The author is deeply indebted to his doctoral advisor, Prof. Dr. Christoph Thiele, for the time devoted to thorough inspection of the arguments, careful understanding of ideas and several suggestions that led to the final version of this manuscript. He would also like to acknowledge the interest in this project manifested by some of his colleagues, among which he would like to thank especially Marco Fraccaroli and Gennady Uraltsev for discussions about related work, as well as Diogo Oliveira e Silva and Vjekoslav Kovaˇc, which, besides sharing their recent project [KOeS18], also took the time carefully read and make valuable suggestions on an early version of this work. Finally, the author thanks the referee for numerous remarks and corrections.
Chapter 5
Low-dimensional maximal restriction principles for the Fourier transform
No creo en una tercera alternativa: creo en muchas.
–G.G.M.
This chapter contains the paper [Ram19b]. Following the ideas from the previous chapter, we prove abstract maximal results for the Fourier transform. Our results deal mainly with maximal operators of convolution-type and r−average maximal functions. As a by-product of our techniques we obtain spherical maximal restriction estimates, as well as restriction estimates for 2−average maximal functions, answering thus points left open by V. Kovaˇc and M¨uller, Ricci and Wright.
5.1 Introduction
The classical restriction problem for the Fourier transforms asks for the largest possible range of exponents 1 ≤ p, q ≤ +∞ so that an inequality of the form
k bf |SkLq(S)≤ Cp,q,d,Skf kLp(Rd) (5.1) holds for any function f ∈ S(Rd). Here, S is taken to be a subset of Rd, endowed with a suitable measure.
The existence of such a priori inequalities allows one to define restrictions of Fourier transforms of Lp functions to smaller sets in the Lq−sense. Recently, effort has been put into extending this definition to a pointwise sense: one has to look instead at
kM( bf )kLq(S)≤ Cp,q,d,Skf kLp(Rd),
where M is a suitable maximal operator. In [MRW19], the authors prove, for the first time, such a statement about restriction to curves. Their techniques adapt the ones in [CS72] to the maximal context. The works of Vitturi [Vit17], Kovaˇc and Oliveira e Silva [KOeS18] and Ramos [Ram18] have subsequently dealt with this problem, extending the maximal restriction property to higher dimensions, considering variational versions of it and sharpening the results in [MRW19].
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More recently, Kovaˇc [Kov19] proved a general, abstract principle for such pointwise statements to hold. One of his results is that, whenever restriction estimates like (5.1) hold with p < q, and whenever µ : B(Rd) → C is a complex measure such that |∇bµ(ξ)| ≤ D(1 + |ξ|)−1−η, for some η > 0, then
k sup
t>0
| bf ∗ µt(x)|kLq(S)≤ Cp,q,S,η· Dkf kLp(Rd). (5.2) Here, µt(E) := µ(t−1E). Note that dµ = χB(0,1)(x)dx satisfies the Fourier decay condition above in any dimension, which generalizes the results of Vitturi [Vit17], M¨uller, Ricci, Wright [MRW19] and Kovaˇc and Oliveira e Silva [KOeS18].
The purpose of this note is to employ the techniques in [Ram18] to extend inequality (5.2) in low-dimensional cases not covered by Kovaˇc’s techniques. Additionally, we simplify the techniques in [Ram18] in order to extend a result from [MRW19].
5.1.1 Two-dimensional results
In (5.2), the main requirement on the measure µ that |∇µ(ξ)| .b η,µ (1 + |ξ|)−1−η, for some η > 0, is only satisfied by the spherical measure dµ = dσSd−1 if d ≥ 4. Therefore, Kovaˇc’s result does not yield bounds for lower-dimensional restrictions of spherical maxi-mal functions of the Fourier transform. This was our motivation for the first result of this paper.
Theorem 5.1. Let µ be a positive, finite Borel measure defined in R2, and suppose that the maximal function
Mµg(x) := sup
t>0
|g| ∗ µt(x).
is bounded from Lr(R2) → Lr(R2), whenever r > 2. Then the following bound holds:
kMµ( bf )kLq(S1)≤ Cp,µkf kLp(R2), where 1 ≤ p < 43, p0 ≥ 3q.
In Proposition 5.9 at the end of this note, we prove that Kovaˇc’s [Kov19] assumptions on the measure imply ours. The spherical maximal function in dimensions 2, 3 is an ex-ample that shows, as elaborated in Section 5.4.1, that Theorems 5.1 and 5.4 are strictly stronger.
On the other hand, in [MRW19], the authors, in the end of their manuscript, make use of the maximal function
M2(h) := M (|h|2)1/2, where M f (x) = supr>0 −R
B(x,r)|f | denotes the usual Hardy–Littlewood maximal function, to prove results about Lebesgue points of the Fourier transform on curves in the range 1 ≤ p < 87. In [Ram18], this author circumvents this problem by considering a suitable linearization instead of working with M2. Our next result combines the two approaches:
Theorem 5.2. Let 1 ≤ r ≤ 2. Define the maximal functions Mrh(x) := (M (|h|r)(x))1/r. The following bound holds:
kMr( bf )kLq(S1)≤ Cp,rkf kLp(R2), where 1 ≤ p < 43, p0 ≥ 3q.
5.1. INTRODUCTION 99 The main feature in the proofs of these Theorems is the linearization method employed in [Ram18] together with Lemmata 5.6 and 5.7. These, on the other hand, provide a way to bypass the interpolation scheme employed in [Ram18, Lemmata 1 and 2]. Also, in the case where one takes dµ = dσS1 to be the arc-length measure in the circle the in-terpolation idea fails due to the lack of L2(R2) bounds for maximal functions, whereas working directly with the aid of the Hausdorff–Young inequality gives us the result, as long as the measure we consider satisfies the above conditions. By the celebrated result of Bourgain [Bou86], this is exactly the case for the circular maximal function in dimension 2.
In Section 5.4.3, we present two different kinds of counterexamples, in order to impose restrictions on r so that Theorem 5.2 can hold. Both the examples yield the same r ≤ 4 bound, whereas Theorem 5.2 only works in the r ≤ 2 case. One is led to pose the following question:
Question 5.3. Can the two-dimensional full range of maximal restriction inequalities hold for Ms, 2 < s ≤ 4?
5.1.2 Three-dimensional results
In dimension 3, our main theorems deal with the Tomas-Stein exponent case, in both the context of measures as well as in the context of Mr−maximal functions:
Theorem 5.4. Let Let µ be a positive, finite Borel measure defined in R3, and suppose that the maximal function
Mµg(x) := sup
t>0
|g| ∗ µt(x).
is bounded from L2(R3) → L2(R3). Then the following bound holds:
kMµ( bf )kL2(S2)≤ Cp,µkf kLp(R3), where 1 ≤ p ≤ 43.
Theorem 5.5. Let 1 ≤ r < 2. Then the following bound holds:
kMr( bf )kL2(S2)≤ Cp,rkf kLp(R3),
where 1 ≤ p ≤ 43. Aditionally, the quadratic maximal function M2 satisfies that kM2( bf )kL2(S2)≤ Cp,rkf kLp(R3),
whenever 1 ≤ p < 43.
We prove these results in Section 5.3 by merging the ideas in Theorems 5.1 and 5.2 with Vitturi’s method. As a by-product, the counterexamples built in Section 5.1.1 provide us with the restriction that s ≤ 2 in order for Theorem 5.5 to hold. In particular, a further use of one of these counterexamples in higher dimensions gives us as a direct corollary that the only dimensions in which a full-range restriction result for the strong maximal function
MSf (x) := sup
R axis parallel, centered at x
− Z
R
|f |
of the Fourier transform could hold are d = 2, 3. We talk about this property in more detail in Proposition 5.12.
5.1.3 Notation
In what follows, we denote A . B to mean that A ≤ C · B, for some universal constant C > 0. If we let C depend on a parameter α, we write A .α B. We suppress this notation in case the specific dependence on α is not important. We also normalize the Fourier transform as F f (ξ) = bf (ξ) =R
Rde−2πix·ξf (x) dx. Finally, we often write −R
Bg := |B|1 R
Bg for the average of g over a set B.
Acknowledgements. The author would like to thank Felipe Gon¸calves for helpful dis-cussions which led to the proof of Theorem 5.1 and to a great deal of inspiration for the other results in this manuscript. He is also indebted to his supervisor Prof. Dr. Christoph Thiele for reading this manuscript and giving advice on how to improve the presenta-tion. The author also acknowledges finantial support by the Deutscher Akademischer Austauschdient (DAAD).