Approximation by the Bessel–Riesz Quotient

Ikemefuna Agbanusi Department of Mathematics and Computer Science, Colorado College, [email protected]

Abstract

How large is the Bessel potential, $G_{\alpha,\mu}f$ , compared to the Riesz potential, $I_{\alpha}f$ , of a given function? We prove that, for certain $f$ and $p$ ,

\|G_{1,\mu}f\|_{p}\approx\omega(I_{1}f,1/\mu)_{p},

where $\omega(f,t)_{p}$ is the $L^{p}$ modulus of continuity. However, for $0<\alpha<1$ ,

\|G_{\alpha,\mu}f\|_{p}\leq C(\omega(I_{\alpha}f,1/\mu)_{p})^{\alpha}\cdot\|I_{\alpha}f\|^{1-\alpha}_{p}.

These estimates are obtained by studying the quotient of the two operators, $E_{\alpha,\mu}:=\frac{(-\Delta)^{\alpha/2}}{(\mu^{2}-\Delta)^{\alpha/2}}$ , and exploiting its approximation theoretic properties. Additionally, $G_{\alpha,\mu}f=\mathcal{O}(\mu^{-\frac{\alpha}{2}})$ if $I_{\alpha}f$ vanishes near a given point. This “localization” result is derived from kernel estimates of $E_{\alpha,\mu}$ .

1 Overview

This paper deals with some topics at the intersection of Fourier analysis, potential theory and approximation theory. Specifically, for $\mu>0$ , we study the Bessel–Riesz quotient, $E_{\alpha,\mu}$ , which is the operator defined by the multiplier

m_{\alpha,\mu}(\xi):=\dfrac{\left|\xi\right|^{\alpha}}{(\mu^{2}+\left|\xi\right|^{2})^{\frac{\alpha}{2}}}.

(1)

Here $\hat{f}(\xi)$ denotes the Fourier transform while $\widehat{G_{\alpha,\mu}f}(\xi):=(\mu^{2}+|\xi|^{2})^{-\frac{\alpha}{2}}\hat{f}(\xi)$ and $\widehat{I_{\alpha}f}(\xi):=|\xi|^{-\alpha}\hat{f}(\xi)$ define the family of Bessel and Riesz potentials respectively. We focus mainly on the case $0<\alpha\leq 1$ .

The Riesz and Bessel potentials are indispensable in potential theory, the theory of Sobolev spaces, elliptic PDEs, and describing the fine structure of functions and sets to name just a few applications. Quantifying the relationship between these two classical operators is also an important problem. For the “standard” case $\mu=1$ , this comparison has been expressed in the literature in several ways—in terms of their capacities as in Ziemer [15, pg. 67] and via fractional maximal functions as in Adams–Hedberg [1, Theorem 3.6.2].

Since $G_{\alpha,\mu}f=E_{\alpha,\mu}I_{\alpha}f$ , norm or pointwise estimates for $E_{\alpha,\mu}$ allow for a direct comparison of the two potentials. This is one reason for introducing and studying this operator. Moreover, the approach using the Bessel–Riesz quotient appears better at handling the behaviour as $\mu\to\infty$ . This is because we can apply some approximation theoretic tools as discussed below.

At least two observations point to the connection with approximation theory. The first is the trivial fact that $m_{\alpha,\mu}(\xi)\to 0$ pointwise as $\mu\to\infty$ . The second observation starts with a formula from Stein [13, §5.3.2]

\dfrac{\left|\xi\right|^{\alpha}}{(\mu^{2}+\left|\xi\right|^{2})^{\frac{\alpha}{2}}}=\left(1-\frac{\mu^{2}}{\mu^{2}+\left|\xi\right|^{2}}\right)^{\frac{\alpha}{2}}=1-\sum_{j=1}^{\infty}a_{\alpha,j}(1+|\xi\mu^{-1}|^{2})^{-j},

for some positive coefficients with $\sum a_{\alpha,j}=1$ . By Fourier inversion we obtain

E_{\alpha,\mu}f(x)=f(x)-T_{\alpha,\mu}f(x),

(2)

where $T_{\alpha,\mu}$ has the convolution kernel $A_{\alpha,\mu}(z)$ defined as

A_{\alpha,\mu}(z):=\mu^{-d}\sum_{j=1}^{\infty}a_{\alpha,j}G_{2j}(\mu z).

(3)

Again, $G_{s}(z)$ is the Bessel kernel whose Fourier transform is $(1+|\xi|^{2})^{-\frac{s}{2}}$ . Their well known properties imply that $A_{\alpha,\mu}(z)$ is a positive, radial, integrable function with $L^{1}$ norm $\|A_{\alpha,\mu}\|_{1}=\|\sum_{j=1}^{\infty}a_{\alpha,j}G_{2j}\|_{1}=1$ . Note that $A_{\alpha,\mu}(z)$ is also singular at the origin since $\lim_{|z|\to 0}G_{2j}(z)=\infty$ when $0<2j\leq d$ . In any case, $T_{\alpha,\mu}$ is an approximate identity and, by (2), $E_{\alpha,\mu}$ is its approximation error.

With any approximation scheme, the main task is quantifying the error, $\|E_{\alpha,\mu}f\|_{p}$ . The error turns out to be related to the $L^{p}$ modulus of continuity defined by

\omega(f,t)_{p}:=\sup_{|h|\leq t}\|f(\cdot+h)-f(\cdot)\|_{p}.

(4)

The exact dependence is our first main result and is summarized next. Here $X\approx Y$ means that $C^{-1}Y\leq X\leq CY$ for some $C>0$ .

Theorem 1.1.

Suppose $f\in L^{p}(\mathbb{R}^{d})$ .

(a)

If $\alpha=1$ and $1<p<\infty$ then

$\|E_{1,\mu}f\|_{p}\approx\omega(f,1/\mu)_{p}.$ (5)
(b)

If $\alpha=1$ and $p=1$ there is a constant $k=k(d)$ such that

$\|E_{1,\mu}f\|_{{}_{1}}\leq k\omega(f,1/\mu)_{1}\log\omega(f,1/\mu)_{1}.$ (6)
(c)

If $0<\alpha<1$ and $1\leq p<\infty$ there is a constant $C=C(\alpha,d)$ such that

$\|E_{\alpha,\mu}f\|_{p}\leq C(\omega(f,1/\mu)_{p})^{\alpha}\cdot\|f\|^{1-\alpha}_{p}$ (7)

The novelty here is the $L^{1}$ estimate (6) and the interpolation type inequality (7). It turns out that (5) is implicit in Colzani [5] and Liu–Lu [10], but we give an independent proof tailored to our operator. We do not know if (6) or (7) is sharp.

Theorem 1.1 and the identity $G_{\alpha,\mu}f=E_{\alpha,\mu}I_{\alpha}f$ have the following corollary.

Corollary 1.2.

(a)

If $1<p<\infty$ and $I_{1}f\in L^{p}(\mathbb{R}^{d})$ , we have

$\|G_{1,\mu}f\|_{p}\approx\omega(I_{1}f,1/\mu)_{p}.$
(b)

If $I_{1}f\in L^{1}(\mathbb{R}^{d})$ , then

$\|G_{1,\mu}f\|_{{}_{1}}\leq k\omega(I_{1}f,1/\mu)_{1}\log\omega(I_{1}f,1/\mu)_{1}.$
(c)

If $0<\alpha<1$ , $1\leq p<\infty$ and $I_{\alpha}f\in L^{p}(\mathbb{R}^{d})$ , then

$\|G_{\alpha,\mu}f\|_{p}\leq C(\omega(I_{\alpha}f,1/\mu)_{p})^{\alpha}\cdot\|I_{\alpha}f\|^{1-\alpha}_{p}.$

This gives the precise sense in which the Bessel potential “fine tunes” the Riesz potential. Note that the Hardy–Littlewood–Sobolev inequality gives conditions for the assumptions of Corollary 1.2 to hold. Corollary 1.2 also connects the Riesz potential and the truncated maximal function. The latter is defined as follows:

M_{\alpha,\delta}f(x)=\sup_{r\leq\delta}\frac{1}{r^{d-\alpha}}\int_{|x-y|\leq r}|f(y)|\,dy.

(8)

Corollary 1.3.

If $p>1$ and $I_{1}f\in L^{p}(\mathbb{R}^{d})$ ,

\omega(I_{1}f,1/\mu)_{p}\leq C\|M_{1,\frac{1}{\mu}}f\|_{p}.

This is an immediate consequence of a result of Schechter [12, Theorem 3.5] which in turn refines the Muckenhoupt–Wheeden [11, Theorem 1] estimate for fractional integrals. Corollary 1.3 is likely a known result, but we have not found it stated anywhere. It would be nice to have a more direct proof.

Returning to the approximation properties of $E_{\alpha,\mu}$ , it is natural to seek the best or worst possible order of approximation. So called “saturation theorems” giving the best possible rate can be found in §3 when $\alpha=1$ . Finding the worst rate of approximation is tantamount to knowing how slow $\omega(f,t)_{p}\to 0$ as $t\to 0$ as $f$ ranges in $L^{p}$ . This is an open problem as far as we know though there are partial results due to Besov–Stechkin for $L^{2}(\mathbb{R})$ in [3].

Theorem 1.1 also gives another characterization of Besov spaces (see section on Notation for the relevant definitions).

Corollary 1.4.

Fix $0<s<1$ . For $1<p<\infty$ and $0<q\leq\infty$ , we have

|f|^{q}_{{B}^{s}_{p,q}}\approx\int_{1}^{\infty}(\mu^{s}||E_{1,\mu}f||_{p})^{q}\,\frac{d\mu}{\mu}.

We now turn to issues of pointwise convergence. Let $\mathcal{M}_{\text{HL}}$ denote the Hardy–Littlewood maximal operator. As $A_{\alpha,1}(z)$ is radially decreasing, we have the maximal inequality $\sup_{\mu>0}|(A_{\alpha,\mu}\star f)(x)|\leq C\mathcal{M}_{\text{HL}}f(x)$ (see Stein [14, §2.2.1]). This implies a weak type $(1,1)$ boundedness from which convergence of $T_{\alpha,\mu}f(x)$ to $f(x)$ for almost all $x$ follows.

However, to deal with convergence at a specified point, or to study the structure of the set where pointwise convergence holds, we need good kernel estimates. A direct attack on the series (3) defining $A_{\alpha,\mu}$ appears unwieldy. In fact, we could not make this approach work because the known estimates for Bessel kernels are not precise enough to be summed in an infinite series.

Note that if $b(\xi)$ is either $m_{\alpha,\mu}(\xi)$ or $1-m_{\alpha,\mu}(\xi)$ , then $b(\xi)$ satisfies

|\partial^{\beta}_{\xi}b(\xi)|\leq C_{\beta}\left|\xi\right|^{-|\beta|};\quad\xi\neq 0.

(9)

Incidentally, this implies the uniform $L^{p}$ boundedness of $E_{\alpha,\mu}$ for $1<p<\infty$ , but still does not give fast enough decay at infinity for the kernel. A more detailed analysis actually shows that

|\partial^{\beta}_{\xi}b(\xi)|\leq C_{\alpha,\beta}|\xi|^{\alpha-|\beta|}(\mu^{2}+|\xi|^{2})^{-\frac{\alpha}{2}}.

(10)

This refinement is the main ingredient in the following result.

Theorem 1.5.

Suppose $b(\xi)\in L^{\infty}$ and satisfies (10) for $0<\alpha\leq 1$ . Then its kernel $B(x)$ satisfies

|\partial_{x}^{\gamma}B(x)|\leq C_{\alpha,\gamma,d}\begin{cases}|2\mu x|^{-\frac{\alpha}{2}}|x|^{-|\gamma|-d};&|\mu x|>1,\\ (|\mu x|^{2}+1)^{-\frac{\alpha}{2}}|x|^{-|\gamma|-d};&|\mu x|\leq 1.\end{cases}

Near the origin, this is the standard estimate for Calderon–Zygmund type kernels. However, the extra decay at infinity has the following quantitative localization principle as a consequence. The proof is short enough to be given here.

Corollary 1.6.

If $f$ in $L^{p}$ vanishes near $x_{0}$ , then $E_{\alpha,\mu}f(x_{0})=\mathcal{O}(\mu^{-\frac{\alpha}{2}})$ . In particular, if $I_{\alpha}f$ is in $L^{p}$ and vanishes near $x_{0}$ , then $G_{\alpha,\mu}f(x_{0})=\mathcal{O}(\mu^{-\frac{\alpha}{2}})$ .

Proof.

By translation invariance, we may assume that $x_{0}=0$ , and $\delta>0$ is such that $f=0$ for $|x|<\delta$ . If $\mu\delta>1$ , Theorem 1.5 applied to $m_{\alpha,\mu}(\xi)$ gives

|E_{\alpha,\mu}f(0)|=\left|\int_{|y|>\delta}K_{\alpha,\mu}(-y)f(y)\,dy\right|\leq\frac{C}{\mu^{\frac{\alpha}{2}}}\int_{|y|>\delta}\frac{|f(y)|}{|y|^{d+\frac{\alpha}{2}}}\,dy,

where $K_{\alpha,\mu}$ is its kernel. By Hölder’s inequality,

|E_{\alpha,\mu}f(0)|\leq C_{d,p}\mu^{-\frac{\alpha}{2}}\delta^{-(\frac{d}{p}+\frac{\alpha}{2})}\|f\|_{p}\leq C_{d,\delta,p}\mu^{-\frac{\alpha}{2}}\|f\|_{p}.

The proof is completed by appealing to the identity $G_{\alpha,\mu}f=E_{\alpha,\mu}I_{\alpha}f$ . ∎

The main idea in this paper is ultimately a simple one—to compare two operators, examine their quotient! Unfortunately, it gets bogged down in a morass of notation and computations. Luckily they are fairly straightforward. In fact, the deepest results used are the (Hörmander–Mikhlin) multiplier theorem and the equivalence of $K$ -functionals as in [9].

It would be interesting to know if similar approximation and kernel estimates hold for, say, $\frac{\sqrt{L(x,D)}}{\sqrt{\mu^{2}+L(x,D)}}$ where $L(x,D)$ is a linear second order differential operator. Indeed, it may be worth seeking sharp kernel estimates for pseudo-differential operators, i.e those with non-constant coefficient symbols, satisfying appropriate variants of (10). We hope to treat these and other issues elsewhere.

Before leaving this section we mention that the Bessel–Riesz quotient also surfaced in the author’s unpublished study of transmission boundary value problems with a large parameter. Models of scattering by high contrast materials are good examples to keep in mind here. In these problems, the unknown Dirichlet and Neumann traces of the solution on the “interface” are related by operators similar to the Bessel–Riesz quotient. This was separate motivation for examining this operator in detail.

An outline of the paper now follows. We first introduce some notation and definitions in the upcoming subsection. The proof of Theorem 1.1 is presented in §2 along with some extensions to Hardy spaces. As indicated earlier, the Saturation Theorems are proven in §3. The “Fourier Analytic” proof of Theorem 1.5 is given in §4 and shown to apply to a wider class of symbols.

Notation

Everything takes place in $d$ –dimensional Euclidean space $\mathbb{R}^{d}$ and for $1\leq p\leq\infty$ , $L^{p}=L^{p}(\mathbb{R}^{d})$ are the usual Lebesgue spaces with norm denoted by $\|f\|_{p}$ .

For a non-negative integer $k$ , the Sobolev space $W^{k}_{p}$ consists of $L^{p}$ functions having distributional derivatives up to order $k$ in $L^{p}$ . In $W^{k}_{p}$ we use the norm $\|f\|_{W^{k}_{p}}=\sum_{|\gamma|\leq k}\|D^{\gamma}f\|_{p}$ , and seminorm $|f|_{W^{k}_{p}}=\sum_{|\gamma|=k}\|D^{\gamma}f\|_{p}$ .

The direct and inverse Fourier transform of $f$ and $\hat{g}$ respectively defined as

\hat{f}(\xi)=\int e^{-ix\cdot\xi}f(x)\,dx;\qquad\check{g}(x)=(2\pi)^{-d}\int e^{ix\cdot\xi}\hat{g}(\xi)\,d\xi.

When convenient, we also use $\mathcal{F}_{x\to\xi}$ and $\mathcal{F}_{\xi\to x}$ for the direct and inverse transform. For suitable functions $a(\xi)$ , we associate the operator

a(D)f(x)=(2\pi)^{-d}\int e^{ix\cdot\xi}a(\xi)\widehat{f}(\xi)\,d\xi.

The Bessel potential space $\mathcal{L}_{p}^{s}$ is defined as $\mathcal{L}_{p}^{s}:=\{G_{s}\star f:f\in L^{p}\}$ . Here $G_{s}(x)$ is the Bessel potential defined by

G_{s}(x)=\mathcal{F}_{\xi\to x}\left[(1+|\xi|^{2})^{-\frac{s}{2}}\right].

Occasionally we need the “dilated” Bessel kernel defined, for $t>0$ , by

\mathcal{J}_{s}(x,t)=t^{d}G_{s}(tx).

For $0<s<1$ , $1\leq p<\infty$ and $1\leq q\leq\infty$ , we define the Besov spaces $B^{s}_{p,q}$ as those $f\in L^{p}$ for which the seminorm

|f|_{B^{s}_{p,q}}:=\left(\int_{0}^{1}(t^{-s}\omega(f,t)_{p})^{q}\frac{dt}{t}\right)^{1/q}<\infty.

Equipped with the norm $\|f\|_{B^{s}_{p,q}}=\|f\|_{p}+|f|_{B^{s}_{p,q}}$ this becomes a Banach space. $\text{Lip}(s,p)$ is the set of $L^{p}$ functions which satisfy $\omega(f,t)_{p}=\mathcal{O}(t^{s})$ . For $0<s<1$ , $\text{Lip}(s,p)=B^{s}_{p,\infty}$ while $\text{Lip}(1,p)=W^{1}_{p}$ .

For more on these function spaces we refer the reader to [13].

2 Order of Approximation

2.1 The $L^{p}$ Case when $0<\alpha\leq 1$

Recall from the Introduction that $A_{\alpha,\mu}(z)>0$ and $\|A_{\alpha,\mu}(z)\|_{1}=1$ . Thus

E_{\alpha,\mu}f(x)=f(x)-\int_{\mathbb{R}^{d}}A_{\alpha,\mu}(x-y)f(y)\,dy=\int_{\mathbb{R}^{d}}A_{\alpha,\mu}(x-y)(f(x)-f(y))\,dy.

Minkowski’s inequality and a change of variables readily show

\|E_{\alpha,\mu}f\|_{p}\leq\int_{\mathbb{R}^{d}}A_{\alpha,1}(y)\|f(\cdot)-f(\cdot-y/\mu)\|_{p}\,dy\leq\int_{\mathbb{R}^{d}}A_{\alpha,1}(y)\omega(f,|y|/\mu)_{p}\,dy.

To simplify the notation we set $A_{\alpha}(z):=A_{\alpha,1}(z)=\sum_{j=1}^{\infty}a_{\alpha,j}G_{2j}(z)$ . Some properties of $a_{\alpha,j}$ and $G_{2j}(z)$ are summarized next.

Lemma 2.1.

(a)

$G_{2j}(z)$ is positive, radial and decreasing with $\|G_{2j}\|_{L^{1}}=1$ . Moreover, $\displaystyle G_{2j}(z)=\frac{1}{2^{\frac{d+2j-2}{2}}\pi^{\frac{d}{2}}\Gamma\left(j\right)}K_{\frac{d-2j}{2}}(|z|)|z|^{\frac{2j-d}{2}}$ , where $K_{\nu}(t)$ is the modified Bessel function of the third kind.
(b)

We have $a_{\alpha,j}>0$ and $a_{\alpha,j}\leq C_{\alpha}j^{-1-\frac{\alpha}{2}}$ with $\sum_{j=1}^{\infty}a_{\alpha,j}=1$ .
(c)

$\displaystyle\int_{\mathbb{R}^{d}}G_{2j}(y)|y|^{s}\,dy=C_{d,s}\frac{\Gamma\left(j+\frac{s}{2}\right)}{\Gamma(j)}$ for some constant $C_{d,s}$ . In addition, for $0\leq s<\alpha$ , $\displaystyle\int_{\mathbb{R}^{d}}A_{\alpha}(y)|y|^{s}\,dy$ converges.

Proof.

(a)

These are proved in Aronszajn–Smith [2, pgs. 413–421].

(b)

We use the binomial expansion $\left(1-t\right)^{\alpha/2}=1-\sum_{j=1}^{\infty}a_{\alpha,j}t^{j}$ , where

a_{\alpha,j}=\left|\binom{\alpha/2}{j}\right|=\frac{1}{\Gamma(-\frac{\alpha}{2})j^{1+\frac{\alpha}{2}}}\left(1+o(1)\right)\leq C_{\alpha}j^{-1-\frac{\alpha}{2}}.

It follows that $\sum_{j=1}^{\infty}a_{\alpha,j}t^{j}$ converges absolutely for $\left|t\right|\leq 1$ . Plugging $t=1$ into $\left(1-t\right)^{\alpha/2}$ shows that $\sum_{j=1}^{\infty}a_{\alpha,j}=1$ .

(c)

For $|\text{Re}(\nu)|<\text{Re}(\beta)$ , we use the formula [7, Eq. 10.43.19]:

\int_{0}^{\infty}t^{\beta-1}K_{\nu}(t)\,dt=2^{\beta-2}\Gamma\left(\frac{\beta+\nu}{2}\right)\Gamma\left(\frac{\beta-\nu}{2}\right).

(11)

A switch to spherical coordinates combined with part (a) and (11) gives

\int_{\mathbb{R}^{d}}G_{2j}(y)|y|^{s}\,dy=\frac{2^{2-j-\frac{d}{2}}}{\Gamma(\frac{d}{2})\Gamma(j)}\int_{0}^{\infty}t^{j+\frac{d}{2}+s-1}K_{j-\frac{d}{2}}(t)\,dt=\frac{2^{s}\Gamma(\frac{d+s}{2})}{\Gamma(\frac{d}{2})}\cdot\frac{\Gamma\left(j+\frac{s}{2}\right)}{\Gamma(j)}.

Since $\Gamma(x+a)\sim\Gamma(x)x^{a}$ for large $x$ , and $a_{j}\leq Cj^{-1-\alpha/2}$ we have

\int_{\mathbb{R}^{d}}A_{\alpha}(y)|y|^{s}\,dy=\sum_{j=1}^{\infty}a_{\alpha,j}\int_{\mathbb{R}^{d}}G_{2j}(y)|y|^{s}\,dy=C_{s,d}\sum_{j=1}^{\infty}a_{\alpha,j}\frac{\Gamma\left(j+\frac{s}{2}\right)}{\Gamma(j)}\leq C\sum_{j=1}^{\infty}j^{-\frac{(2+\alpha-s)}{2}},

which converges when $0\leq s<\alpha$ .

∎

We warm–up with a computation in the case $\alpha=1$ . Set $B(\xi,\mu)=\ln(1+|\xi/\mu|^{2})$ . The inequality $1-e^{-t}\leq\min\{1,t\}$ for $t>0$ shows

\displaystyle m_{1,\mu}(\xi)=\sum a_{1,j}\left(1-(1+|\xi/\mu|^{2})^{-j}\right)=\sum a_{1,j}\left(1-e^{-jB}\right)\leq\sum a_{1,j}\min\{1,jB\},

and split the resulting sum to see that

m_{1,\mu}(\xi)\leq\sum_{j\leq B^{-1}}a_{1,j}jB+\sum_{j\geq B^{-1}}a_{1,j}.

The bound for $a_{1,j}$ in Lemma 2.1(b) and comparison with an integral now yields

\displaystyle m_{1,\mu}(\xi)

\displaystyle\leq B\int_{0}^{\frac{1}{B}}\frac{1}{\sqrt{t}}\,dt+\int_{\frac{1}{B}}^{\infty}\frac{1}{t^{\frac{3}{2}}}\,dt\leq 2\sqrt{B(\xi,\mu)}.

This suggests a Kolmogorov–Seliverstov–Plessner type result.

Proposition 2.2.

If $\|\sqrt{\ln(1+|\xi|^{2})}\widehat{f}(\xi)\|_{2}<\infty$ , then $\|E_{1,\mu}f\|_{{}_{2}}=\mathcal{O}((\ln\mu)^{-\frac{1}{2}})$ as $\mu\to\infty$ .

Proof.

By Plancherel’s formula

\displaystyle\|E_{1,\mu}f\|^{2}_{{}_{2}}\leq\int\frac{|\xi|^{2}}{(|\xi|^{2}+\mu^{2})\ln(1+|\xi|^{2})}\ln(1+|\xi|^{2})|\widehat{f}(\xi)|^{2}\,d\xi.

${r^{2}}/((r^{2}+\mu^{2})\ln(1+r^{2}))$ attains its maximum when $r\approx\mu$ completing the proof. ∎

Our next result extends this exercise and contains Theorem 1.1 (b) and (c).

Theorem 2.3.

Assume $1\leq p<\infty$ .

(i)

If $\alpha=1$ , there is a $C>0$ depending only on $d$ such that

\|E_{1,\mu}f\|_{p}\leq C\omega(f,1/\mu)_{p}\left(3+2\ln\left(\frac{||f||_{p}}{2\omega(f,1/\mu)_{p}}\right)\right).

(ii)

If $0<\alpha<1$ , there is a constant $C$ depending only on $d$ and $\alpha$ such that

$\|E_{\alpha,\mu}f\|_{p}\leq C\left(\omega(f,1/\mu)_{p}+(\omega(f,1/\mu)_{p})^{\alpha}\cdot||f||_{p}^{1-\alpha}\right)$

Proof.

Recall that $\displaystyle A_{\alpha}(z)=\sum_{j=1}^{\infty}a_{\alpha,j}G_{2j}(z)$ and that $\displaystyle\|E_{\alpha,\mu}f\|_{p}\leq\int A_{\alpha}(y)\omega(f,|y|/\mu)_{p}\,dy$ . Let $R>0$ be a large number to be chosen shortly. It follows that

	$\displaystyle\\|E_{\alpha,\mu}f\\|_{p}$	$\displaystyle\leq\sum_{j=1}^{\infty}a_{\alpha,j}\int_{\mathbb{R}^{d}}\omega\left(f,\|y\|/\mu\right)_{p}G_{2j}(y)\,dy$
		$\displaystyle\leq\sum_{j\leq R}a_{\alpha,j}\int_{\mathbb{R}^{d}}\omega\left(f,\|y\|/\mu\right)_{p}G_{2j}(y)\,dy+\sum_{j>R}a_{\alpha,j}\int_{\mathbb{R}^{d}}\omega\left(f,\|y\|/\mu\right)_{p}G_{2j}(y)\,dy.$

For the first sum we use the inequality $\omega(f,\gamma t)_{p}\leq(1+|\gamma|)\omega(f,t)_{p}$ . In the second sum we use $\omega(f,t)_{p}\leq 2||f||_{p}$ . Altogether

\|E_{\alpha,\mu}f\|_{p}\leq\omega\left(f,1/\mu\right)_{p}\sum_{j\leq R}a_{\alpha,j}\int_{\mathbb{R}^{d}}(1+|y|)G_{2j}(y)\,dy+2\|f\|_{p}\sum_{j>R}a_{\alpha,j}\int_{\mathbb{R}^{d}}G_{2j}(y)\,dy.

By Lemma 2.1(c),

\|E_{\alpha,\mu}f\|_{p}\leq c_{d}\omega\left(f,1/\mu\right)_{p}\sum_{j\leq R}a_{\alpha,j}(1+j^{\frac{1}{2}})+2\|f\|_{p}\sum_{j\geq R}a_{\alpha,j}.

We can now split the argument into the two cases.

(i)

The case $\alpha=1$ .

We know that $a_{1,j}\leq cj^{-3/2}$ from Lemma 2.1(b) and can compare sums to integrals to deduce

$\|E_{1,\mu}f\|_{p}\leq c_{d}\left(\omega\left(f,1/\mu\right)_{p}(1+\ln R)+2\|f\|_{p}R^{-\frac{1}{2}}\right).$ (12)

The choice $R=\left(||f||_{p}/2\omega(f,1/\mu)_{p}\right)^{2}$ minimizes (12) and completes the proof in this case.

(ii)

The case $0<\alpha<1$ .

Here $a_{\alpha,j}\leq c_{\alpha}j^{-1-\frac{\alpha}{2}}$ and this time the integral test yields

\|E_{\alpha,\mu}f\|_{p}\leq c_{\alpha,d}\left(\omega\left(f,1/\mu\right)_{p}(1+R^{\frac{1}{2}-\frac{\alpha}{2}})+\|f\|_{p}R^{-\frac{\alpha}{2}}\right).

(13)

This is minimized by the choice $R=\left(\dfrac{\alpha||f||_{p}}{(1-\alpha)\omega(f,1/\mu)_{p}}\right)^{2}$ .

∎

2.2 Proof of Theorem 1.1 (a)

In this and the next two sections, we take $\alpha=1$ and to simplify the notation we set $E_{\mu}:=E_{1,\mu}$ , $T_{\mu}:=T_{1,\mu}$ and $A(x):=A_{1}(x)$ . To motivate the proof, let us quickly estimate the order of approximation for functions in Besov spaces. Assume first that $f\in B^{s}_{p,\infty}:=\text{Lip}(s,p)$ with $0<s<1$ and $1\leq p<\infty$ . Recall that functions in $\text{Lip}(s,p)$ satisfy $\omega(f,t)_{p}=\mathcal{O}(t^{s})$ . Arguing as in the above proof, but using Lemma 2.1(c)

	$\displaystyle\\|E_{\mu}f\\|_{p}\leq\int A(y)\\|f(\cdot)-f(\cdot-y/\mu)\\|_{p}\,dy$	$\displaystyle\leq\mu^{-s}\|f\|_{B^{s}_{p,\infty}}\int A(y)\|y\|^{s}\,dy$
		$\displaystyle\leq\mu^{-s}C_{d,s}\|f\|_{B^{s}_{p,\infty}}.$

As the embedding $B^{s}_{p,q}\hookrightarrow B^{s}_{p,\infty}$ holds, the same estimate applies to $f\in B^{s}_{p,q}$ .

To handle the case $s=1$ we restrict to $1<p<\infty$ . Note that $\omega(f,t)_{p}=O(t)$ implies $f\in\mathcal{L}_{p}^{1}=W^{1}_{p}$ (see [13, pgs. 135–139]). Thus, for some $g\in L^{p}$ , $(1+|\xi|^{2})^{1/2}\widehat{f}(\xi)=\widehat{g}(\xi)$ and $E_{\mu}f=\mu^{-1}E_{1}(\mathcal{J}_{1}(\cdot,\mu)\star g)$ where $\mathcal{J}_{s}(x,t)=t^{d}G_{s}(tx)$ is the dilated Bessel kernel defined in the Notation section. From this

\|E_{\mu}f\|_{p}=\mu^{-1}\|E_{1}(\mathcal{J}_{1}(\cdot,\mu)\star g))\|_{p}\leq C\mu^{-1}\|g\|_{p}\leq C\mu^{-1}\|f\|_{W^{1}_{p}(\mathbb{R}^{d})}.

The first inequality follows from the $L^{p}$ boundedness of both $E_{1}$ and convolution with $\mathcal{J}_{s}(x,t)$ . This argument combined with Theorem 2.3 imply results for function in various spaces. For simplicity, we state those that apply to functions in the Lipschitz classes $\text{Lip}(s,p)$ .

Corollary 2.4.

If $f\in\text{Lip}(s,p)$ then

\|E_{\mu}f\|_{p}\leq C\begin{cases}\mu^{-s},&p>1,\quad 0<s\leq 1;\\ \mu^{-s},&p=1,\quad 0<s<1;\\ \mu^{-1}\ln(\mu),&p=1,\quad s=1.\end{cases}

Corollary 2.4 improves on Theorem 2.3, but the sharp result is Theorem 1.1 as it asserts the equivalence $\|E_{\mu}f\|_{p}\approx\omega(f,1/\mu)_{p}$ for $1<p<\infty$ . The idea is to use $K$ –functionls as an intermediary. We establish the equivalence between the order of approximation and a $K$ –functional. Known relationships between $K$ –functionals and the modulus of continuity allow us to complete the proof.

As in Ditzian–Ivanov [6], we introduce the $K$ –functional

K(t,f,|D|)_{p}:=\inf_{\begin{subarray}{c}g\in L^{p}\\ |D|g\in L^{p}\end{subarray}}\left(\|f-g\|_{p}+t\||D|g\|_{p}\right).

(14)

Here is the main step in proving the equivalence theorem.

Lemma 2.5.

$K(1/\mu,f,|D|)_{p}\approx\|E_{\mu}f\|_{p}$ , for $1<p<\infty$ .

Proof.

If both $g,|D|g\in L^{p}$ ,

\|E_{\mu}g\|_{p}=\frac{1}{\mu}\|\mathcal{J}_{1}(\cdot,\mu)\star|D|g\|_{p}\leq\frac{1}{\mu}\||D|g\|_{p}\leq\mu^{-1}\||D|g\|_{p}.

This in turn implies

\|E_{\mu}f\|_{p}\leq\|E_{\mu}(f-g)\|_{p}+\|E_{\mu}g\|_{p}\leq\|f-g\|_{p}+\mu^{-1}\||D|g\|_{p}.

Taking the infimum over such $g$ gives $\|E_{\mu}f\|_{p}\leq K(1/\mu,f,|D|)_{p}$ which is one direction of the result.

We turn to the opposite inequality. Set $g=T_{\mu}f$ . We only need show that $\mu^{-1}\||D|g\|_{p}:=\mu^{-1}\||D|T_{\mu}f\|_{p}\leq C\|f-T_{\mu}f\|_{p}$ . On the Fourier transform side

	$\displaystyle\mu^{-1}\widehat{\|D\|T_{\mu}f}(\xi)=\frac{\|\xi\|}{\mu}\left(1-\frac{\|\xi\|}{(\mu^{2}+\|\xi\|^{2})^{\frac{1}{2}}}\right)\widehat{f}(\xi)$	$\displaystyle=\frac{\mu}{((\mu^{2}+\|\xi\|^{2})^{\frac{1}{2}}+\|\xi\|)}\cdot\frac{\|\xi\|\widehat{f}(\xi)}{(\mu^{2}+\|\xi\|^{2})^{\frac{1}{2}}}$
		$\displaystyle:=r(\xi)\widehat{E_{\mu}f}(\xi),$

and we only need show that $r(\xi)$ defines a bounded operator on $L^{p}$ . A direct computation shows that for $\xi\neq 0$

\left|\frac{\partial r}{\partial\xi_{k}}\right|=\left|-\mu((\mu^{2}+|\xi|^{2})^{\frac{1}{2}}+|\xi|)^{-2}\cdot\left(\frac{\xi_{k}}{|\xi|}+\frac{\xi_{k}}{(\mu^{2}+|\xi|^{2})^{\frac{1}{2}}}\right)\right|\leq\frac{2}{|\xi|}.

For any multi-index $\gamma$ , this can be extended to $|\partial^{\gamma}r(\xi)|\leq C_{\gamma}|\xi|^{-|\gamma|}$ . The multiplier theorem (see [14]) shows that $r(D)$ is $L^{p}$ bounded for $1<p<\infty$ . Hence, for $1<p<\infty$ , we obtain $\mu^{-1}\||D|T_{\mu}f\|_{p}\leq C\|f-T_{\mu}f\|_{p}$ . Thus

K(1/\mu,f,|D|)_{p}\leq\|f-T_{\mu}f\|_{p}+\mu^{-1}\||D|T_{\mu}f\|_{p}\leq C\|E_{\mu}f\|_{p},

concluding the proof. ∎

We need one additional result.

Lemma 2.6.

For $1<p<\infty$ , $g\in W^{1}_{p}$ if and only if $g$ and $|D|g$ are in $L^{p}$ .

Proof.

Using the Riesz transforms $R_{j}$ , we can write $D_{j}g=R_{j}(|D|g)$ , and the boundedness of $R_{j}$ implies that $D_{j}g\in L^{p}$ whenever $|D|g\in L^{p}$ if $1<p<\infty$ . Note that $R_{j}$ is unbounded in $L^{1}$ and $L^{\infty}$ and so we cannot include the case $p=1,\infty$ . For the converse, suppose that $g\in W^{1}_{p}$ . Then $g=G_{1}\star h$ for some $h\in L^{p}$ . By definition, $\widehat{|D|g}(\xi)=|\xi|(|\xi|^{2}+1)^{-\frac{1}{2}}\widehat{h}(\xi)$ , so that $|D|g=E_{1}h$ . As $E_{1}$ is $L^{p}$ bounded, $\||D|g\|_{p}<\infty$ . ∎

Proof of Theorem 1.1(a).

We apply the result of Johnen–Scherer, [9], on the equivalence of moduli of continuity and $K$ –functionals. If we define

K(t,f,L^{p},W^{1}_{p})=\inf_{g\in W^{1}_{p}}\left(||f-g||_{p}+t\sup_{|\gamma|=1}||D^{\gamma}g||_{p}\right),

their result is that $K(t,f,L^{p},W^{1}_{p})\approx\omega(f,t)_{p}$ for $1\leq p\leq\infty$ . However, Lemma 2.6 shows that when $g\in W^{1}_{p}$ , we have $\sup_{|\gamma|=1}\|D^{\gamma}g\|_{p}\approx\||D|g\|_{p}$ for $1<p<\infty$ . This implies $K(t,f,|D|)\approx K(t,f,L^{p},W^{1}_{p})$ , and we have shown

\|E_{\mu}f\|_{p}\approx K(1/\mu,f,|D|)\approx K(1/\mu,f,L^{p},W^{1}_{p})\approx\omega(f,1/\mu)_{p}.

∎

2.3 Extension to $H^{p}$ when $\alpha=1$

We denote the real Hardy spaces by $H^{p}(\mathbb{R}^{d})$ . For $p>1$ , they coincide with the $L^{p}$ spaces. For $0<p\leq 1$ , it is a normed space of distributions. We denote the norm by $\|\cdot\|_{H^{p}}$ in this section. A far more thorough exposition on these spaces can be found in [14, Chap. 3] and we only state the bare minimum required.

The analog of Theorem 1.1 for functions in Hardy spaces is the following.

Theorem 2.7.

For $0<p\leq 1$ , we have $\displaystyle\|E_{\mu}f\|_{{H}^{p}}\leq C_{p}\omega(f,1/\mu)_{{H}^{p}}$ .

For $f$ in $H^{p}(\mathbb{R}^{d})$ , $\omega(f,t)_{{H}^{p}}:=\sup_{|h|\leq t}||f(\cdot+h)-f(\cdot)||_{H^{p}}$ is the $H^{p}$ modulus of continuity. We require a result of Colzani [5, Theorem 4.1] in the proof and to state it we need some notation. Let $\phi\in C^{\infty}(\mathbb{R}^{d})$ be supported in $|\xi|\leq 2$ , with $\phi(\xi)=1$ for $|\xi|\leq 1$ . and define $\displaystyle\widehat{\Phi_{s}\star f}(\xi)=\phi\left(\xi/s\right)\widehat{f}(\xi)$ .

Lemma 2.8.

$||f-\Phi_{\mu}\star f||_{{H}^{p}}\leq C\omega(f,1/\mu)_{{H}^{p}}$ ,

We also repeatedly use the “multiplier theorem for Hardy spaces” [13, §7.4.9].

Lemma 2.9.

Let $k\in\mathbb{N}$ . Then $m(\xi)$ is $H^{p}$ bounded if $m\in L^{\infty}$ and, for $|\beta|\leq k$ and $k>d/p$ , satisfies

\sup_{0<A<\infty}A^{2|\beta|-d}\int_{A<|\xi|\leq 2A}|\partial^{\beta}m(\xi)|^{2}\,d\xi\leq B.

Proof of Theorem 2.7.

$E_{\mu}$ is ${H}^{p}$ bounded as the estimate $|\partial^{\beta}m_{\mu}(\xi)|\leq C_{\beta}|\xi|^{-\beta}$ and the pointwise boundedness $m_{\mu}(\xi)$ are enough to verifiy the conditions of Lemma 2.9. From here we check that

	$\displaystyle\\|E_{\mu}f\\|_{{H}^{p}}=\\|E_{\mu}(f-\Phi_{\mu}\star f+\Phi_{\mu}\star f)\\|_{{H}^{p}}$	$\displaystyle\leq C\\|f-\Phi_{\mu}\star f\\|_{{H}^{p}}+\\|E_{\mu}(\Phi_{\mu}\star f)\\|_{{H}^{p}}$
		$\displaystyle\leq C\omega(f,1/\mu)_{{H}^{p}}+\\|E_{\mu}(\Phi_{\mu}\star f)\\|_{{H}^{p}},$

where we used the $H^{p}$ boundedness of $E_{\mu}$ in the second line and Colzani’s result, Lemma 2.8, in the third line. Thus the proof reduces to showing that $\|E_{\mu}(\Phi_{\mu}\star f)\|_{{H}^{p}}\leq C\omega(f,1/\mu)_{{H}^{p}}$ .

To prove this, we modify another argument of Colzani [5, Theorem 5.1]. As in that paper, we can assume that $\widehat{f}$ is supported in $(\overline{R_{+}})^{d}=\{x_{k}\geq 0;\,1\leq k\leq d\}$ . Let $\theta\in S^{d-1}$ be a fixed unit vector with $\theta$ in a proper conic subset of $(R_{+})^{d}$ . We can thus find an $\epsilon>0$ , such that $\theta^{\perp}$ , the plane through the origin orthogonal to $\theta$ , lies outside the $\epsilon$ conic neighborhood of $(\overline{R_{+}})^{d}$ . Let $\chi(\xi)$ be smooth, homogenous of degree $0$ , identically 1 on $(\overline{R_{+}})^{d}$ and vanish outside of the $\epsilon/2$ conic neighborhood of $(\overline{R_{+}})^{d}$ . We use Colzani’s trick and write

\widehat{E_{\mu}(\Phi_{\mu}\star f)}(\xi)=\chi(\xi)m_{\mu}(\xi)\phi\left(\xi/\mu\right)(e^{i\frac{\xi\cdot\theta}{\mu}}-1)^{-1}\cdot(e^{i\frac{\xi\cdot\theta}{\mu}}-1)\widehat{f}(\xi)

We exploit the homogeneity of $\chi$ and define two auxilary functions

\psi(z):=\frac{|z|\chi(z)\phi(z)}{(|z|^{2}+1)^{\frac{1}{2}}(e^{iz\cdot\theta}-1)},\quad\widetilde{\psi}(z):=\psi(z/\mu).

If $\widetilde{\psi}(\xi)$ were $H^{p}$ bounded, we would have

\|E_{\mu}(\Phi_{\mu}\star f)\|_{{H}^{p}}=\|\widetilde{\psi}(D)\left[f(\cdot+\theta/\mu)-f(\cdot)\right]\|_{{H}^{p}}\leq C_{p}\|f(\cdot+\theta/\mu)-f(\cdot)\|_{{H}^{p}},

which of course implies that $\|E_{\mu}(\Phi_{\mu}\star f)\|_{{H}^{p}}\leq C_{p}\omega(f,1/\mu)_{{H}^{p}}$ by the definition of the $H^{p}$ modulus of continuity. The proof thus reduces to checking that $\widetilde{\psi}$ satisfies the conditions of Lemma 2.9. Those conditions are dilation invariant so it is enough to check them for $\psi$ . To show boundedness, we focus on the behavior near $\theta^{\perp}$ where $e^{iz\cdot\theta}-1=0$ . Excluding the origin, $\chi$ vanishes on $\theta^{\perp}$ by construction. As $\lim_{z\to 0}\psi(z)$ exists, we see that $\psi$ is bounded. For the derivatives, we use the product rule

\partial^{\beta}\psi(z)=\sum_{\beta_{1}+\ldots+\beta_{4}=\beta}\binom{\beta}{\beta_{1}\cdots\beta_{4}}\partial^{\beta_{1}}\chi\partial^{\beta_{2}}\phi\partial^{\beta_{3}}\left(\frac{1}{e^{iz\cdot\theta}-1}\right)\partial^{\beta_{4}}\left(\frac{|z|}{(1+|z|^{2})^{\frac{1}{2}}}\right).

Straightforward estimates using the support of $\phi$ , and the homogeneity of $\chi$ , yield $|\partial^{\beta}\psi(z)|\leq C_{\epsilon,\beta}|z|^{-\beta}$ . Applying Lemma 2.9 completes the proof. ∎

3 Saturation Theorems for $\alpha=1$

In this section we show there is a limit to how well $T_{1,\mu}$ approximates functions in $L^{p}$ . We also characterize the class of functions which achieve the optimal approximation rate. To see the main idea, note that

m_{1,\mu}(\xi)\widehat{f}(\xi)=\dfrac{\left|\xi\right|}{(\mu^{2}+\left|\xi\right|^{2})^{\frac{1}{2}}}\widehat{f}(\xi)=\frac{|\xi|\widehat{f}(\xi)}{(|\xi|^{2}+\mu^{2})^{\frac{1}{2}}}.

The numerator is bounded in $L^{p}$ when $\nabla f$ is in $L^{p}$ while the denominator is $\mathcal{O}(1/\mu)$ . We thus expect that functions with a derivative in $L^{p}$ should have approximation error that is $\mathcal{O}(1/\mu)$ . Though not rigorous, it is not far from the truth. The arguments to follow dress this observation in functional analytic clothing. Parts are adapted from Butzer–Nessel [4]. Once again $E_{\mu}:=E_{1,\mu}$ .

Theorem 3.1.

For $1\leq p<\infty$ , $\|E_{\mu}f\|_{p}=o(1/\mu)$ implies $f=0$ in $L^{p}$ .

Proof.

The proof is broken into cases.

(a)

When $p=1$ , both $\widehat{f}(\xi)$ and $\widehat{E_{\mu}f}(\xi)$ are continuous functions. By assumption, $\mu|\widehat{E_{\mu}f}(\xi)|\leq\mu\|E_{\mu}f\|_{1}$ so $\lim_{\mu\to\infty}\mu|\widehat{E_{\mu}f}(\xi)|=0$ holds pointwise. A calculus argument gives $\lim_{\mu\to\infty}\mu\widehat{E_{\mu}f}(\xi)=|\xi|\widehat{f}(\xi)$ . This implies $\widehat{f}(\xi)=0$ almost everywhere finishing the proof. Incidentally, we showed that $\lim_{\mu\to\infty}\mu E_{\mu}f=|D|f$ . This remark is used below.
(b)

For the case $1<p\leq 2$ , we use the Hausdorff–Young inequality to arrive at $\|\mu\widehat{E_{\mu}f}\|_{p^{\prime}}\leq\|\mu E_{\mu}f\|_{p}=o(1)$ . If we define the sequence $g_{n}(\xi):=n\widehat{E_{n}f}(\xi)$ , we see that $\lim g_{n}=0$ in $L^{p^{\prime}}$ and some subsequence satisfies $\lim g_{n_{k}}=0$ almost everywhere. But the earlier calculus argument and the uniqueness of limits implies $0=\lim_{k}g_{n_{k}}=\lim_{k}n_{k}\widehat{E_{n_{k}}f}=|\xi|\widehat{f}(\xi)$ , and again $\widehat{f}(\xi)=0$ almost everywhere.

(c)

For the case $p>2$ we use a duality argument. As $1<p^{\prime}<2$ , we know that $\varphi\in W^{1}_{p^{\prime}}$ implies $|D|\varphi\in L^{p^{\prime}}$ . The functional on $W^{1}_{p^{\prime}}$ defined by $L_{\mu,f}(\varphi):=\int\mu E_{\mu}f(x)\varphi(x)\,dx$ is easily seen to be a bounded linear functional with norm $o(1)$ . Two application of the dominated convergence theorem shows that

0=\lim_{\mu\to\infty}L_{\mu,f}(\varphi)=\lim_{\mu\to\infty}\int\mu E_{\mu}f\varphi=\lim_{\mu\to\infty}\int f\mu E_{\mu}\varphi=\int f\cdot|D|\varphi.

As a linear functional, $f$ vanishes on the dense subspace $W^{1}_{p^{\prime}}$ so $f=0$ .

∎

We now describe the functions which attain the optimal order $\|E_{\mu}f\|_{p}=\mathcal{O}(1/\mu)$ .

Theorem 3.2.

For $f\in L^{p}$ , $\|E_{\mu}f\|_{p}=\mathcal{O}(1/\mu)$ holds if and only if

(a)

$f\in W^{1}_{p}$ when $1<p<\infty$ .
(b)

$|\xi|\widehat{f}(\xi)=\widehat{\nu}(\xi)$ , for some bounded measure $\nu$ when $p=1$ .

Proof.

(a)

In view of Theorem 1.1, such a function satisfies $\omega(f,t)_{p}=\mathcal{O}(t)$ for $1<p<\infty$ . This defines $W^{1}_{p}$ when $1<p<\infty$ .
(b)

In proving Theorem 3.1, we showed that $\lim\limits_{\mu\to\infty}\mu E_{\mu}f=|D|f$ . If $\|E_{\mu}f\|_{1}=\mathcal{O}(1/\mu)$ , then $d\nu_{n}=nE_{n}f(x)\,dx$ is a bounded sequence of Radon measures. This sequence satisfies $\sup_{n}|\nu_{n}|(\mathbb{R}^{d})<\infty$ . By weak compactness (see [8, pgs. 54–55]), we can extract a limit, call it $\nu$ , also satisfying $|\nu|(\mathbb{R}^{d})<\infty$ . As a result, $|D|f=\nu$ or $|\xi|\widehat{f}(\xi)=\widehat{\nu}(\xi)$ which is one direction of (b).

If $f\in L^{1}$ and $|D|f=\nu$ , for some bounded measure $\nu$ , then

$\|E_{\mu}f\|_{1}=\mu^{-1}\|(\mathcal{J}_{1}(\cdot,\mu)\star\nu\|_{1}\leq\mu^{-1}|\nu|(\mathbb{R}^{d}),$

which proves the other half of (b).

∎

4 Kernel Estimates and Localization

We prove Theorem 1.5 in this section. The proof is based on a Littlewood–Paley type argument as in Stein [14, pgs. 241-247]. We modify the standard argument by replacing the Hörmander–Mikhlin condition (9) with the condition (10): $\,\,|\partial^{\beta}b(\xi)|\leq C_{\beta}|\xi|^{\alpha-|\beta|}(\mu^{2}+|\xi|^{2})^{-\frac{\alpha}{2}}$ .

Proof of Theorem 1.5.

Let $1=\sum_{j\in\mathbb{Z}}\phi(2^{-j}\xi)$ be a Littlewood–Paley partition of unity. Put

B_{j}(x)=\int e^{ix\cdot\xi}\phi(2^{-j}\xi)b(\xi)\,d\xi

For any multi-indices $\beta$ and $\gamma$ we see

\left|x^{\beta}(-i\partial_{x})^{\gamma}B_{j}(x)\right|=\left|\int x^{\beta}e^{ix\xi}\xi^{\gamma}\phi(2^{-j}\xi)b(\xi)\,d\xi\right|\leq\int\left|\partial_{\xi}^{\beta}(\xi^{\gamma}\phi(2^{-j}\xi)b(\xi))\right|\,d\xi.

The product rule, (10), and direct integration gives

\left|x^{\beta}(-i\partial_{x})^{\gamma}B_{j}(x)\right|\leq C_{\gamma,\beta,d}2^{j(d+|\gamma|-|\beta|)}\frac{2^{\alpha(j-1)}}{(2^{2(j-1)}+\mu^{2})^{\alpha/2}},

which can be rearranged to the derivative estimate

|\partial_{x}^{\gamma}B_{j}(x)|\leq C_{\gamma,M,d}2^{j(d+|\gamma|-M)}\frac{2^{\alpha(j-1)}}{(2^{2(j-1)}+\mu^{2})^{\alpha/2}}|x|^{-M}.

(15)

We now split the sum as

\partial_{x}^{\gamma}B(x)=\sum_{2^{j-1}\leq|x|^{-1}}\partial_{x}^{\gamma}B_{j}(x)+\sum_{2^{j-1}>|x|^{-1}}\partial_{x}^{\gamma}B_{j}(x).

To estimate the first sum, set $M=0$ in (15) to find that

\sum_{2^{j-1}\leq|x|^{-1}}|\partial_{x}^{\gamma}B_{j}(x)|\leq C_{\gamma,d}\sum_{2^{j-1}\leq|x|^{-1}}\frac{2^{j(d+|\gamma|)}}{(1+\left(\mu/2^{j-1}\right)^{2})^{\frac{\alpha}{2}}}.

(16)

When $2^{j-1}\leq|x|^{-1}$ , we see that $(1+(\mu/2^{j-1})^{2})^{-\frac{\alpha}{2}}\leq(1+|\mu x|^{2})^{-\frac{\alpha}{2}}$ and summing the geometric series (16) we obtain

\sum_{2^{j-1}\leq|x|^{-1}}|\partial_{x}^{\gamma}B_{j}(x)|\leq\frac{C_{\gamma,d}}{(1+|\mu x|^{2})^{\frac{\alpha}{2}}}\frac{1}{|x|^{d+|\gamma|}}.

(17)

For the second sum, we set $M$ to be the smallest integer greater than $|\gamma|+d+1/2$ and arrive at

	$\displaystyle\sum_{2^{j-1}>\|x\|^{-1}}\|\partial_{x}^{\gamma}B_{j}(x)\|$	$\displaystyle\leq C_{\gamma,d,M}\|x\|^{-M}\sum_{2^{j-1}>\|x\|^{-1}}2^{j(d+\|\gamma\|-M)}\frac{2^{\alpha(j-1)}}{\left(\mu 2^{j-1}\left(\frac{\mu}{2^{j-1}}+\frac{2^{j-1}}{\mu}\right)\right)^{\frac{\alpha}{2}}}$
		$\displaystyle\leq C_{\gamma,d,M}\|x\|^{-M}\mu^{-\alpha/2}\sum_{2^{j-1}>\|x\|^{-1}}\frac{2^{j(d+\|\gamma\|+\frac{\alpha}{2}-M)}}{\left(\frac{\mu}{2^{j-1}}+\frac{2^{j-1}}{\mu}\right)^{\frac{\alpha}{2}}}.$		(18)

Setting $t=\mu/2^{j-1}$ and $L=|\mu x|$ . If $2^{j-1}>|x|^{-1}$ , we see that $0<t\leq L$ . A direct calculation shows

\sup\limits_{0<t\leq L}(t+t^{-1})^{-\frac{\alpha}{2}}=\begin{cases}2^{-\frac{\alpha}{2}};&L>1,\\ (L+L^{-1})^{-\frac{\alpha}{2}};&L\leq 1.\end{cases}

We use this to sum the geometric series in (4) and obtain

\sum_{2^{j-1}>|x|^{-1}}|\partial_{x}^{\gamma}B_{j}(x)|\leq C\begin{cases}|2\mu x|^{-\frac{\alpha}{2}}|x|^{-|\gamma|-d};&|\mu x|>1,\\ (|\mu x|^{2}+1)^{-\frac{\alpha}{2}}|x|^{-|\gamma|-d};&|\mu x|\leq 1.\end{cases}

(19)

Combining (19) with the earlier estimate (17) completes the proof. ∎

A similar argument establishes a Hörmander condition which we include here for completeness and to contrast with the usual one.

Theorem 4.1.

If $b(\xi)$ satisfies (10), then its kernel satisfies

\int\limits_{|x|\geq 2|y|}\left|B(x+y)-B(x)\right|\,dx\leq C\begin{cases}|2\mu y|^{-1/2};&|\mu y|>1,\\ (|\mu y|^{2}+1)^{-1/2};&|\mu y|\leq 1.\end{cases}

(20)

We present a refinement of Corollary 1.6 from the Introduction.

Theorem 4.2.

Fix $\delta>0$ and suppose $f\in L^{\infty}$ vanishes for $|x|<\delta$ . If $\sigma<\delta$ , there is a constant $C=C(d,\sigma,\delta)$ such that the “uniform maximal” estimate holds:

\sup_{|x|\leq\sigma}\sup_{\mu\geq 1}|E_{\alpha,\mu}f(x)|\leq C\|f\|_{{}_{L^{\infty}}}.

(21)

The behavior of the constant in (21) is of interest here. Our proof gives a logarithmic dependence on the distance $\delta-\sigma$ which is independent of $\alpha$

Proof.

Let $K_{\alpha,\mu}(z)$ be the associated kernel. For any $|x|<\sigma$ it is enough to estimate

\int_{|y|\geq\delta}|K_{\alpha,\mu}(x-y)|\,dy\leq\int_{|x-y|\geq\delta-\sigma}|K_{\alpha,\mu}(x-y)|\,dy.

Since we are taking a supremum in $\mu$ , we cannot assume that $1/\mu$ is small compared to $\delta-\sigma$ . Instead we split the integral over two regions: $|x-y|\geq 1/\mu$ and $\delta-\sigma\leq|x-y|\leq 1/\mu$ . By Theorem 1.5

\int\limits_{|y|\geq\delta}|K_{\alpha,\mu}(x-y)|\,dy\leq\int\limits_{|x-y|\geq\frac{1}{\mu}}\frac{\mu^{-\frac{\alpha}{2}}}{|x-y|^{d+\frac{\alpha}{2}}}\,dy\,+\int\limits_{\delta-\sigma\leq|x-y|\leq\frac{1}{\mu}}\frac{(1+(\mu|x-y|)^{2})^{-\frac{\alpha}{2}}}{|x-y|^{d}}\,dy.

After the integration dust settles, we see that

\sup_{\mu\geq 1}\int\limits_{|y|\geq\delta}|K_{\alpha,\mu}(x-y)|\,dy\leq C\sup_{\mu\geq 1}\left(1+\frac{|\ln\mu(\delta-\sigma)|}{(1+(\mu(\delta-\sigma))^{2})^{\frac{\alpha}{2}}}\right)\leq C(1+|\ln(\delta-\sigma)|).

∎

Acknowledgement

I would like to thank Professor L. Colzani for clarifying some points in [5]. This improved the argument in §2.2. Professor A. Larrain–Hubach also made helpful comments on an earlier draft. Any remaining errors are, of course, mine.

References

[1] D. Adams and L. Hedberg, Function Spaces and Potential Theory, Springer Berlin Heidelberg, 1999.
[2] N. Aronszajn and K.T Smith, Theory of Bessel Potentials. I, Annales de l’institut Fourier 11 (1961), 385–475.
[3] O. V. Besov and S. B. Stechkin, Description of Moduli of Continuity in $L_{2}$ , Proceedings of the Steklov Institute of Mathematics 134 (1975).
[4] P. L. Butzer and R. J. Nessel, Contributions To The Theory Of Saturation For Singular Integrals In Several Variables. I: General Theory, Indagationes Mathematicae (Proceedings) 69 (1966), 513–531.
[5] L. Colzani, Jackson Theorems in Hardy Spaces and Approximation by Riesz Means, Journal of Approximation Theory 49 (1987), 240–251.
[6] Z. Ditzian and K.G Ivanov, Strong Converse Inequalities, Journal d’Analyse Mathématique 62 (1993), 61–111.
[7] NIST Digital Library of Mathematical Functions, http://dlmf.nist.gov/, Release 1.1.8 of 2022-12-15, F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, B. V. Saunders, H. S. Cohl, and M. A. McClain, eds.
[8] L. C. Evans and R. F. Gariepy, Measure theory and fine property of functions, CRC Press, Florida, USA, 1992.
[9] H. Johnen and K. Scherer, On the equivalence of the $K$ -functional and moduli of continuity and some applications, Constructive Theory of Functions of Several Variables (W. Schempp and K. Zeller, eds.), Springer Berlin Heidelberg, 1977, pp. 119–140.
[10] Z. Liu and S. Lu, Applications of Hörmander multiplier theorem to approximation in real Hardy spaces, Harmonic Analysis (MT. Cheng, DG. Deng, and XW. Zhou, eds.), Lecture Notes in Mathematics, Springer, Berlin, 1991, pp. 119–129.
[11] B. Muckenhoupt and R. Wheeden, Weighted Norm Inequalities for Fractional Integrals, Trans. Amer. Math. Soc. 192 (1974), 261–274.
[12] M. Schecter, The Spectrum of the Schrodinger Operator, Trans. Amer. Math. Soc. 312 (1989), no. 1, 115–128.
[13] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1970.
[14] , Harmonic Analysis: Real-Variable Methods, Orthogonality and Oscillatory integrals, Princeton University Press, 1993.
[15] W. P. Ziemer, Weakly Differentiable Functions, Springer Berlin Heidelberg, 1989.

	$\displaystyle\sum_{2^{j-1}>\|x\|^{-1}}\|\partial_{x}^{\gamma}B_{j}(x)\|$	$\displaystyle\leq C_{\gamma,d,M}\|x\|^{-M}\sum_{2^{j-1}>\|x\|^{-1}}2^{j(d+\|\gamma\|-M)}\frac{2^{\alpha(j-1)}}{\left(\mu 2^{j-1}\left(\frac{\mu}{2^{j-1}}+\frac{2^{j-1}}{\mu}\right)\right)^{\frac{\alpha}{2}}}$
		$\displaystyle\leq C_{\gamma,d,M}\|x\|^{-M}\mu^{-\alpha/2}\sum_{2^{j-1}>\|x\|^{-1}}\frac{2^{j(d+\|\gamma\|+\frac{\alpha}{2}-M)}}{\left(\frac{\mu}{2^{j-1}}+\frac{2^{j-1}}{\mu}\right)^{\frac{\alpha}{2}}}.$		(18)

Approximation by the Bessel–Riesz Quotient

Abstract

1 Overview

Theorem 1.1.

Corollary 1.2.

Corollary 1.3.

Corollary 1.4.

Theorem 1.5.

Corollary 1.6.

Proof.

Notation

2 Order of Approximation

2.1 The LpL^{p} Case when 0<α≤10<\alpha\leq 1

Lemma 2.1.

Proof.

Proposition 2.2.

Proof.

Theorem 2.3.

Proof.

2.2 Proof of Theorem 1.1 (a)

Corollary 2.4.

Lemma 2.5.

Proof.

Lemma 2.6.

Proof.

Proof of Theorem 1.1(a).

2.3 Extension to HpH^{p} when α=1\alpha=1

Theorem 2.7.

Lemma 2.8.

Lemma 2.9.

Proof of Theorem 2.7.

3 Saturation Theorems for α=1\alpha=1

Theorem 3.1.

Proof.

Theorem 3.2.

Proof.

4 Kernel Estimates and Localization

Proof of Theorem 1.5.

Theorem 4.1.

Theorem 4.2.

Proof.

Acknowledgement

References

2.1 The $L^{p}$ Case when $0<\alpha\leq 1$

2.3 Extension to $H^{p}$ when $\alpha=1$

3 Saturation Theorems for $\alpha=1$