The Disk Multiplier and the Spherical Maximal Operator

Spain. 03.02.2022
Author: Javier Minguillón Sánchez

Introduction

We can define an operator between function spaces by modifying the frequencies in which a function decomposes via the Fourier transform. If the modification consists in multiplying the Fourier transform of a function by a certain other function (called the multiplier), the operator is called a Fourier multiplier operator.

Throughout the text, we denote the dimension of the underlying real space by $d\in \N$. Additionally, $\|\cdot\|_p$, $1\leq p \leq \infty$ denotes the usual norm of the Lebesgue space $L^p (\R^d)$. We denote the Fourier transform of a function $f\in L^1(\R^d)$ by
$\widehat f (\xi) = \mathfrak F f (\xi) = \int_{\R^d} f(x)e^{-2\pi i x \cdot \xi}. (1)$

We use this same notation for the extension that the Fourier transform operator admits. We refer to the extension from $L^1(\R^d)\cap L^2 (\R^d)$ to the whole of $L^2 (\R^d)$, which is made possible by means of the density of $L^1\cap L^2$ within $L^2$ and Plancherel’s identity.

Now we can define the notion of Fourier multiplier rigorously.
Definition 1.
Fix a function $m\in L^\infty(\R^d).$ Consider the linear operator $T_m:L^2(\R^d)\to L^2(\R^d)$ given by
$\widehat{T_mf}(\xi)= m(x)\widehat f(\xi). (2)$

The function $m$ is a Fourier multiplier and the associated operator $T_m$ is called a Fourier multiplier operator.

One example of the above type of operator is the one that translates the original function, with $m_\tau(\xi) = e^{2\pi i \tau\cdot \xi},$ $\tau \in \R^d$, as the multiplier. In other words, given $f\in L^2(\R^d)$ and $\tau \in \R^d$, we have that
$T_{m_\tau} f(x)= {f(x-\tau)} = \mathfrak F^{-1}\left( e^{2\pi i \tau \cdot \xi } \widehat f(\xi ) \right). (3)$

The above multiplier operator is particularly well behaved and we only need to rely on properties of the translation of functions to see it. One such property is that, for any $f\in L^2(\R^d)\cap L^p(\R^d),$ we have $\|T_{m_\tau}f\|_p = \|f\|_p$.

This implies that the operator norm,
$\|T_{m_\tau}\|_{p,L^2\cap L^p} = \sup_{\substack{f\in L^2\cap L^p\\ \|f\|_p \neq 0}} \frac{\|T_{m_\tau}f\|_p}{\|f\|_p} ,(4)$

is finite for all $1\leq p \leq \infty$. Thus, $T_{m_\tau}:L^2\cap L^p\to L^2\cap L^p$ is continuous or, equivalently, bounded. Thanks to the density of $L^2\cap L^p$ within $L^p$, the operator can be extended to $T_{m_\tau}: L^p\to L^p$, $1\leq p < \infty$.

The reader would be right to think that the above is a rather complicated way of proving the existence of the obvious extension of $T_{m_\tau}$. We chose this way because it illustrates the general method of taking a bounded operator on a dense subset and extending the operator to the whole set.

The Disk Multiplier Operator

Another natural multiplier operator is one that gets rid of higher frequencies using an indicator function on the unit ball of the frequency domain. Or a ball of an arbitrary radius for that matter. This one is known as the disk multiplier operator.

We write $\mathscr S(\R^d)$ to denote the Schwartz class of rapidly decreasing smooth functions.

Definition 2.
Let $d\in \N,$ $f\in \mathscr S (\R^d)$ and $r>0$. The $\mathbf {r-}$disk multiplier is the operator $D_r: \mathscr S(\R^d)\to L^\infty(\R^d)$ given as
$D_rf(x) = \int_{\mathbb{R}^d} { \chi_{ \{| \xi| \lt r \} } (\xi) \hat f(\xi) e^{2\pi i x \cdot \xi} d \xi.} (5)$

The above operator provides a sort of ‘partial’ inverse Fourier transform. One would expect to have a convergence of $D_rf\xrightarrow[r\to \infty]{} f$ in some sense. In fact, thanks to Plancherel’s identity, we have the convergence in the $L^2$ norm as
$\|D_rf-f\|_2 = \lVert \chi_{\{|\xi| \lt r\}}\hat f-\hat f\mathbb \rVert_2 \xrightarrow[r \to \infty]{} 0. (6)$

In order to be able to extend the disk multplier as we have done above with $T_{m_\tau}$, we would want to prove that
, given $r>0$ and $1<p<\infty$, the operator norm is
$\|D_r \|_{p,\mathscr S}= \sup_{\substack{f\in \mathscr S \\ \|f\|_p \neq 0}} \frac{\|D_rf\|_p}{\|f\|_p} <\infty. (7)$

Unfortunately, the above is not always finite.

If we fix $d=1,$ and $r>0,$ the $r-$disk multiplier operator is bounded for all $1< p <\infty$ (see [12]). For any other dimension $d>1$, the operator is unbounded for any $1<p<\infty$ except $p=2.$ Fefferman [4] proved this in 1971 by constructing a counterexample.

There are three important reductions for the counterexample. First of all, given $r>0$, we have $\|D_r\|_{p,\mathscr S} = \|D_1\|_{p,\mathscr S}.$ This can be proven by a change of variables. Secondly, the operator $D_1$ is self-adjoint with respect to the usual inner product of $L^2$. This, coupled with the extremal case of Hölder’s inequality, lets us deduce that, whenever $\frac1p+\frac1q =1,$ we have $\|D_1\|_{p,\mathscr S} = \|D_1\|_{q,\mathscr S}.$ Thirdly, there is a theorem of De Leeuw (See Proposition 3.2 in [3]) for multiplier operators that we can apply to obtain that
$\|D_1\|_{p,\mathscr S(\mathbb R^{d-1})}\leq \|D_1\|_{p,\mathscr S(\mathbb R^d)}. (8)$

From the above three facts, we know it suffices to construct the counterexample for $D_1$ in the $L^p$-norm for $p>2$ in dimension $d=2.$

The Disk Operator is Unbounded

The construction of the counterexemple that we outline here follows the one in [12] and is based in the following two lemmas.

These lemmas are formulated in terms of rectangles, that we use to build the functions that provide a counterexample to the boundedness of the disk operator. Given a rectangle $R,$ we denote by $\tilde R$ a translation of two times the length of the rectangle. Then we perform a reduction to one third of the width (so that the short sides of $\tilde R$ and $R^*$ have the same midpoints). This construction can be seen in the following Figure.

Figure. The rectangle $R^*$ to scale. The midpoints $M_1,M_2$ determine the direction of the translation

The first lemma gives us functions $f_R$ which act as building blocks.

Lemma 1.
Let $p > 2$. There exist constants $0 \lt C_1\leq C_2$ such that, given a rectangle $R$ of dimensions $A^2\times a A$ with large enough constants $A, a>0$, the following holds. There is a smooth $f_R:\mathbb R^2\to \mathbb R^2$ supported inside $R$ such that $|f_R | \leq 1$ on $R$ and
$C_1 \leq |D_1f_R| \leq C_2, (9)$ on $R^*$.

The second lemma affirms there is a way to pack narrow enough rectangles $R$ with a high degree of overlap while the modified rectangles $R^*$ are all disjoint. It exploits the properties of Besicovitch sets.

Lemma 2.
Let $\epsilon > 0$, $a>0$. If $A>0$ is large enough, then there is a finite collection $\mathcal R$ of $A^2\times aA$ rectangles that are pairwise disjoint and satisfy
$\big\lvert \bigcup_{R\in \mathcal R} R^* \big\rvert \leq \epsilon \lvert \bigcup_{R\in \mathcal R} R \big\rvert. (10)$

We fix $\epsilon>0$. Using both lemmas, we build the compactly supported, smooth function
$f_\epsilon(x) = f(x) = \sum_{R\in \mathcal R}\sigma_Rf_R(x),$ where the $\sigma_R\in \{-1,1\}$. Then, thanks to the lemmas and an argument that involves Khintchine’s inequality, one can prove the existence of a choice of signs for the $\sigma_R,$ $R\in \mathcal R$ and the existence of a constant $C$ only depending on $p$ such that
$\|D_1f\|_p^p\geq C\epsilon^{1-\frac p2}\|f\|_p^p. (11)$

Since $p>2$ and $\epsilon$ can be chosen to be arbitrarily small, we conclude that $\|D_1\|_{p,\mathscr S}=\infty.$

The Spherical Means and the Spherical Maximal Operator

The spherical means opearator is a rather well behaved operator that calculates the mean of a function on a sphere of radius $t>0$ centered around a point $x$ in the domain of the function. One would expect these means to approach the value of the function at $x$ as $t\to 0$ for continuous functions. And there is pointwise convergence in that case.
But the good behaviour is even broader.
We are talking convergence in $L^p$ norm and almost everywhere for functions merely in $L^p$ (with a condition on $p$ that depends on the dimension $d$). The latter convergence is related to Stein’s spherical maximal operator.

Let $d\sigma$ be the normalized spherical measure over $S^{d-1}\subset \mathbb R^d$. Consider its Fourier transform (as a finite measure),
$\widehat{d\sigma}(\xi) = \int_{S^{d-1}}d\sigma(x). (12)$

Let $d\geq 2,$ $t>0$ and $f\in\mathscr S$. The $\mathbf{t-}$spherical means operator is defined as
$S_t f (x) = \int_{S^{d-1}} f(x-ty) d\sigma(y) = \mathfrak F^{-1} \left( \widehat{f} (\xi) \widehat{d\sigma} (t\xi) \right)(x). (13)$

The above equation makes it clear that $S_t$ is a multiplier operator.
It is not hard to show that $\|S_t\|_{p,\mathscr S}\leq 1$ for all $t>0$ and $1\leq p \leq \infty$. This lets us extend the operator to $L^p$ and deduce that
$S_t\xrightarrow[t\to 0]{} f , (14)$

in $L^p$ norm for all $f\in L^p.$ In fact, through the boundedness of the associated maximal operator, one can find out that
$S_t\xrightarrow[t\to 0]{}f, (15)$

almost everywhere for all $p>\frac{d}{d-1}.$ The maximal operator associated to $\{S_t\}_{t>0}$ is the following.

Definition 3.
Let $d\geq 2$, $t>0$, $f\in \mathscr S(\mathbb R^d).$ The spherical maximal operator is given by
$M_{\sigma}f(x) = \sup_{t>0}|S_tf(x)| = \sup_{t>0} \big\lvert \int_{S^{d-1}}f(x-ty)d\sigma(y) \big\rvert. (16)$

As it turns out, this non-linear operator is bounded.

Theorem 1.
Given $d\geq 2$ and $p>\frac{d}{d-1}$, there exists $C>0$ such that, for all $f\in \mathscr S(\mathbb R^d),$ $\|M_\sigma f\|_p \leq C\|f\|_p. (17)$

The case $d>2$ was proven by Stein [9],[10] and the case $d=2$ is due to Bourgain [1],[2]. One could consider the case $d=1$ by defining the spherical means with a discrete spherical measure $d\sigma$ supported on $S^0=\{-1,1\}$ but the nature of this measure gives us an unbounded spherical means operator, and, of course, an unbounded spherical maximal operator.

The Bound for the Spherical Maximal Operator

The boundedness of the spherical maximal operator in the case $d>2$ can be proven as a corollary of the following theorem due to Rubio de Francia [7].

Theorem 2.
Let $T_t$ be a family of multiplier operators defined on $\mathscr S (\mathbb R^d)$ as
$\widehat{T_tf}(\xi) = \widehat f(\xi) \widehat{d\mu}(t\xi), (18)$

where $d\mu$ is a compactly supported finite Borel measure.
Consider the maximal operator $Mf= \sup\limits_{t>0} |T_tf|$.

If there is $C>0$ so that $| \widehat{d\mu}(\xi)|\leq C{|\xi|^{-a} }$ for all $\xi\in \mathbb R^d$, then the extension $M : L^p(\mathbb R^d)\to L^p(\mathbb R^d)$ exists and is bounded for all $p>\frac{2a+1}{2a}$ and all $a>\frac12$.

One can prove that $|\widehat{d\sigma}(\xi)|\leq C {|\xi|^{-\frac{d-1}{2}}}$ and apply Theorem 2 to prove Theorem 1.

The decay of $\widehat{d\sigma}$ can be proven in the following way. First, we use the invariance by orthogonal transformations that $d\sigma$ satisfies and notice that it suffices to prove the bound for
$\widehat{d\sigma}(\lambda e_d) = \int_{S^{d-1}} e^{-2\pi i \lambda x_d} d\sigma(x), (19)$

where $e_d=(0,0,...0,1)\in \mathbb R^d$ and $x=(x_1,x_2,...,x_d)$. After that, we parametrize the sphere caps as graphs and integrate by parts to arrive at
$\widehat{d\sigma}(\lambda e_d) = 2 C_d \int_0^1 \cos(\lambda t) ( 1- t^2 )^{\frac{d-3}{2}} dt. (20)$

Then, we would only need to prove the following proposition, that arises from the oscillatory nature of the trigonometric functions.

Proposition.
Let $n\in \mathbb N$ and $g(x) = \cos(x)$ or $g(x) = \sin(x)$ for $x\in \mathbb R.$ Then,
$\big\lvert \int_0^1 g(\lambda x) (1-x^2)^{\frac{d-3}{2}}dx \big\rvert = O\left(\lambda ^{-\frac{d-1}{2}} \right), \quad \lambda\to \infty. (21)$

Regarding the proof of Theorem 2, the proof is rather long and we are going to briefly summarize it in what remains of the section. We take $\phi_j$ smooth functions on $\mathbb R^d$ supported on $\{x\in \mathbb R^d : 2^{j-1}\leq |x|\leq 2^j \}$ for $j\in \mathbb N$ and $\phi_0$ supported on $\{x\in \mathbb R^d: |x|\leq 2\}$ while imposing that $\sum_{j=0}^{\infty} \phi_j(x)$ is constantly equal to one. After that, we define some partial versions of $T_t$, given by
$\widehat{T_t^j}f(x) = \phi_j (t\xi) \widehat{d\mu} (t\xi) \widehat f (\xi), \quad j\in \mathbb N. (22)$

Then, we define the associated maximal operators $M_jf(x) = \sup_{t>0} |T_t^jf(x)|,$ $j\in \mathbb N\cup \{0\}$. And we observe that
$M_\mu f(x) \leq \sum_{j=0}^\infty M_jf(x). (23)$

In order to bound the above sum we use the following three lemmas. The operators can actually be extended so that the lemmas need not be stated merely for functions in $\mathscr S.$

Lemma 3
Consider the hypotheses of Theorem 2. Let $1 \lt p \leq \infty$. Then, there exists $C>0$ such that
$\| M_0f \|_{p} \leq C \|f\|_{p}, (24)$

for all functions. $f \in L^p(\mathbb R^d)$.

Lemma 4
Consider the hypotheses of Theorem 2. There exists $C>0$ such that, for all $j \geq 1$, the estimate
$\| M_jf\|_{2} \leq C \dfrac{1}{2^{j(a-\frac 1 2)}}\|f\|_{2}, (25)$

holds for all functions $f \in L^2(\mathbb R^d)$.

The above lemma is the reason for Theorem 2 to be stated for $a>\frac 12.$ In the following lemma, $\| \cdot \|_{L^{1,\infty}}$ denotes the quasi-norm of the weak-$L^1$ space, defined as
$\| f \|_{L^{1,\infty}} = \inf \bigg\lbrace C>0: \bigg\lvert \{x\in \mathbb R^d : |g(x)| > \lambda\} \bigg\rvert \leq \dfrac{C}{\lambda} \text{ for all } \lambda> 0 \bigg\rbrace. (26)$

Lemma 5
Consider the hypotheses of Theorem 2. There exists $C > 0$ such that, for all $j\geq 1$, the estimate
$\|M_jf\|_{L^{1,\infty}} \leq Cj2^j \|f\|_{1}, (27)$

holds for all functions $f \in L^1(\mathbb R^d).$

If we use Marcinkiewicz’s interpolation theorem on the last two lemmas, we obtain, for all $1 \lt p \lt 2,$ $\|M_\sigma f\|_p \leq \sum_{j=0}^\infty \|M_jf\|_p \leq \left(\sum_{j=0}^\infty C_{j,p,a} \right) \|f\|_p, (28)$

where the constants are positive and the sum is finite if and only if $p>\frac{2a+1}{2a}.$

After that, we prove that
$\|M_\mu f \|_\infty \leq \int_{\mathbb R^d} \|f\|_\infty d\mu(x) = d\mu(\mathbb R^d)\|f\|_\infty. (29)$

Which allows us to use Marcinkiewicz’s interpolation theorem yet again (over the last two equations this time) to obtain a bound for the $M_j, j\in \mathbb N$, for all $\frac{2a+1}{2a} \lt p \lt \infty.$

References

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