Some sharp and endpoint inequalities in Harmonic Analysis

Abstract

This cumulative dissertation is dedicated to the study of certain sharp and endpoint inequalities in Harmonic Analysis. <br /> The first part of this thesis focuses on sharp Fourier extension inequalities on spheres. Our objective is to determine the optimal constant for these inequalities, as well as to characterize maximizers, that is, functions that attain the optimal constant. In this part of the dissertation, we present a sharp Fourier extension inequality on the circle, in the Stein–Tomas endpoint case, for functions whose Fourier support satisfies certain arithmetic properties. These properties correspond to a generalization of the definition of B(3)-sets. Among functions in this class, we show that constant functions are the unique maximizers. Then, we consider sharp mixed-norm Fourier extension inequalities on the sphere. First, we extend the range of exponents for which constant functions are known to be maximizers for these inequalities. Next, we show that, in this very same range of exponents, constant functions are local maximizers for the L^p(S^d-1) to L^p(R^d) Fourier extension inequality. <br /> The second part of this dissertation focuses on weak-type endpoint estimates for certain Marcinkiewicz multiplier operators. Marcinkiewicz multipliers on the real line are bounded functions of uniformly bounded variation on each Littlewood–Paley dyadic interval. Optimal weak-type endpoint estimates for the associated multiplier operators have been obtained by Tao and Wright showing that these operators map locally Llog^1/2L to weak L¹. Here, we consider higher order Marcinkiewicz multipliers, that is, multipliers of uniformly bounded variation on each interval arising from a higher order lacunary partition of the real line. We present optimal weak-type endpoint estimates for the corresponding multiplier operators, recovering for the first order case the result of Tao and Wright.