Equivalent norms on L[superscript p] and E[superscript p] (T) speces

Abstract

In this paper we will be considering two measures on sorae measure space (X, A). For general p > o we can consider the tiro norms |jf|[p,y and }|fj|p,v. We are interested in conditions on y and v that make ||f||p, y - | [f | |p,v for all f in sorae class of functions, i.e. when there exist positive constants 1^ and k2 ^^ tliat L !|f||p,y < k2 ||f||p,v (1) In chapter I we will five necessary and sufficient condition for (l) to hold for all measurable functions on an arbitrary measure space (X, A). The techniques used in this will be standard techniques in neasirrc theory and integration theory. In Chapter II we will restrict ourselves to the real line with the B^rel sigraa field. The class of functions we are interested in is E^(T), entire functions of exponential type T whose restrictions to R are in L^(R,dx). We mil give conditions on y and v that make (1) hold for all feEp(T). The present work is largely an extension of LIN's work in He analyzed the p = 2 case in M dimensions. We will consider arbitrary p (o < p < °°) in the one dimensional case. Although we have not done so here, there are M dimensional versions of all of our results. When p f 2 we no longer have a Hilbert space and when o < p < 1, we are not even in a Banach space. The techniques used ffo back to Plincberel and Polya in [4] . Methods from functional analysis, complex analysis and real analysis will be used.