Function Spaces with BoundedLpMeans and Their Continuous Functionals

Abstract

This paper studies typical Banach and complete seminormed spaces of locally summable functions and their continuous functionals. Such spaces were introduced long ago as a natural environment to study almost periodic functions (Besicovitch, 1932; Bohr and Fölner, 1944) and are defined by boundedness of suitableLpmeans. The supremum of such means defines a norm (or a seminorm, in the case of the full Marcinkiewicz space) that makes the respective spaces complete. Part of this paper is a review of the topological vector space structure, inclusion relations, and convolution operators. Then we expand and improve the deep theory due to Lau of representation of continuous functional and extreme points of the unit balls, adapt these results to Stepanoff spaces, and present interesting examples of discontinuous functionals that depend only on asymptotic values.

Multi-dimensional Besicovitch almost periodic type functions and applications

“…a multi-dimensional generalization can be introduced analogously. Fairly complete analysis of the generalized Besicovitch class B α,β (R : Y ) and its multi-dimensional analogues is without scope of this paper (let us only observe here that the space W α , considered by M. A. Picardello [58] in the usual one-dimensional setting with 0 < α ≤ 1, is nothing else but the space B 1,α (R : C)).…”

“…Clearly, B p (R n : X) is a closed subspace of M p (R n : X) and therefore a Banach space itself. Concerning the Banach space M p (R n : X), we would like to recall that this space is not separable for any finite exponent p ≥ 1; see, e.g., [58,Theorem 18] which concerns the one-dimensional case.…”

“…For further information about Besicovitch almost periodic functions, Besicovitch almost automorphic functions and their applications, we refer the reader to [1,2,4,5,6,7,9,11,12], [13,14,15,16,29,31,32,35], [36,49,51,53,54,56,57,58,60] and references cited therein; we would like to specially emphasize here the important research monograph [55] by A. A. Pankov.…”

Multi-dimensional besicovitch almost periodic type functions and applications

“…Clearly, B p (R n : X) is a closed subspace of M p (R n : X) and therefore a Banach space itself. Concerning the Banach space M p (R n : X), we would like to recall that this space is not separable for any finite exponent p ≥ 1; see, e.g., [40,Theorem 18] which concerns the one-dimensional case.…”

“…For further information about Besicovitch almost periodic functions, Besicovitch almost automorphic functions and their applications, we refer the reader to [2,3,4,6,7,8,9,22,33,35,36,38,39,40,42] and references cited therein; we would like to specially emphasize here the important research monograph [37] by A. A. Pankov.…”