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Journal of Function Spaces and Applications
Volume 4, Issue 3, Pages 329-341

Modulation spaces Mp,q for 0<p, q

Department of Mathematics, Tokyo University of Science, Kagurazaka 1-3, Shinjuku-ku, Tokyo 162-8601, Japan

Received 1 September 2005

Academic Editor: Hans Triebel

Copyright © 2006 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.


The purpose of this paper is to construct modulation spaces Mp,q(Rd) for general 0<p, q, which coincide with the usual modulation spaces when 1p,q, and study their basic properties including their completeness. Given any gS(Rd) such that supp ĝ  {ξ|ξ|1} and kZdĝ(ξ-αk)1, our modulation space consists of all tempered distributions f such that the (quasi)-norm fM[g]p,q:≔(Rd(Rd|f*(Mωg)(x)|pdx)qpdω)1q is finite.