Duality Theory for p-th Power Factorable Operators

The class of p-th power factorable operators was developed in 2008 by Okada, Ricker and Sánchez Pérez. This is a family of Banach space valued (linear and continuous) operators defined on a Banach function space over a finite measure, which can be extended to the p-th power space of its original domain. This class of operators has been applied to obtain generaliztions of Maurey-Rosenthal's Theorem, and also to the study of the largest (by inclusion) p-convex domain of the convolution operators and the Fourier transform. Here we develop the duality of this class and obtain generalizations of these results by means of factorizations through p-convex and q'-concave spaces, these spaces have optimal properties in its domain and range, respectively. This technique is very useful, since now we have shown that these properties are invariant by complex interpolation and also we see how we can easily apply to kernel operators, as the Laplace transform. This memoir arises from the Ph.D. thesis of the author presented at the Universitat Politècnica de València and supervised by the professors Fernando Mayoral Masa and Enrique A. Sánchez Pérez, with whom the author is greatly indebted.