Weighted Fourier inequalities and uncertainty principle relations

Abstract

[en] The main actor of this project is the Fourier transform, which, for $f$ integrable is defined as $$ \hat{f}(\xi)=\int_{\mathbb{R}} f(x) e^{2 \pi i x \xi} d x $$ This object, and its discrete counterpart, the Fourier series, are extremely important, not only in Mathematics but also in Physics and Engineering. In spite of its simple definition, the relation between $f$ and $\hat{f}$ is not at all easy to understand. To this end, we can use norm inequalities, like the Hausdorf-Young inequality, which, for $1 \leq p \leq 2$ states that $$ \|\hat{f}\|_{p ;} \leq K\|f\|_p $$ or Uncertainty Principle relations, which, roughly speaking, assert that the Fourier transform of a localized function is not localized. The most famous of these is the Heisenberg Uncertainty Principle: $$ \|x f\|_2\|\xi \hat{f}\|_2 \geq K\|f\|_2^2 $$ From a qualitative point of view, these inequalities tell us that that the Fourier transform of an integrable function can not have important blow-ups and that the transform of a concentrated function can not be concentrated around one point. The goal of this thesis is to present generalizations of the aforementioned inequalities. First, to obtain relations between the distribution of $f$ and $\hat{f}$ in their domain, we study Weighted Fourier inequalities, that is, inequalities of the form $$ \|\hat{f}\|_{q, u} \leq K\|f\|_{p, v} $$ where $u, v$ are weights, that is, non-negative measurable functions. Observe that inequality $(3$ is clearly a generalization of (1). Second, there are many ways in which the idea behind the Uncertainty Principle, that is, transforms of localized functions must be spread over their domain, can be quantified. For instance, we can measure the degree of localization of a function by studying its rate of decay, by computing the fraction of its mass which lies outside of some region, or by studying generalizations of inequality 22 , namely, $$ \|f\|_{p, u}\|\hat{f}\|_{q, v} \geq K\|f\|_r $$ The work is devoted to surveying known results and obtaining new ones on the previously mentioned problems. The thesis is organized as follows. Chapter 1 is devoted to introducing some preliminary results and concepts which are used in this work. In Chapter 2 we review the conditions on $u$ and $v$ obtained in [4) which guarantee that inequality (3) holds. Next, we review classical necessary conditions and obtain new ones (Theorems 2.3.6 and 2.3.7), thereby showing that the conditions in 44 are necessary when $u$ and $v$ satisfy a natural monotonicity condition. To conclude this chapter, we further explore the topic by studying the case of non-monotonous $u, v$ and obtain new results. Finally, in Chapter 3 we survey several forms of the Uncertainty Principle (UP): the Hardy UP, the Amrein-Berthier UP and the Nazarov UP. We also study UP of the type (4), extending the results obtained in 28 , for the whole range of parameters, (see Theorem 3.4 .2 . Moreover, we fully characterize a symmetric Heisenberg type UP with broken power weights, see Theorem 3.5.1. Finally, I would like to thank my supervisor, Sergey Tikhonov, for his guidance and also Kristina Oganesyan for numerous helpful remarks.