Abstract and Applied Analysis
Volume 2011 (2011), Article ID 947908, 15 pages
http://dx.doi.org/10.1155/2011/947908
Research Article

On the Generalized Weighted Lebesgue Spaces of Locally Compact Groups

I. Akbarbaglu and S. Maghsoudi

Department of Mathematics, Zanjan University, Zanjan 45195-313, Iran

Received 11 June 2011; Accepted 16 August 2011

Academic Editor: Narcisa C. Apreutesei

Copyright © 2011 I. Akbarbaglu and S. Maghsoudi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let 𝐺 be a locally compact group with a fixed left Haar measure 𝜆 and Ω be a system of weights on 𝐺 . In this paper, we deal with locally convex space 𝐿 𝑝 ( 𝐺 , Ω ) equipped with the locally convex topology generated by the family of norms ( ‖ . ‖ 𝑝 , 𝜔 ) 𝜔 ∈ Ω . We study various algebraic and topological properties of the locally convex space 𝐿 𝑝 ( 𝐺 , Ω ) . In particular, we characterize its dual space and show that it is a semireflexive space. Finally, we give some conditions under which 𝐿 𝑝 ( 𝐺 , Ω ) with the convolution multiplication is a topological algebra and then characterize its closed ideals and its spectrum.