Transference theory between quasi-Banach function spaces with applications to the restriction of Fourier multipliers.

Abstract

In the early 1970 fs, R. Coifman and G. Weiss, generalizing the techniques introduced by A. Calderon, developed a method for transferring abstract convolution type operators, defined on general topological groups, and their respective bounds, to the so called gtransferred operators h, which are operators defined on general measure spaces. To be specific, if G is a topological group and R_x is a representation of G on some Banach space B and K is a convolution operator on G given by Kf= çk(x-y) f(y) dy with k an L^1 function, the transferred operator T is defined by letting Tf= çk(x-y) R_xf(y) dy. Transfer methods deal with the study of the preservation of properties of K that are still valid for T, mostly focusing on the preservation of boundedness on Lebesgue spaces Lp. These methods has been applied to several problems in Mathematical Analysis, and especially to the problem of restrict Fourier multipliers to closed subgroups. These techniques have been extended by other authors as N. Asmar, E. Berkson and A. Gillespie, among many others. It is worth noting however, that these prior developments have always been focused on inequalities for operators on Lebesgue spaces Lp. In this thesis there are developed several transference techniques for quasi-Banach spaces more general than Lebesgue spaces Lp, as Lorentz spaces Lp, q, Orlicz-Lorentz, Lorentz-Zygmund spaces as well as for weighted Lebesgue spaces Lp(w). The most significant applications are obtained in the field of restriction of Fourier multipliers for rearrangement invariant spaces and weighted Lebesgue spaces Lp(w). Specifically, we get generalizations of the results obtained by K. De Leeuw for Fourier multipliers. There are also developed similar techniques in the context of multilinear operators of convolution type, where the basic example is the bilinear Hilbert transform, as well as for modular inequalities and inequalities arising in extrapolation