\eta|^{2}))^{s}}(1+|\xi|^{2}+|\eta|^{2}))^{s}\hat{u}(\eta,\xi)

By hypothesis, $u\in H^{s}$ , and the term $\frac{(1+|\xi|^{2})^{s_{1}}(1+|\eta|^{2})^{s_{2}}}{(1+|\xi|^{2}+|\eta|^{2}))^{s}}$ is a bounded Fourier multiplier for $s_{1}$ , $s_{2}\geq 0$ . Thus

(21)

\displaystyle\lVert u_{s}\rVert_{L^{2}_{x,y}}\leq C_{s_{1},s_{2}}\lVert u\rVert_{H^{s}_{x,y}}

We return to the analysis of $u$ using (20). Write formally

(22)

\displaystyle u(x,y)=\!\left<\partial_{x}\right>^{-s_{1}}\!\left<\partial_{y}\right>^{-s_{2}}u_{s}(x,y),

and define, $g(x)$ as an inverse Fourier transform of $\!\left<\partial_{x}\right>^{-s_{1}}$ , i.e.

\displaystyle\hat{g}(\xi)=(1+|\xi|^{2})^{-s_{1}}\in L^{2}_{\xi}\text{ for }s_{1}>\frac{1}{2}.

Note that from the definition $\lVert g\rVert_{L^{2}_{x}}\leq C_{s_{1}}$ . With this notation we rewrite (22) as

(23)

\displaystyle u(x,y)=g(x)\ast_{x}\!\left<\partial_{y}\right>^{-s_{2}}u_{s}.

Applying Cauchy-Schwartz, (21) and the $L^{2}$ estimate for $g$ implies

(24)

\displaystyle\lVert u\rVert_{L^{\infty}_{x}H^{s_{2}}_{y}}\leq C_{s_{1}}\lVert\!\left<\partial_{y}\right>^{-s_{2}}u_{s}\rVert_{L^{2}_{x}H^{s_{2}}_{y}}=C_{s_{1}}\lVert u_{s}\rVert_{L^{2}_{x,y}}\leq C_{s_{1},s_{2}}\lVert u\rVert_{H^{s}_{x,y}},

which concludes (19). Furthermore, to demonstrate continuity we use (23)

\displaystyle\lVert u(x,\cdot)-u(x^{\prime},\cdot)\rVert_{H^{s_{2}}_{y}}\leq\int\left|g(x-z)-g(x^{\prime}-z)\right|\lVert\!\left<\partial_{y}\right>^{-s_{2}}u_{s}(z,\cdot)\rVert_{H^{s_{2}}_{y}}dz

Then by Cauchy-Schwartz

\displaystyle\lVert u(x,\cdot)-u(x^{\prime},\cdot)\rVert_{H^{s_{2}}_{y}}\leq\left(\int|g(x-z)-g(x^{\prime}-z)|^{2}dz\right)^{1/2}\lVert u_{s}\rVert_{L^{2}_{z,y}}

As $x^{\prime}\to x$ , the first factor on the right converges to zero by properties of the translation operator in $L^{2}$ .∎

We conclude the section by combining Lemmas 16 and 18.

Proposition 19.

Let $s=s_{1}+s_{2}$ with $s_{1}>\frac{1}{2}$ and $s_{2}>0$ . Suppose the operator $L$ from (8) is $H^{s}$ hypoelliptic at $(x_{0},y_{0})$ . Then for some $\Omega^{\prime}\Subset\Omega\Subset\operatorname{\mathbb{R}}^{n}_{y}$ and $r>0$ small enough,

(25)

\displaystyle\lVert w(x_{0},\cdot)\rVert_{H^{s_{2}}(\Omega^{\prime})}\leq\tilde{C}(s_{1},s_{2},\Omega,\Omega^{\prime},r)\lVert w\rVert_{L^{2}({I_{r}}\times\Omega)}

4. Hypoellipticity with spectral projections

As mentioned above, for generic $u\in L^{2}_{y}$ or even $C^{\infty}_{0}$ , we may not be able to guarantee the hypothesis of Lemma 13. We show below that for hypoelliptic operators, Lemma 13 applies for any $P_{j}u$ . In particular, this will imply that for hypoelliptic operator $L$ the spectral projection $P_{j}$ from (48) is smoothing. We state this in a form that is crucial for the proof of Theorem 4.

Proposition 20.

Let $s=s_{1}+s_{2}$ with $s_{1}>\frac{1}{2}$ and $s_{2}>0$ . Suppose $L$ from (8) is $H^{s}$ hypoelliptic near $(x_{0},y_{0})$ and let the operator $P_{j}$ be defined by (48). Let $\varepsilon>0$ and $y_{0}\in\Omega^{\prime}\Subset\Omega\Subset\operatorname{\mathbb{R}}^{n}_{y}$ , then

(26)

\displaystyle\lVert P_{j}u\rVert_{H^{s_{2}}(\Omega^{\prime})}\leq\tilde{C}(\varepsilon,x_{0},s_{1},s_{2},\Omega,\Omega^{\prime})\exp(\varepsilon\sqrt{e^{j}})\lVert P_{j}u\rVert_{L^{2}(\Omega)}

for all $u(y)\in L^{2}_{y}(\Omega)$ and all $j\geq 0$ . The constant $\tilde{C}\to\infty$ as $\varepsilon\to 0$ , and similarly as $d(\partial\Omega^{\prime},\partial\Omega)\to 0$ , see Remark 17.

Proof.

Let $u\in L^{2}$ and define $u_{j}=P_{j}u$ and

w_{j}=v(x,B)u_{j}=\psi_{j}(B)v(x,B)u.

Note by (10) $w_{j}(x_{0})=u_{j}$ . By Lemma 26

(27)

\displaystyle\lVert e^{\varepsilon\sqrt{B}}u_{j}(y)\rVert_{L^{2}_{y}}\leq\sum_{|j^{\prime}-j|\leq 1}e^{\varepsilon\sqrt{e^{j^{\prime}}}}\lVert P_{j^{\prime}}P_{j}u\rVert<\infty

Therefore, Lemma 13 applies to $u_{j}$ and gives

Lw_{j}(x,y)=0\text{ for }|x-x_{0}|\leq r_{0}.

Thus hypoellipticity of $L$ implies that $w_{j}\in C^{\infty}_{loc}$ . Note, in particular, $P_{j}u$ is locally smooth. It remains to establish (26) - a quantitative control of this smoothness.

We now invoke Proposition 19. By (25)

(28)

\displaystyle\lVert w_{j}(x_{0},\cdot)\rVert_{H^{s_{2}}(\Omega^{\prime})}\leq\tilde{C}\lVert w_{j}\rVert_{L^{2}(I_{r}\times\Omega)}

where the constant $\tilde{C}$ is independent of $w_{j}$ . In particular, as $r\to 0$ , $\tilde{C}$ may go to infinity by Remark 17.

Recall that $w_{j}(x_{0},y)=P_{j}u$ . Therefore we can restate (28) as

(29)

\displaystyle\lVert P_{j}u\rVert_{H^{s_{2}}(\Omega^{\prime})}\leq\tilde{C}\lVert w_{j}\rVert_{L^{2}(I_{r}\times\Omega)}

Proposition 23 and the support of $\psi_{j}$ imply

(30)

\displaystyle\lVert w_{j}(x,y)\rVert_{L^{2}(I_{r}\times\Omega)}^{2}\leq\int_{x_{0}-r}^{x_{0}+r}\int_{e^{j-1}}^{e^{j+1}}|v(x,\lambda)|^{2}\psi_{j}^{2}(\lambda)d\lVert E_{\lambda}u\rVert_{L^{2}_{y}}^{2}dx

From Lemma 12 we get

\displaystyle|v(x,\lambda)|\leq e^{C_{x_{0}}\!\left<\lambda\right>^{\frac{1}{2}}|x-x_{0}|}

Note, that $C_{x_{0}}$ , unlike $\tilde{C}$ is independent of $\varepsilon$ . Hence for $\lambda\in[{e^{j-1}},{e^{j+1}}]$ and $x\in[{x_{0}-r},{x_{0}+r}]$ :

	$\displaystyle\|v(x,\lambda)\|^{2}\leq e^{2C_{x_{0}}e^{\frac{j+1}{2}}r}=e^{C_{x_{0}}e^{\frac{j}{2}}r},\text{ where }C_{x_{0}}\text{ is replaced with a larger constant}$
	that is still independent of $j$ and $\varepsilon$ in the last equality.

For $\varepsilon>0$ in the hypothesis of our Proposition choose

\displaystyle r=C_{x_{0}}^{-1}\varepsilon.

Then

\displaystyle|v(x,\lambda)|^{2}\leq\exp(\varepsilon\sqrt{e^{j}}).

We now return to (30) with this estimate of $v$ on our projected band:

\displaystyle\lVert w_{j}(x,y)\rVert_{L^{2}(I_{r}\times\Omega)}^{2}\leq\int_{x_{0}-r}^{x_{0}+r}\exp(\varepsilon\sqrt{e^{j}})\int_{e^{j-1}}^{e^{j+1}}\psi_{j}^{2}(\lambda)d\lVert E_{\lambda}u\rVert_{L^{2}_{y}}^{2}dx

Since $\lVert P_{j}u\rVert_{L^{2}(\Omega)}^{2}=\int_{e^{j-1}}^{e^{j+1}}\psi_{j}^{2}(\lambda)d\lVert E_{\lambda}u\rVert_{L^{2}(\Omega)}^{2}$ , we rewrite this as

\displaystyle\lVert w_{j}\rVert_{L^{2}(I_{r}\times\Omega)}^{2}\leq\varepsilon\exp(\varepsilon\sqrt{e^{j}})\lVert P_{j}u\rVert^{2}_{L^{2}(\Omega)}

This estimate together with (29) gives

\displaystyle\lVert P_{j}u\rVert_{H^{s_{2}}(\Omega^{\prime})}^{2}\leq\tilde{C}\varepsilon\exp(\varepsilon\sqrt{e^{j}})\lVert P_{j}u\rVert^{2}_{L^{2}(\Omega)}

Note, that this estimate does not imply that $P_{j}u\to 0$ , as $\varepsilon\to 0$ , since $\tilde{C}$ depends on $\varepsilon$ itself. Absorbing, $\varepsilon$ into a new constant $\tilde{C}$ concludes the proof. ∎

5. Interpolation and proof Theorem 4

Lemma 21.

Let $\varepsilon>0$ , $s_{2}>0$ . For each $\xi\in\operatorname{\mathbb{R}}^{n}$ let $R=R(\xi)$ be defined by

(31)

\displaystyle e^{R}=\left(\frac{s_{2}\log\!\left<\xi\right>}{2\varepsilon}\right)^{2}.

Consider a sequence of functions $\alpha_{k}(y)\in H^{s_{2}}_{y}$ for $k=0,1,2\ldots$ Then

(32)			$\displaystyle\lVert\sum_{k\leq R(\xi)}\log\!\left<\xi\right>\widehat{\alpha_{k}}(\xi)\rVert_{L^{2}_{\xi}}^{2}\leq C(\varepsilon,s_{2})\sum_{k=0}^{\infty}e^{-2\varepsilon\sqrt{e^{k}}}\lVert\alpha_{k}\rVert_{H^{s_{2}}_{y}}^{2}$
(33)			$\displaystyle\lVert\sum_{k\geq R}\log\!\left<\xi\right>\widehat{\alpha_{k}}(\xi)\rVert_{L^{2}_{\xi}}^{2}\leq C\left(\frac{\varepsilon}{s_{2}}\right)^{2}\sum_{k=0}^{\infty}e^{k}\lVert\alpha_{k}\rVert_{L^{2}_{y}}^{2}$

Proof.

Both of the inequalities are proved by a clever use of the Cauchy-Schwartz inequality.

By the Cauchy-Schwartz inequality

\displaystyle\sum_{k\leq R}\log\!\left<\xi\right>|\widehat{\alpha_{k}}(\xi)|\leq\left(\sum_{k\leq R}\frac{\log^{2}\!\left<\xi\right>}{\!\left<\xi\right>^{2s_{2}}}e^{2\varepsilon\sqrt{e^{k}}}\right)^{\frac{1}{2}}\cdot\left(\sum_{k\leq R}\frac{\!\left<\xi\right>^{2s_{2}}}{e^{2\varepsilon\sqrt{e^{k}}}}|\widehat{\alpha_{k}}|^{2}\right)^{\frac{1}{2}}

The terms of the first sum are increasing in $k$ , so the sum can be estimated

\displaystyle\sum_{k\leq R}\frac{\log^{2}\!\left<\xi\right>}{\!\left<\xi\right>^{2s_{2}}}e^{2\varepsilon\sqrt{e^{k}}}\leq\frac{\log^{2}\!\left<\xi\right>}{\!\left<\xi\right>^{2s_{2}}}R\cdot e^{2\varepsilon\sqrt{e^{R}}}

Substitution of the identity (31) (which is equivalent to $\!\left<\xi\right>^{s_{2}}=e^{2\varepsilon\sqrt{e^{R}}}$ ) implies

\displaystyle\sum_{k\leq R}\frac{\log^{2}\!\left<\xi\right>}{\!\left<\xi\right>^{2s_{2}}}e^{2\varepsilon\sqrt{e^{k}}}\leq\frac{\log^{2}\!\left<\xi\right>(2\log s_{2}+2\log\log\!\left<\xi\right>-2\log 2\varepsilon)}{\!\left<\xi\right>^{s_{2}}}\leq C(\varepsilon,s_{2})

Combining the last three estimates we obtain

\displaystyle\lVert\sum_{k\leq R}\log\!\left<\xi\right>\widehat{\alpha_{k}}(\xi)\rVert_{L^{2}_{\xi}}^{2}\leq C(\varepsilon,s_{2})\int\sum_{k\leq R}\frac{\!\left<\xi\right>^{2s_{2}}}{e^{2\varepsilon\sqrt{e^{k}}}}|\widehat{\alpha_{k}}|^{2}d\xi

Observe

\displaystyle\sum_{k\leq R}\frac{\!\left<\xi\right>^{2s_{2}}}{e^{2\varepsilon\sqrt{e^{k}}}}|\widehat{\alpha_{k}}|^{2}\leq\sum_{k=0}^{\infty}\frac{\!\left<\xi\right>^{2s_{2}}}{e^{2\varepsilon\sqrt{e^{k}}}}|\widehat{\alpha_{k}}|^{2}

Interchanging the sum and integration using Fubini concludes the proof of (32).

Similarly for (33), by Cauchy-Schwartz

\displaystyle\left(\sum_{k\geq R}\log\!\left<\xi\right>|\widehat{\alpha_{k}}(\xi)|\right)^{2}\leq\left(\sum_{k\geq R}\frac{\log^{2}\!\left<\xi\right>}{e^{k}}\right)\cdot\left(\sum_{k\geq R}e^{k}|\widehat{\alpha_{k}}|^{2}\right)

The first sum is a geometric series in $k$ , so

\displaystyle\sum_{k\geq R}\frac{\log^{2}\!\left<\xi\right>}{e^{k}}=\frac{\log^{2}\!\left<\xi\right>}{e^{R}}\frac{e}{e-1}\leq 3\left(4\frac{\varepsilon}{s_{2}}\right)^{2}

Integrating in $\xi$ gives

\displaystyle\lVert\sum_{k\geq R}\log\!\left<\xi\right>\widehat{\alpha_{k}}\rVert_{L^{2}}^{2}\leq C\left(\frac{\varepsilon}{s_{2}}\right)^{2}\int\sum_{k}e^{k}|\hat{\alpha}_{k}|^{2}d\xi

Applying Fubini concludes (33) ∎

Proof of Theorem 4.

Suppose $L$ from (8) is $H^{s}$ hypoelliptic for $s>\frac{1}{2}$ , then Proposition 20 applies for $s_{2}=s-s_{1}>0$ with some $s_{1}>\frac{1}{2}$ .

Let $u(y)\in C^{\infty}_{0}(\Omega^{\prime})$ , and define $\alpha_{j}=P_{j}u$ . Since $P_{j}$ ’s are a resolution of identity by Lemma 26, we can decompose

\displaystyle u=\sum_{j\geq 0}P_{j}u=\sum_{j\geq 0}\alpha_{j}

By the triangle inequality

\lVert\log\!\left<\xi\right>\hat{u}\rVert_{L^{2}}\leq\lVert\log\!\left<\xi\right>\sum_{j\leq R}\hat{\alpha}_{j}\rVert_{L^{2}}+\lVert\log\!\left<\xi\right>\sum_{j\geq R}\hat{\alpha}_{j}\rVert_{L^{2}}

By Cauchy-Schwartz

\displaystyle\lVert\log\!\left<\xi\right>\hat{u}\rVert_{L^{2}}^{2}\leq 2\left(\lVert\log\!\left<\xi\right>\sum_{j\leq R}\hat{\alpha}_{j}\rVert_{L^{2}}^{2}+\lVert\log\!\left<\xi\right>\sum_{j\geq R}\hat{\alpha}_{j}\rVert_{L^{2}}^{2}\right)

We are now set to use Lemma 21 for our choice of $\alpha_{j}$ (replacing $k$ with $j$ ):

(34)

\displaystyle\lVert\log\!\left<\xi\right>u\rVert_{L^{2}}^{2}\leq C\frac{\varepsilon}{s_{2}}^{2}\sum_{j=0}^{\infty}e^{j}\lVert P_{j}u\rVert_{L_{2}}^{2}+C_{\varepsilon}\sum_{j=0}^{\infty}e^{-2\varepsilon\sqrt{e^{j}}}\lVert P_{j}u\rVert_{H^{s_{2}}}^{2}:=I+II

We estimate $I$ and $II$ separately. For $I$ we estimate the first sum via (49) for $f(\lambda)=\sqrt{\lambda}$ . As $f(\lambda)>0$ and increasing for $\lambda\geq 1$

\displaystyle I=C\varepsilon^{2}\sum_{j\geq R}e\cdot|\sqrt{e^{j-1}}|^{2}\lVert P_{j}u\rVert_{L_{2}}^{2}\leq Ce\varepsilon^{2}\lVert\sqrt{B}u\rVert^{2}

Choose $\varepsilon>0$ small enough, so that $Ce\varepsilon\leq 1$ . Then

\displaystyle I\leq\varepsilon\lVert\sqrt{B}u\rVert^{2}

Using $(Bu,u)=(L_{2}u,u)$ and the fact that $B$ is an extension of the operator $L_{2}$ and $u\in C^{\infty}_{0}$ we obtain

\displaystyle I\leq\varepsilon(L_{2}u,u)

For $II$ we observe from Proposition 20

\displaystyle\lVert P_{j}u\rVert_{H^{s_{2}}}^{2}\leq C(\varepsilon,s)e^{2\varepsilon\sqrt{e^{j}}}\lVert P_{j}u\rVert_{L^{2}}^{2}

Hence

\displaystyle II\leq C(\varepsilon,s)\sum_{j}\lVert P_{j}u\rVert_{L^{2}}^{2}\leq C_{\varepsilon}\lVert u\rVert^{2}

Combining $I$ and $II$ into (34) we obtain

\displaystyle\lVert\log\!\left<\xi\right>u\rVert_{L^{2}(\Omega^{\prime})}^{2}\leq\varepsilon(L_{2}u,u)+C(\varepsilon,s)\lVert u\rVert^{2}_{L^{2}(\Omega)}\hskip 20.0pt\qed

Appendix A Spectral projections for the operator

We start with recalling the relevant facts about the spectral projection and constructing Littlewood-Paley projections adapted to the operator.

Note that by (2), $L_{2}$ is an operator bounded below. That is, given $\Omega\Subset\operatorname{\mathbb{R}}^{n}$ there is a constant $c_{\Omega}\in\operatorname{\mathbb{R}}$ so that

(35)

\displaystyle(L_{2}u,u)\geq c_{\Omega}\lVert u\rVert^{2}\text{ for all }u\in C_{0}^{\infty}(\Omega)

Fix a neighborhood of $(x_{0},y_{0})$ with compact closure. Since Theorem 4 is a local result, we may change the operator $L$ from (8) away from this neighborhood to assume (35) holds with a constant $c=c_{\Omega_{(x_{0},y_{0})}}$ .

Remark 22.

We may assume without loss of generality that (35) holds for $c=1$ and hence $L_{2}$ is a positive operator:

(36)

\displaystyle(L_{2}u,u)\geq\lVert u\rVert^{2}

Proof.

If $c<1$ in (35), we replace $L_{2}$ with $L_{2}:=L_{2}-c+1$ for $c$ from (35) and $a_{0}(x)$ with $a_{0}(x)+g(x)[c-1]$ . Note that with this change the operator $L$ in (8) remains unchanged. Such a replacement makes $L_{2}$ a positive operator, which we would still call $L_{2}$ . ∎

We now extend the domain of $L_{2}$ from $C^{\infty}_{0}(\Omega)$ to a unique maximal domain inside $L^{2}(\Omega)$ to make it self adjoint. More precisely, let $B$ be the Friedreich extension of $L_{2}$ (c.f. Theorem 2 on p. 317 of [Yos95]). We then have that $B$ satisfies

(37)

\displaystyle(Bu,u)\geq\lVert u\rVert^{2},

since $Bu=L_{2}u$ for $u\in C^{\infty}_{0}$ . Moreover, as $B$ is a self-adjoint operator, the spectral theorem/calculus applies to it. We summarize the relevant parts from [Yos95] ch XI sec. 5 and 10 (p.300-343)]

Proposition 23.

There exists a spectral projection $E_{B}(\lambda):L^{2}(\Omega)\to L^{2}(\Omega)$ , that is a resolution of identity and allows to form functions of the operator $B$ as follows:

(1)

$E_{B}$ is a projection, i.e.

(38) $\displaystyle E_{B}(\lambda)E_{B}(\lambda^{\prime})=E_{B}\left[\min(\lambda,\lambda^{\prime})\right]$

And a resolution of identity: $E_{B}(\infty)=I$ , $E_{B}(-\infty)=0$ ; Moreover $E$ is right-continuous.
(2)

For a measurable function $f(x)$ we can define the function $f(B)$ of the operator $B$ as follows:

(39) $\displaystyle f(B)u=\int_{-\infty}^{\infty}f(\lambda)dE_{B}(\lambda)u$

Where $\operatorname{dom}f(B)=\{u\in L^{2}(\Omega):\lVert f(B)u\rVert^{2}:=\int_{-\infty}^{\infty}|f(\lambda)|^{2}d||E_{B}(\lambda)u||^{2}<\infty\}.$

(3)

In particular, for $f(\lambda)=1$ and $\lambda$ gives the following identities

(40)

\displaystyle u=\int_{-\infty}^{\infty}dE_{B}(\lambda)u

\displaystyle\hskip 10.0ptBu=\int_{-\infty}^{\infty}\lambda dE_{B}(\lambda)u

are defined on $L^{2}(\Omega)$ and $\operatorname{dom}(B)$ respectively.

(4)

For any measurable functions $f$ and $g:\operatorname{\mathbb{R}}\to\operatorname{\mathbb{R}}$ , $f(B)$ and $g(B)$ commute on the $dom\left(f(B)g(B)\right)$ defined via (39). Moreover,

(41)

\displaystyle(f(B)g(B)u,v)=(f(B)u,g(B)v)=\int f(\lambda)\overline{g(\lambda)}d(E_{B}(\lambda)u,v)

Proof.

Existence of spectral projection is given Theorem XI.6.1 on p. 313 of [Yos95] via unitary operators. Part 2 and Part 3 are proved in Theorem XI.5.2. on p.311 using Part 1 and orthogonality. Part 4 is proved in Corollary XI.5.2 and for measurable functions in Theorem XI.12.3 on p. 343 of [Yos95]. ∎

We follow the notation of [Yos95] and denote

(42)

\displaystyle E(\lambda,\mu):=E(\lambda)-E(\mu)\text{ for }\lambda\geq\mu

One of the important consequences of part 4 of Proposition 23 is the following equality

\lVert\sqrt{B}u\rVert^{2}=(Bu,u).

Lemma 24.

Suppose (37) holds for $B$ . Then the projection $E_{B}$ vanishes below $\lambda=1$ , i.e. $E_{B}(1^{-})=0$ ,

Proof.

It suffices to show that $u:=E(\mu)v=0$ for all $\mu<1$ and all $v\in L^{2}(\Omega)$ . To achieve that we substitute $u$ into (37) and (40) to obtain

0\leq\int_{-\infty}^{\infty}(\lambda-1)d(E_{B}(\lambda)u,u)

We first consider the large frequencies and decompose the integral into a Riemann sum as follows

	$\displaystyle\int^{\infty}_{\mu}(\lambda-1)d(E(\lambda)u,u)$
	$\displaystyle=\sup_{\mu=\lambda_{0}<\lambda_{1}<\ldots<\lambda_{N}=M}\sum_{j=0}^{N}(\lambda_{j}-1)(E(\lambda_{j},\lambda_{j-1})E(\mu)v,E(\mu)v),$

Here the supremum is taken over all partitions $\mu=\lambda_{0}<\lambda_{1}<\ldots<\lambda_{N}=M$ of all intervals $[\mu,M]\subset[\mu,\infty)$ . By (42) and (38) all terms in the sum above vanish. We conclude that

\displaystyle\int^{\infty}_{\mu}(\lambda-1)d(E(\lambda)u,u)=0

Hence the integral over the low frequencies must be non-negative. We proceed in the low frequency case as before

\displaystyle\int_{-\infty}^{\mu}(\lambda-1)d(E(\lambda)u,u)=\sup_{-M=\lambda_{0}<\lambda_{1}<\ldots<\lambda_{N}=\mu}\sum_{j=1}^{N}(\lambda_{j}-1)(E(\lambda_{j},\lambda_{j-1})E(\mu)v,E(\mu)v)

with the supremum taken over all partitions $-M=\lambda_{0}<\lambda_{1}<\ldots<\lambda_{N}=\mu$ of all intervals $[-M,\mu]\subset(-\infty,\mu]$ . Observe that (38) implies orthogonality of projections localized at different frequencies, i.e. $E(\lambda_{j},\lambda_{j-1})E(\lambda_{k},\lambda_{k-1})u=0$ for different $j$ and $k$ . Moreover, using (41) with $f=\chi_{(0,\lambda_{j})}-\chi_{(0,\lambda_{j-1})}$ and $g=\chi_{(0,\lambda_{k})}-\chi_{(0,\lambda_{k-1})}$ we have

	$\displaystyle\int_{-\infty}^{\mu}(\lambda-1)d(E(\lambda)u,u)=\sup_{-M=\lambda_{0}<\lambda_{1}<\ldots<\lambda_{N}=\mu}\sum_{j=0}^{N}(\lambda_{j}-1)\lVert E(\lambda_{j},\lambda_{j-1})v\rVert^{2}$
	$\displaystyle\leq(\mu-1)\sup_{-M=\lambda_{0}<\lambda_{1}<\ldots<\lambda_{N}=\mu}\sum_{j=0}^{N}\lVert E(\lambda_{j},\lambda_{j-1})v\rVert^{2}$

Using orthogonality of distinct frequency interval again we obtain

\displaystyle\int_{-\infty}^{\mu}(\lambda-1)d(E(\lambda)u,u)\leq(\mu-1)\lVert E(\mu)v\rVert^{2}

Combining all the estimates established in this proof we obtain

\displaystyle 0\leq\int_{-\infty}^{\infty}(\lambda-1)d(E_{B}(\lambda)u,u)\leq(\mu-1)\lVert E(\mu)v\rVert^{2}\leq 0,\quad\text{since}\ \ \mu<1.

We must conclude that $E(\mu)v=0$ . ∎

In light of Lemma 24, we only need to consider $\lambda\geq 1$ . For the analysis below it is convenient to localize solutions to specific frequencies. Littlewood-Paley decomposition is ideal for that. Let $\phi$ be a fixed cut-off function satisfying

\displaystyle\phi(\lambda)\in C^{\infty}_{0}([-e,e]),\,\,\phi(\lambda)\equiv 1\text{ for }|\lambda|\leq 1\text{ and }0\leq\phi\leq 1.

Moreover, we assume that $\phi$ is even and nonincreasing in $[0,e]$ . Now, for any integer $j$ let

\displaystyle\phi_{j}(\lambda)\equiv\phi(\lambda\cdot e^{-j}).

Then $\phi_{j}(\lambda)=1$ for $|\lambda|\leq e^{j}$ and $\phi_{j}(\lambda)=0$ for $|\lambda|\geq e^{j+1}$ . We further define cut-offs localized to $|\lambda|\approx e^{j}$ as follows:

(43)

\displaystyle\psi_{j}(\lambda)=\phi_{j}(\lambda)-\phi_{j-1}(\lambda)\text{ for }j\geq 0,

so that $\operatorname{supp}{\psi_{j}(|\lambda|)}=[e^{j-1},e^{j+1}]$ and $0\leq\psi_{j}\leq 1$ .
We summarize properties of such cutoffs in this range of $\lambda\geq 1$ as follows.

Lemma 25.

Let $\psi_{j}$ be as defined in (43) and consider $\lambda\geq 1$ . We then have

(44)			$\displaystyle\operatorname{supp}\psi_{j}(\|\lambda\|)=[e^{j-1},e^{j+1}]\text{ for }j\geq 0;\hskip 10.0pt\sum_{j=0}^{\infty}\psi_{j}(\lambda)=1$
(45)			$\displaystyle\psi_{j}(\lambda)\cdot\psi_{k}(\lambda)>0\text{ if and only if }\|j-k\|\leq 1;\hskip 10.0pt\frac{1}{2}\leq\sum_{j=0}^{\infty}\psi_{j}^{2}\leq 1\$

Furthermore, for any positive increasing function $f:[1,\infty)\to\operatorname{\mathbb{R}}^{+}$

(46)

\displaystyle\sum_{j=0}^{\infty}f(e^{j-1})\psi_{j}^{2}(\lambda)\leq f(\lambda)\leq 2\sum_{j=0}^{\infty}f(e^{j+1})\psi_{j}^{2}(\lambda)

Proof.

From the definition of $\psi_{j}$ and supports of $\phi_{j}(\lambda)$ , we immediately get the support property (44) from (43). A telescoping sum argument establishes the series identity in (44). Indeed

(47)

\displaystyle\sum_{j=0}^{n}\psi_{j}(\lambda)=\phi_{n}(\lambda)-\phi_{-1}(\lambda)

Since $\phi_{n}(\lambda)=1$ for $|\lambda|\leq e^{n}$ and $\phi_{-1}(\lambda)=0$ for $\lambda\geq 1$ , (44) follows.

For (45), the orthonormal property is immediate from the fact that $\psi_{j}(\lambda)$ is nonzero only when $|\lambda|\in[e^{j-1},e^{j+1}]$ . From the size of $\psi_{j}$ , $\sum_{j=0}^{\infty}\psi_{j}^{2}\leq\sum_{j=0}^{\infty}\psi_{j}=1$ . By (45) and the Cauchy-Schwartz inequality,

1=(\sum_{j=0}^{\infty}\psi_{j}(\lambda))^{2}\leq\sum_{j=0}^{\infty}\sum_{k=j-1}^{j+1}\psi_{j}\psi_{k}\leq\sum_{j=0}^{\infty}(\psi_{j}^{2}+\frac{1}{2}\psi_{j-1}^{2}+\frac{1}{2}\psi_{j+1}^{2})

which concludes (45).

On the support of each $\psi_{j}$ for $j\geq 1$ , monotonicity of $f$ implies

f(e^{j-1})\leq f(\lambda)\leq f(e^{j+1}).

Multiplying by $\psi_{j}(\lambda)^{2}$ makes the inequality valid for all $\lambda$ :

f(e^{j-1})\psi_{j}^{2}(\lambda)\leq f(\lambda)\psi_{j}^{2}\leq f(e^{j+1})\psi_{j}^{2}(\lambda).

Summing up gives

\displaystyle\sum_{j=0}^{\infty}f(e^{j-1})\psi_{j}^{2}(\lambda)\leq\sum_{j=0}^{\infty}f(\lambda)\psi_{j}^{2}\leq\sum_{j=0}^{\infty}f(e^{j+1})\psi_{j}^{2}(\lambda)

Finally, from (45)

\displaystyle\sum_{j=0}^{\infty}f(e^{j-1})\psi_{j}^{2}(\lambda)\leq f(\lambda)\leq 2\sum_{j=0}^{\infty}f(e^{j+1})\psi_{j}^{2}(\lambda)

∎

With $\psi_{j}$ as above, we define a frequency localization adapted to the operator via Proposition 23

(48)

\displaystyle P_{j}u:=\psi_{j}(B)u=\int\psi_{j}(\lambda)dE_{B}(\lambda)u

for $j\in\mathbb{N}$ . Combining Proposition 23, Lemma 24 and Lemma 25 we get

Lemma 26.

Let $P_{j}$ be defined by (48). Then for any $u\in L^{2}(\Omega)$ there holds

	$\displaystyle u=\sum_{j=0}^{\infty}P_{j}u\hskip 15.0pt\lVert u\rVert^{2}\approx\sum_{j=0}^{\infty}\lVert P_{j}u\rVert^{2};$
	$\displaystyle\int P_{j}uP_{j^{\prime}}vdx=0\text{ for }\|j-j^{\prime}\|>1\text{ and }v\in L^{2}$

Finally, for $f:[1,\infty)\to\mathbb{R}_{+}$ nondecreasing

(49)

\displaystyle\sum_{j=0}^{\infty}|f\left(e^{j-1}\right)|^{2}\lVert P_{j}u\rVert^{2}\leq\lVert f(B)u\rVert^{2}\leq 2\sum_{j=0}^{\infty}|f\left(e^{j+1}\right)|^{2}\lVert P_{j}u\rVert^{2}

Proof.

By Proposition 23 Parts 2 and 4, Lemma 24 and (44)

\displaystyle u=\int_{-\infty}^{\infty}dE(\lambda)u=\int_{1}^{\infty}\sum_{j=0}^{\infty}\psi_{j}(\lambda)dE(\lambda)u=\sum_{j=0}^{\infty}P_{j}u

Similarly, using (45)

\displaystyle\lVert u\rVert^{2}=\int_{-\infty}^{\infty}(dE(\lambda)u,u)\approx\int_{1}^{\infty}\sum_{j=0}^{\infty}\psi_{j}^{2}(\lambda)(dE(\lambda)u,u)=\sum_{j=0}^{\infty}\lVert P_{j}u\rVert^{2}

Orthogonality part of (45) for $|j-j^{\prime}|>1$ and (41) establish

\int P_{j}uP_{j^{\prime}}vdx=0

Finally, from (46) and Fubini we conclude

\displaystyle\lVert f({B})u\rVert^{2}=\int_{1}^{\infty}|f(\lambda)|^{2}d(E_{\lambda}u,u)\leq 2\sum_{j=0}^{\infty}\int_{1}^{\infty}f(e^{j+1})^{2}\psi_{j}^{2}(\lambda)d(E_{\lambda}u,u)

The lower bound is established similarly, completing the proof. ∎

2. Acknowledgments

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

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