Analytical perspectives on ko¨ the amalgams and newton-Sobolev spaces

Abstract

Function spaces have not only played a crucial role in the genesis of functional analysis from its very beginning but have also been instrumental in the development of modern theories, including the analysis of partial differential equations, distribution theory, interpolation theory, approximation theory, and beyond. The classical Lebesgue spaces are among the earliest classes of measurable functions that have have played a central role in analysis. However, by the first half of the 20th century, it became clear that these spaces were insufficient for describing the mapping properties of certain operators. Consequently, several new classes of measurable function spaces, such as Lorentz and Orlicz spaces, were introduced. In parallel, C.S. Herz introduced another generalization of Lebesgue spaces called the Herz spaces. Over recent decades, Herz spaces have seen significant advancements due to their extensive applications. Nevertheless, even Herz spaces are sometimes insufficient for capturing finer properties of functions and operators. To overcome these drawbacks, we introduce a novel class of functions called K¨othe amalgam spaces. These spaces not only extend various classical function spaces in an abstract framework but also unify them under a single umbrella. These spaces consist of both local and global components, where the local components are infinite direct sums of quasi-Banach spaces, and the global components are the appropriate sequence spaces. Our primary objective is to investigate key properties of these spaces, including integral inequalities, topological and geometric properties, interpolation properties, and the boundedness of operators. The study of Lorentz-Herz and Orlicz-Herz spaces, built upon this foundation, offers an enhanced structure to classical Herz spaces. These spaces are particularly suited for capturing refined operator properties that Herz spaces cannot fully describe. Furthermore, this thesis examines some fundamental properties of distribution functions and decreasing rearrangements, which play a central role in the study of function spaces, particularly Lorentz spaces. Necessary and sufficient conditions for their continuity are established, and a classical result concerning their integral norms, referred to as the rearrangement identity , is extended to a more general setting. Additionally, the rearrangement identity is employed to derive several inequalities that bound certain linear combinations of tail probabilities. Another key aspect of this dissertation is the study of pointwise properties of Sobolev functions defined on metric measure spaces. Through the development of a Wiener-type integral condition, we show that Sobolev functions exhibit a strong form of the Lebesgue point property. Moreover, we provide explicit criteria for pð€€€fine continuity of Sobolev functions in non-Euclidean settings that support a Poincar´e inequality.