Hardy-type operators involving suprema have turned out to be a useful tool in the theory of interpolation, for deriving Sobolev-type inequalities, for estimates of the non-increasing rearrangements of fractional maximal functions or for the description of norms appearing in optimal Sobolev embeddings. This thesis deals with the compactness of these operators on weighted Banach function spaces. We de ne a category of pairs of weighted Banach function spaces and formulate and prove a criterion for the compactness of a Hardy-type operator involving supremum which acts between a couple of spaces belonging to this category. Further, we show that the category contains speci c pairs of weighted Lebesgue spaces determined by a relation between the exponents. Besides, we bring an extension of the criterion to all weighted Lebesgue spaces, in proof of which we use characterization of the compactness of operators having the range in the cone of non-negative non-increasing functions, introduced as a separate result.

In this thesis we shall provide the reader with results in the field of classical Lorentz spaces. These spaces have been studied since the 50's and have many applications in partial differential equations and interpolation theory. This work includes five papers. First paper studies the properties of Generalized Gamma spaces. Second paper provides an alternative proof of normability characterization of classical Lorentz spaces. The third paper discus conditions of linearity and quasi-norm property of rearrangement-invariant lattices. The following paper gives a characterization of normability of Gamma spaces. And finally the last paper characterizes the embeddings between Generalized Gamma spaces. Powered by TCPDF (www.tcpdf.org)

In this thesis we introduce the concept of Wiener-Luxemburg amalgam spaces which are a modification of the more classical Wiener amalgam spaces intended to address some of the shortcomings the latter face in the context of rearrangement invariant Banach function spaces. We first provide some new results concerning quasinormed spaces. Then we illustrate the asserted shortcomings of Wiener amalgam spaces by provid- ing counterexamples to certain properties of Banach function spaces as well as rearrangement invariance. We introduce the Wiener-Luxemburg amalgam spaces and study their properties, including (but nor limited to) their normability, em- beddings between them and their associate spaces. Finally we provide some applications of this general theory. 1

El presente trabajo está dedicado al análisis de una clase particular de operadores (lineales y continuos) entre espacios de Banach de funciones. El objetivo es avanzar en la teoría de los llamados operadores factorizables a la p-potencia analizando todos los aspectos de la dualidad. Esta clase de operadores ha demostrado ser de utilidad tanto en la teoría de factorización de operadores sobre espacios de Banach de funciones (teoría de Maurey-Rosenthal) como en el Análisis Armónico (dominios óptimos de la transformada de Fourier y operadores de convolución). A ¿n de desarrollar esta teoría de dualidad y sus aplicaciones, se de¿ne y estudia una nueva clase de operadores con propiedades de extensión que involucran al operador y a su adjunto. Ésta es la familia de operadores factorizables a la (p,q)- potencia, 1 · p,q Ç 1, y pueden caracterizarse mediante un esquema de factorización a través del espacio de p-potencias del dominio y el dual del espacio de q-potencias del dual del codominio. También se obtiene una equivalencia mediante un diagrama de factorización a través de espacios L p (m) y L q (n) 0 , donde m y n son medidas vectoriales adecuadas y ésta será nuestra principal herramienta. Para esta construcción resultan necesarios algunos resultados preliminares relativos a las p-potencias de los espacios de Banach de funciones que intervienen y que también se estudian. Con estos útiles se dan algunos resultados para caracterizar el rango óptimo ¿el menor espacio de Banach de funciones en el que puede tomar valores el operador¿ para operadores que van de un espacio de Banach a un espacio de Banach de funciones. Además, se desarrolla y presenta formalmente la idea de factorización óptima de un operador que optimiza una factorización previa, en términos del diagrama que debe satisfacer un operador factorizable a su (p,q)-potencia. Todos estos resultados extienden los actuales cálculos del dominio óptimo mediante medidas vectoriales para operadores sobre espacios de Banach de funciones. Dichos cálculos han dado resultados relevantes en diversas áreas del análisis matemático mediante una descripción del mayor espacio de Banach de funciones al cual, operadores relevantes ¿como la transformada de Fourier o el operador de Hardy¿ se pueden extender. La teoría se aplica para encontrar nuevos resultados en determinados campos: como la teoría de interpolación de operadores entre espacios de Banach de funciones, los operadores de núcleo y en particular, la transformada de Laplace. / Galdames Bravo, OE. (2013). Duality theory for p-th power factorable operators and kernel operators [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/31523 / TESIS

We study the behaviour of logarithmically convex combinations of operators given by Tf = |S1f| 1 θ |S2f|1− 1 θ , where S1, S2 are some (usually quasi-linear) operators acting on spaces of measurable functions and θ ∈ (1, ∞) is a parameter. We develop two, quite different in nature, interpolation theories, each of which enables us to obtain a rather com- prehensive information about the behavior of such operators on function spaces. The first one is completely general and is based on abstract interpolation and Calderón spaces. We illustrate the theoretical results by a wide variety of examples of pairs of spaces X, Y such that T: X → Y is bounded, these in particular include the so-called Calderón-Lozanovskiı̌ construction. The second theory departs from pointwise estimates by Calderón operators and is particularly tailored for obtaining boundedness results between Orlicz spaces given weak-type estimates that arise in applications. A common feature of both theories is an approach, apparently new, involving interpolation of four spaces. The input data in each case consists of two reasonable separate endpoint estimates for the operators S1 and S2. 1

The present thesis is devoted to the study of various properties of Banach func- tion spaces, with a particular emphasis on applications in the theory of Sobolev spaces and in harmonic analysis. The thesis consists of four papers. In the first one we investigate higher-order embeddings of Sobolev-type spaces built upon rearrangement-invariant Banach function spaces. In particular, we show that optimal higher-order Sobolev embeddings follow from isoperimetric inequal- ities. In the second paper we focus on the question when the above-mentioned Sobolev-type space is a Banach algebra with respect to a pointwise multiplica- tion of functions. An embedding of the Sobolev space into the space of essentially bounded functions is proved to be the answer to this question in several standard as well as nonstandard situations. The third paper is devoted to the problem of validity of the Lebesgue differentiation theorem in the context of rearrangement- invariant Banach function spaces. We provide a necessary and sufficient condition for the validity of this theorem given in terms of concavity of certain functional depending on the norm in question and we find also alternative characterizations expressed in terms of properties of a maximal operator related to the norm. The object of the final paper is the boundedness of the...