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 . Additionally, , denotes the usual norm of the Lebesgue space . We denote the Fourier transform of a function by
We use this same notation for the extension that the Fourier transform operator admits. We refer to the extension from to the whole of , which is made possible by means of the density of within and Plancherel’s identity.
Now we can define the notion of Fourier multiplier rigorously.
Definition 1.
Fix a function
Consider the linear operator given by
The function is a Fourier multiplier and the associated operator is called a Fourier multiplier operator.
One example of the above type of operator is the one that translates the original function, with , as the multiplier. In other words, given and , we have that
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 we have .
This implies that the operator norm,
is finite for all . Thus, is continuous or, equivalently, bounded. Thanks to the density of within , the operator can be extended to , .
The reader would be right to think that the above is a rather complicated way of proving the existence of the obvious extension of . 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 to denote the Schwartz class of rapidly decreasing smooth functions.
Definition 2.
Let and . The disk multiplier is the operator given as
The above operator provides a sort of ‘partial’ inverse Fourier transform. One would expect to have a convergence of in some sense. In fact, thanks to Plancherel’s identity, we have the convergence in the norm as
In order to be able to extend the disk multplier as we have done above with , we would want to prove that
, given and , the operator norm is
Unfortunately, the above is not always finite.
If we fix and the disk multiplier operator is bounded for all (see [12]). For any other dimension , the operator is unbounded for any except Fefferman [4] proved this in 1971 by constructing a counterexample.
There are three important reductions for the counterexample. First of all, given , we have This can be proven by a change of variables. Secondly, the operator is self-adjoint with respect to the usual inner product of . This, coupled with the extremal case of Hölder’s inequality, lets us deduce that, whenever we have Thirdly, there is a theorem of De Leeuw (See Proposition 3.2 in [3]) for multiplier operators that we can apply to obtain that
From the above three facts, we know it suffices to construct the counterexample for in the -norm for in dimension
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 we denote by 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 and have the same midpoints). This construction can be seen in the following Figure.
Figure. The rectangle to scale. The midpoints determine the direction of the translation
The first lemma gives us functions which act as building blocks.
Lemma 1.
Let . There exist constants such that, given a rectangle of dimensions with large enough constants , the following holds. There is a smooth supported inside such that on and
on .
The second lemma affirms there is a way to pack narrow enough rectangles with a high degree of overlap while the modified rectangles are all disjoint. It exploits the properties of Besicovitch sets.
Lemma 2.
Let , . If is large enough, then there is a finite collection of rectangles that are pairwise disjoint and satisfy
We fix . Using both lemmas, we build the compactly supported, smooth function
where the . 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 and the existence of a constant only depending on such that
Since and can be chosen to be arbitrarily small, we conclude that
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 centered around a point in the domain of the function. One would expect these means to approach the value of the function at as for continuous functions. And there is pointwise convergence in that case.
But the good behaviour is even broader.
We are talking convergence in norm and almost everywhere for functions merely in (with a condition on that depends on the dimension ). The latter convergence is related to Stein’s spherical maximal operator.
Let be the normalized spherical measure over . Consider its Fourier transform (as a finite measure),
Let and . The spherical means operator is defined as
The above equation makes it clear that is a multiplier operator.
It is not hard to show that for all and . This lets us extend the operator to and deduce that
in norm for all In fact, through the boundedness of the associated maximal operator, one can find out that
almost everywhere for all The maximal operator associated to is the following.
Definition 3.
Let , , The spherical maximal operator is given by
As it turns out, this non-linear operator is bounded.
Theorem 1.
Given and , there exists such that, for all
The case was proven by Stein [9],[10] and the case is due to Bourgain [1],[2]. One could consider the case by defining the spherical means with a discrete spherical measure supported on 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 can be proven as a corollary of the following theorem due to Rubio de Francia [7].
Theorem 2.
Let be a family of multiplier operators defined on as
where is a compactly supported finite Borel measure.
Consider the maximal operator .
If there is so that for all , then the extension exists and is bounded for all and all .
One can prove that and apply Theorem 2 to prove Theorem 1.
The decay of can be proven in the following way. First, we use the invariance by orthogonal transformations that satisfies and notice that it suffices to prove the bound for
where and . After that, we parametrize the sphere caps as graphs and integrate by parts to arrive at
Then, we would only need to prove the following proposition, that arises from the oscillatory nature of the trigonometric functions.
Proposition.
Let and or for Then,
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 smooth functions on supported on for and supported on while imposing that is constantly equal to one. After that, we define some partial versions of , given by
Then, we define the associated maximal operators . And we observe that
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
Lemma 3
Consider the hypotheses of Theorem 2. Let . Then, there exists such that
for all functions. .
Lemma 4
Consider the hypotheses of Theorem 2. There exists such that, for all , the estimate
holds for all functions .
The above lemma is the reason for Theorem 2 to be stated for In the following lemma, denotes the quasi-norm of the weak- space, defined as
Lemma 5
Consider the hypotheses of Theorem 2. There exists such that, for all , the estimate
holds for all functions
If we use Marcinkiewicz’s interpolation theorem on the last two lemmas, we obtain, for all
where the constants are positive and the sum is finite if and only if
After that, we prove that
Which allows us to use Marcinkiewicz’s interpolation theorem yet again (over the last two equations this time) to obtain a bound for the , for all
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/