Adjacent and random dyadic systems and their applications to metric, Euclidean and vector-valued analysis
Year: 2016
Type: dissertation
Abstract: Different dyadic techniques are an inseparable part of modern-day harmonic analysis both in the Euclidean space and in metric spaces. In this dissertation, we improve adjacent and random dyadic techniques in metric spaces and apply these and previously known techniques for questions related to metric, Euclidean and vector-valued analysis. The dissertation consists of an introductory part and four research articles. In the first article, we present a general randomization procedure for dyadic systems in metric spaces which can be used for constructing both random and adjacent dyadic systems. As an application of the new random systems, we improve the continuity properties of metric wavelets of P. Auscher and T. Hytönen by exploiting the improved ``smallness of boundary'' property of our random cubes. In the third article, we prove some additional properties for our adjacent dyadic systems to prove a decomposition result for dyadic systems in metric spaces. With its help, we give an alternative proof for the quantitative bound of the Lp norm of shift operators acting on vector-valued functions in metric spaces. In the second article, we explore certain properties of the Muckenhoupt weight classes, the class of Reverse Hölder weights and their weakened versions in spaces of homogeneous type. In the Euclidean setting, the Muckenhoupt weight classes have numerous different equivalent definitions but in spaces of homogeneous type some of those equivalences break down. We show that although certain definitions are no longer equivalent in this context, their weakened versions still define the same weight classes. We also show that every weak Reverse Hölder weight has a self-improving property. In the literature, these types of weak weights appear especially in the theory of partial differential equations. In the fourth article, we prove quantitative weighted bounds for so called rough homogeneous singular integrals by combining older techniques with a quantitative version of M. Lacey's recent extension of the A2 theorem. The proof of this extension is based on a domination technique which provides a way to dominate Calderón-Zygmund operators pointwise with the help of a finite number of simple sparse operators associated with adjacent dyadic systems.
Source: Työväentutkimus Vuosikirja
Author: Olli Tapiola
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2342970108