Translation invariant operators

Abstract

[en] The theory of translation invariant operators is a branch that involves techniques and concepts from a wide variety of fields such as functional analysis, harmonic analysis or Fourier analysis, and whose main goal is to find and describe boundedness conditions and norm estimates on such objects. Its study started to firmly develop around the second half of the last century, thanks to the work of Hörmander, Fefferman, Jodeit, Stein, Marcinkiewicz, and so many others, but it is still a subject of great interest nowadays, not only because of its applications and relations to other fields, such as partial differential equations or summability methods for Fourier coefficients, but also because of the large amount of open questions that the theory still has to answer. For instance, it is not known yet a complete characterization on when such an operator is bounded between two Lebesgue spaces. Maybe that is one of its most surprising aspects: even though its main object of study is a class of operators which are initially defined with a very simple property (namely, commuting with translations), it turns out that this condition gives rise to a rich structure that requires of much more sophisticated tools in order to understand it. The current development of the theory tries to follow that path, using those tools, such as the theory of distributions and Fourier transform, to translate the problem of boundedness of operators to a problem of studying properties of certain functions, which in this context are called multipliers. This is indeed a helpful step, for one can then find symmetry properties on the classes of multipliers of bounded operators between two Lebesgue spaces, and even arrive to the conclusion that when both spaces are actually the same, every multiplier is a bounded function. Yet, starting from a bounded function, finding which of its properties analytical, geometrical,...) might be relevant in order to have a multiplier is a tough task that, as we say, is still barely solved. It is known, for instance, that when the multiplier is a characteristic function of a set, the geometry of the set plays a huge role. The maximum exponent of this fact is probably the, now disproven, multiplier problem for the ball. Even when the geometry of the set gives good boundedness properties (such as the case of a polygon), there is still much work to do on finding sharp constants for the norm of such operators. On the other hand, regularity conditions seem to affect as well, thanks to a couple of results due to Hörmander and Marcinkiewicz. Our goal in this Master’s Thesis is to give an overview of the most relevant as- pects of the theory, starting from the most basic definitions, and ending precisely in the study of Hörmander’s and Marcinkiewicz’s results and the multiplier problem for the ball. Chapters 1 and 2 cover the initial tools that one needs in order to attack this topic, such as the essential elements of the theory of function spaces and distributions, and the definition of translation invariant operators, together with their identification with multipliers and the properties of multipliers’ classes, given mostly by Theorem 2.15. We then move to Chapter 3 in order to review a theory due to Littlewood and Paley about decomposition of functions in the frequency domain using multipliers, which gives estimates on such decompositions in terms of the original function. The idea behind this theory is to extend estimates which are natural on the space of square integrable functions to other spaces of integrable functions, as done in Theorem 3.4. This theory is later used in Chapter 4 to prove the mentioned results of Hörmander and Marcinkiewicz (Theorems 4.2 and 4.5), and we see some applications of these results to multipliers with simple homogeneity properties. Finally, we focus on Chapter 5 to review this influence of geometrical properties on boundedness of this class of operators, where we have exposed Fefferman’s work on the ball multiplier problem. This work, which culminates with the conclusion of Theorem 5.6, represents a nice example on how the theory of translation invariant operators intersects with other seemingly unrelated branches, for it uses a geometrical construction originated around the first half of the past century known as Perron’s tree, which was originally proposed for solving Kakeya’s needle problem.