Fourier Transform and Convolutions on L p of a Vector Measure on a Compact Hausdorff Abelian Group
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Fourier Transform and Convolutions on Lp of a Vector Measure on a Compact Hausdorff Abelian Group J.M. Calabuig · F. Galaz-Fontes · E.M. Navarrete · E.A. Sánchez-Pérez
Received: 22 November 2011 / Published online: 4 December 2012 © Springer Science+Business Media New York 2012
Abstract Let ν be a countably additive vector measure defined on the Borel subsets B(G) of a compact Hausdorff abelian group G. In this paper we define and study a vector valued Fourier transform and a vector valued convolution for functions which are (weakly) integrable with respect to ν. A form of the Riemann Lebesgue Lemma and a Uniqueness Theorem are established in this context. In order to study the vector valued convolution we discuss the invariance under reflection in G of these spaces of integrable functions. Finally we present a Young’s type inequality in this setting and several relevant examples, namely related with the vector measure associated to different important classical operators coming from Harmonic Analysis. Keywords Countably additive vector measure · Space of p-integrable functions · Fourier transform · Convolution · Pettis integrability Mathematics Subject Classification (2000) 46G10 · 42A38 · 44A35
Communicated by Paul Butzer. J.M. Calabuig () · E.A. Sánchez-Pérez Instituto Universitario de Matemática Pura y Aplicada, Universidad Politécnica de Valencia, Camino de Vera s/n, 46022 Valencia, Spain e-mail: [email protected] E.A. Sánchez-Pérez e-mail: [email protected] F. Galaz-Fontes · E.M. Navarrete Centro de Investigaciones Matemáticas, A.C., Jalisco S/N, Col. Valenciana, 36240 Guanajuato, Gto., México F. Galaz-Fontes e-mail: [email protected] E.M. Navarrete e-mail: [email protected]
J Fourier Anal Appl (2013) 19:312–332
313
1 Introduction Let G be a compact Hausdorff abelian group and mG its normalized Haar measure, defined on the Borel subsets B(G) of G. Take X to be a complex Banach space, ν : B(G) → X a (countably additive) vector measure and consider L1w (ν), the space of weakly integrable functions with respect to ν. The main goal of this article is to introduce and study a vector valued Fourier transform and a vector valued convolution for (weakly) integrable functions with respect to a vector measure. The paper consists of four sections. After this preliminary first section, in the second one we start by introducing the notion of vector valued Fourier transform. First for integrable functions with respect to ν (see Definition 2.1) and then for weakly integrable functions with respect to ν (see Definition 2.6). After giving some important examples we prove that a natural version of the Riemann-Lebesgue’s Lemma is not true in this setting. However, always assuming the vector measure ν is absolutely continuous with respect to mG , we characterize those measures for which this important lemma is satisfied (see Theorem 2.5). We finish this section with a version of a Uniqueness Theorem (see Theorem 2.7 and Corollary 2.8). The definition of the convolution, that is the main goal of the last section, takes us na
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