Resumen de Operator means and spectral integration of Fourier multipliers
Earl Berkson, T. Alastair Gillespie 
Let (Y,µ) be a sigma-finite measure space. Suppose that p is a real number greater than 1, and T is an absolutely mean-bounded, invertible, separation-preserving linear mapping of Lp(µ) onto itself. In this setting Martín-Reyes and de la Torre established the existence of a uniform Ap weight estimate in terms of discrete weights generated by the pointwise action on Y of certain measurable functions canonically associated with T. By proceeding from this fact we have previously used the interaction between real analysis methods and Ap weights to show that the spectral structure of T transfers to the space Lp(µ) the multiplier actions of the class M1(C), consisting of the classical Marcinkiewicz multiplier functions defined on the unit circle C in the complex plane. The purpose of the present article is to study the overall state of affairs for the function classes Mq(C), where q > 1. Mq(C) consists of the bounded complex-valued functions on C whose q-variations on the dyadic arcs are uniformly bounded, and the source of the multiplier properties of Mq(C) in weighted Lp-spaces can ultimately be found in Rubio de Francia¿s weighted Littlewood-Paley inequality for arbitrary intervals, which requires p to exceed 2, and treats weights belonging to A(p/2). Accordingly, we show here that the relevant operator class for achieving the transference of appropriate Mq(C) multiplier classes to Lp(µ), p at least 2, consists of the mean2-bounded operators on Lp(µ) which are separation-preserving. When p > 2 the spectral structure of such operators also transfers to Lp(µ) the aforementioned weighted Littlewood-Paley Inequality for Arbitrary Intervals.
