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Journal of Function Spaces and Applications
Volume 8 (2010), Issue 2, Pages 167-179
http://dx.doi.org/10.1155/2010/524036

The convolution algebra H1(R)

University of Maryland, College Park, MD 20742-4105, USA

Received 1 October 2008

Academic Editor: Sten Kaijser

Copyright © 2010 Hindawi Publishing Corporation. 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

H1(R) is a Banach algebra which has better mapping properties under singular integrals than L1(R) . We show that its approximate identity sequences are unbounded by constructing one unbounded approximate identity sequence {vn}. We introduce a Banach algebra Q that properly lies between H1 and L1, and use it to show that c(1 + ln n) ≤ ||vn||H1Cn1/2. We identify the maximal ideal space of H1 and give the appropriate version of Wiener's Tauberian theorem.