This thesis deals with the problem of localization for the Riesz means for eigenfunction expansions of the Laplace-Beltrami operator. The classical Riemann localization principle states that if an integrable function of one variable vanishes in an open set, then its trigonometric Fourier expansion converges to zero in this set. This localization principle fails in R^d with d>=2. In order to recover localization one has to use suitable summability methods, such as the Bochner-Riesz means. In the first chapter of the thesis we focus on the compact rank one symmetric spaces case. While in the second chapter we show how some of the results obtained in the first chapter can be generalize to smooth compact and connected Riemannian manifolds. Consider a d-dimensional compact rank one symmetric space. To every square integrable function, and more generally tempered distribution, one can associate a Fourier series, i.e. an eigenfunction expansion of the Laplace-Beltrami operator. These Fourier series converge in the metric of L^2 and in the topology of distributions, but in general one cannot ensure the pointwise convergence. For this reason we introduce the summability method of Bochner-Riesz means: S_R^\alpha f(x). When \alpha=0 one obtains the spherical partial sums, which are a natural analogue of the partial sums of one-dimensional Fourier series in the Euclidean space. There are examples of the failure of localization in Holder, Lebesgue and Sobolev spaces. Despite the negative results, it has been proved by A.I. Basatis and C. Meaney that there is an almost everywhere localization principle for square integrable functions on compact rank one symmetric spaces; that is, if a square integrable function vanishes almost everywhere in an open set, then its Fourier series is equal to zero for almost every point in this open set. On the other hand, it is known that for square integrable functions localization holds everywhere above the so-called critical index \alpha=(d-1)/2, while for integrable functions the critical index is \alpha=d-1. In this work we continue this line of research in the area of exceptional sets in harmonic analysis. In particular we prove that for Bochner-Riesz means of order \alpha of p-integrable functions on compact rank one symmetric spaces localization holds, with a possible exception in a set of point of suitable Hausdorff dimension. More generally we consider localization for distributions in Sobolev spaces. Some of the result on compact rank one symmetric spaces can be generalize to a general smooth, connected and compact Riemannian manifold without boundary.
|Data di pubblicazione:||18-feb-2016|
|Titolo:||Localization for Riesz Means on compact Riemannian manifolds|
|Settore Scientifico Disciplinare:||MAT/05 - ANALISI MATEMATICA|
|Corso di dottorato:||MATEMATICA PURA E APPLICATA - 23R|
|Citazione:||(2016). Localization for Riesz Means on compact Riemannian manifolds. (Tesi di dottorato, Università degli Studi di Milano-Bicocca, 2016).|
|Parole Chiave (Inglese):||Harmonic analysis, localization, Bochner-Riesz means|
|Appare nelle tipologie:||07 - Tesi di dottorato Bicocca post 2009|