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Journal of Function Spaces and Applications
Volume 3, Issue 3, Pages 263-286
http://dx.doi.org/10.1155/2005/393050

Domains of pseudo-differential operators: a case for the Triebel-Lizorkin spaces

Department of Mathematics, Aalborg University, Fredrik Bajers Vej 7G, DK-9220 Aalborg Øst, Denmark

Received 1 September 2004

Academic Editor: Victor Burenkov

Copyright © 2005 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

The main result is that every pseudo-differential operator of type 1, 1 and order d is continuous from the Triebel-Lizorkin space Fp,1d to Lp, 1p, and that this is optimal within the Besov and Triebel-Lizorkin scales. The proof also leads to the known continuity for sd, while for all real s the sufficiency of Hörmander's condition on the twisted diagonal is carried over to the Besov and Triebel-Lizorkin framework. To obtain this, type 1, 1-operators are extended to distributions with compact spectrum, and Fourier transformed operators of this type are on such distributions proved to satisfy a support rule, generalising the rule for convolutions. Thereby the use of reduced symbols, as introduced by Coifman and Meyer, is replaced by direct application of the paradifferential methods. A few flaws in the literature have been detected and corrected.