The Disk Multiplier and the Spherical Maximal Operator

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 dN d\in \N . Additionally, p \|\cdot\|_p , 1p 1\leq p \leq \infty denotes the usual norm of the Lebesgue space Lp(Rd) L^p (\R^d) . We denote the Fourier transform of a function fL1(Rd) f\in L^1(\R^d) by
f^(ξ)=Ff(ξ)=Rdf(x)e2πixξ.(1) \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 L1(Rd)L2(Rd) L^1(\R^d)\cap L^2 (\R^d) to the whole of L2(Rd) L^2 (\R^d), which is made possible by means of the density of L1L2 L^1\cap L^2 within L2 L^2 and Plancherel’s identity.

Now we can define the notion of Fourier multiplier rigorously.
Definition 1.
Fix a function mL(Rd). m\in L^\infty(\R^d). Consider the linear operator Tm:L2(Rd)L2(Rd) T_m:L^2(\R^d)\to L^2(\R^d) given by
Tmf^(ξ)=m(x)f^(ξ).(2) \widehat{T_mf}(\xi)= m(x)\widehat f(\xi). (2)

The function m m is a Fourier multiplier and the associated operator Tm 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 mτ(ξ)=e2πiτξ, m_\tau(\xi) = e^{2\pi i \tau\cdot \xi}, τRd \tau \in \R^d , as the multiplier. In other words, given fL2(Rd) f\in L^2(\R^d) and τRd \tau \in \R^d , we have that
Tmτf(x)=f(xτ)=F1(e2πiτξf^(ξ)).(3) 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 fL2(Rd)Lp(Rd), f\in L^2(\R^d)\cap L^p(\R^d), we have Tmτfp=fp \|T_{m_\tau}f\|_p = \|f\|_p .

This implies that the operator norm,
Tmτp,L2Lp=supfL2Lpfp0Tmτfpfp,(4) \|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 1p 1\leq p \leq \infty . Thus, Tmτ:L2LpL2Lp T_{m_\tau}:L^2\cap L^p\to L^2\cap L^p is continuous or, equivalently, bounded. Thanks to the density of L2Lp L^2\cap L^p within Lp L^p , the operator can be extended to Tmτ:LpLp T_{m_\tau}: L^p\to L^p , 1p< 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 Tmτ 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 S(Rd) \mathscr S(\R^d) to denote the Schwartz class of rapidly decreasing smooth functions.

Definition 2.
Let dN, d\in \N, fS(Rd) f\in \mathscr S (\R^d) and r>0 r>0 . The r \mathbf {r-} disk multiplier is the operator Dr:S(Rd)L(Rd) D_r: \mathscr S(\R^d)\to L^\infty(\R^d) given as
Drf(x)=Rdχ{ξ<r}(ξ)f^(ξ)e2πixξdξ.(5) 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 Drfrf D_rf\xrightarrow[r\to \infty]{} f in some sense. In fact, thanks to Plancherel’s identity, we have the convergence in the L2 L^2 norm as
Drff2=χ{ξ<r}f^f^2r0.(6) \|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 Tmτ T_{m_\tau} , we would want to prove that
, given r>0 r>0 and 1<p< 1<p<\infty , the operator norm is
Drp,S=supfSfp0Drfpfp<.(7) \|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, d=1, and r>0, r>0, the r r- disk multiplier operator is bounded for all 1<p< 1< p <\infty (see [12]). For any other dimension d>1 d>1 , the operator is unbounded for any 1<p< 1<p<\infty except p=2. 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 r>0 , we have Drp,S=D1p,S. \|D_r\|_{p,\mathscr S} = \|D_1\|_{p,\mathscr S}. This can be proven by a change of variables. Secondly, the operator D1 D_1 is self-adjoint with respect to the usual inner product of L2 L^2 . This, coupled with the extremal case of Hölder’s inequality, lets us deduce that, whenever 1p+1q=1, \frac1p+\frac1q =1, we have D1p,S=D1q,S. \|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
D1p,S(Rd1)D1p,S(Rd).(8) \|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 D1 D_1 in the Lp L^p -norm for p>2 p>2 in dimension d=2. 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, R, we denote by R~ \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 R~ \tilde R and R R^* have the same midpoints). This construction can be seen in the following Figure.


Figure. The rectangle R R^* to scale. The midpoints M1,M2 M_1,M_2 determine the direction of the translation

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

Lemma 1.
Let p>2 p > 2 . There exist constants 0<C1C2 0 \lt C_1\leq C_2 such that, given a rectangle R R of dimensions A2×aA A^2\times a A with large enough constants A,a>0 A, a>0 , the following holds. There is a smooth fR:R2R2 f_R:\mathbb R^2\to \mathbb R^2 supported inside R R such that fR1 |f_R | \leq 1 on R R and
C1D1fRC2,(9) C_1 \leq |D_1f_R| \leq C_2, (9) on R R^* .

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

Lemma 2.
Let ϵ>0 \epsilon > 0 , a>0 a>0 . If A>0 A>0 is large enough, then there is a finite collection R \mathcal R of A2×aA A^2\times aA rectangles that are pairwise disjoint and satisfy
RRRϵRRR.(10) \big\lvert \bigcup_{R\in \mathcal R} R^* \big\rvert \leq \epsilon \lvert \bigcup_{R\in \mathcal R} R \big\rvert. (10)

We fix ϵ>0 \epsilon>0 . Using both lemmas, we build the compactly supported, smooth function
fϵ(x)=f(x)=RRσRfR(x), f_\epsilon(x) = f(x) = \sum_{R\in \mathcal R}\sigma_Rf_R(x), where the σR{1,1} \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 σR, \sigma_R, RR R\in \mathcal R and the existence of a constant C C only depending on p p such that
D1fppCϵ1p2fpp.(11) \|D_1f\|_p^p\geq C\epsilon^{1-\frac p2}\|f\|_p^p. (11)

Since p>2 p>2 and ϵ \epsilon can be chosen to be arbitrarily small, we conclude that D1p,S=. \|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 t>0 centered around a point x x in the domain of the function. One would expect these means to approach the value of the function at x x as t0 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 Lp L^p norm and almost everywhere for functions merely in Lp L^p (with a condition on p p that depends on the dimension d d ). The latter convergence is related to Stein’s spherical maximal operator.

Let dσ d\sigma be the normalized spherical measure over Sd1Rd S^{d-1}\subset \mathbb R^d . Consider its Fourier transform (as a finite measure),
dσ^(ξ)=Sd1dσ(x).(12) \widehat{d\sigma}(\xi) = \int_{S^{d-1}}d\sigma(x). (12)

Let d2, d\geq 2, t>0 t>0 and fS f\in\mathscr S . The t\mathbf{t-}spherical means operator is defined as
Stf(x)=Sd1f(xty)dσ(y)=F1(f^(ξ)dσ^(tξ))(x).(13) 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 St S_t is a multiplier operator.
It is not hard to show that Stp,S1 \|S_t\|_{p,\mathscr S}\leq 1 for all t>0 t>0 and 1p 1\leq p \leq \infty . This lets us extend the operator to Lp L^p and deduce that
Stt0f,(14) S_t\xrightarrow[t\to 0]{} f , (14)

in Lp L^p norm for all fLp. f\in L^p. In fact, through the boundedness of the associated maximal operator, one can find out that
Stt0f,(15) S_t\xrightarrow[t\to 0]{}f, (15)

almost everywhere for all p>dd1. p>\frac{d}{d-1}. The maximal operator associated to {St}t>0 \{S_t\}_{t>0} is the following.

Definition 3.
Let d2 d\geq 2 , t>0 t>0 , fS(Rd). f\in \mathscr S(\mathbb R^d). The spherical maximal operator is given by
Mσf(x)=supt>0Stf(x)=supt>0Sd1f(xty)dσ(y).(16) 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 d2 d\geq 2 and p>dd1 p>\frac{d}{d-1} , there exists C>0 C>0 such that, for all fS(Rd), f\in \mathscr S(\mathbb R^d), MσfpCfp.(17) \|M_\sigma f\|_p \leq C\|f\|_p. (17)

The case d>2 d>2 was proven by Stein [9],[10] and the case d=2 d=2 is due to Bourgain [1],[2]. One could consider the case d=1 d=1 by defining the spherical means with a discrete spherical measure dσ d\sigma supported on S0={1,1} 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 d>2 can be proven as a corollary of the following theorem due to Rubio de Francia [7].

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

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

If there is C>0 C>0 so that dμ^(ξ)Cξa | \widehat{d\mu}(\xi)|\leq C{|\xi|^{-a} } for all ξRd \xi\in \mathbb R^d , then the extension M:Lp(Rd)Lp(Rd) M : L^p(\mathbb R^d)\to L^p(\mathbb R^d) exists and is bounded for all p>2a+12a p>\frac{2a+1}{2a} and all a>12 a>\frac12 .

One can prove that dσ^(ξ)Cξd12 |\widehat{d\sigma}(\xi)|\leq C {|\xi|^{-\frac{d-1}{2}}} and apply Theorem 2 to prove Theorem 1.

The decay of dσ^ \widehat{d\sigma} can be proven in the following way. First, we use the invariance by orthogonal transformations that dσ d\sigma satisfies and notice that it suffices to prove the bound for
dσ^(λed)=Sd1e2πiλxddσ(x),(19) \widehat{d\sigma}(\lambda e_d) = \int_{S^{d-1}} e^{-2\pi i \lambda x_d} d\sigma(x), (19)

where ed=(0,0,...0,1)Rd e_d=(0,0,...0,1)\in \mathbb R^d and x=(x1,x2,...,xd) x=(x_1,x_2,...,x_d) . After that, we parametrize the sphere caps as graphs and integrate by parts to arrive at
dσ^(λed)=2Cd01cos(λt)(1t2)d32dt.(20) \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 nN n\in \mathbb N and g(x)=cos(x) g(x) = \cos(x) or g(x)=sin(x) g(x) = \sin(x) for xR. x\in \mathbb R. Then,
01g(λx)(1x2)d32dx=O(λd12),λ.(21) \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 ϕj \phi_j smooth functions on Rd \mathbb R^d supported on {xRd:2j1x2j} \{x\in \mathbb R^d : 2^{j-1}\leq |x|\leq 2^j \} for jN j\in \mathbb N and ϕ0 \phi_0 supported on {xRd:x2} \{x\in \mathbb R^d: |x|\leq 2\} while imposing that j=0ϕj(x) \sum_{j=0}^{\infty} \phi_j(x) is constantly equal to one. After that, we define some partial versions of Tt T_t , given by
Ttj^f(x)=ϕj(tξ)dμ^(tξ)f^(ξ),jN.(22) \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 Mjf(x)=supt>0Ttjf(x), M_jf(x) = \sup_{t>0} |T_t^jf(x)|, jN{0} j\in \mathbb N\cup \{0\} . And we observe that
Mμf(x)j=0Mjf(x).(23) 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 S. \mathscr S.

Lemma 3
Consider the hypotheses of Theorem 2. Let 1<p 1 \lt p \leq \infty. Then, there exists C>0 C>0 such that
M0fpCfp,(24) \| M_0f \|_{p} \leq C \|f\|_{p}, (24)

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

Lemma 4
Consider the hypotheses of Theorem 2. There exists C>0 C>0 such that, for all j1 j \geq 1 , the estimate
Mjf2C12j(a12)f2,(25) \| M_jf\|_{2} \leq C \dfrac{1}{2^{j(a-\frac 1 2)}}\|f\|_{2}, (25)

holds for all functions fL2(Rd) f \in L^2(\mathbb R^d) .

The above lemma is the reason for Theorem 2 to be stated for a>12. a>\frac 12. In the following lemma, L1, \| \cdot \|_{L^{1,\infty}} denotes the quasi-norm of the weak-L1 L^1 space, defined as
fL1,=inf{C>0:{xRd:g(x)>λ}Cλ for all λ>0}.(26) \| 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 C > 0 such that, for all j1 j\geq 1 , the estimate
MjfL1,Cj2jf1,(27) \|M_jf\|_{L^{1,\infty}} \leq Cj2^j \|f\|_{1}, (27)

holds for all functions fL1(Rd). f \in L^1(\mathbb R^d).

If we use Marcinkiewicz’s interpolation theorem on the last two lemmas, we obtain, for all 1<p<2, 1 \lt p \lt 2, Mσfpj=0Mjfp(j=0Cj,p,a)fp,(28) \|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>2a+12a. p>\frac{2a+1}{2a}.

After that, we prove that
MμfRdfdμ(x)=dμ(Rd)f.(29) \|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 Mj,jN M_j, j\in \mathbb N , for all 2a+12a<p<. \frac{2a+1}{2a} \lt p \lt \infty.

References

1] J. Bourgain. Estimations de certaines fonctions maximales. C.R. Acad. Sci. Paris Sér. I Math., 301: no. 2, 499502, 1985. https://mathscinet.ams.org/mathscinet-getitem?mr=812567
[2] J. Bourgain. Averages in the plane over convex curves and maximal operators. J. Analyse Math., 47: no. 1, 6985, 1986. https://mathscinet.ams.org/mathscinet-getitem?mr=874045
[3] Karel de Leeuw. On Lp multipliers. Annals of Mathematics, 81: no. 2,364379, 1965. https://mathscinet.ams.org/mathscinet-getitem?mr=174937
[4] Charles Feerman. The multiplier problem for the ball. Annals of Mathematics, 94: no. 2, 330336, 1971. https://mathscinet.ams.org/mathscinet-getitem?mr=296602
[5] Loukas Grafakos. Classical Fourier analysis, volume 249 of Graduate texts in mathematics. Springer, New York, second edition, 2008. https://mathscinet.ams.org/mathscinet-getitem?mr=2445437
[6] Javier Minguillón. Some examples of the Theory of Fourier Multipliers:The Disk Multiplier and the Spherical Maximal Operator. Master’s thesis directed by F. Soria, Universidad Autónoma de Madrid, 2021.
[7] José L. Rubio de Francia. Maximal functions and Fourier transforms. Duke Math. J., 53: no. 2, 395404, 1986. https://mathscinet.ams.org/mathscinet-getitem?mr=850542
[8] Fernando Soria. Five lectures on Harmonic Analysis, chapter 12 of Topics in mathematical analysis. Edited by P.Ciatti et. al. , World Scientic, 2008. https://www.worldscientific.com/worldscibooks/10.1142/6806
[9] Elias M. Stein. Maximal functions: Spherical means. Proceedings of the National Academy of Sciences of the United States of America, 73(7):21742175, 1976. https://mathscinet.ams.org/mathscinet-getitem?mr=420116
[10] Elias M. Stein and Jan O. Strömberg. Behavior of maximal functios in R n for large n. Arkiv för Matematik, 21(1-2):259269, 1983. https://mathscinet.ams.org/mathscinet-getitem?mr=727348
[11] Elias M. Stein. Harmonic analysis: Real-variable methods, orthogonality, and oscillatory integrals, volume 43 of Princeton mathematical series. Princeton University Press, Princeton N.J., 1993. https://mathscinet.ams.org/mathscinet-getitem?mr=1232192
[12] Terence Tao. Lecture notes 3 for 254b: Bochner-Riesz summation,Feerman’s disc multiplier counterexample, 1999. https://www.math.ucla.edu/~tao/254b.1.99s/