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Smooth Molecular Decompositions of Functions and Singular Integral Operators
J. E. Gilbert, University of Texas, Austin, TX, Y. S. Han, Auburn University, AL, J. A. Hogan, University of Arkansas, Fayetteville, AR, J. D. Lakey, New Mexico State University, Las Cruces, NM, D. Weiland, Austin, TX, and G. Weiss, Washington University, St. Louis, MO
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Memoirs of the American Mathematical Society
2002; 74 pp; softcover
Volume: 156
ISBN-10: 0-8218-2772-3
ISBN-13: 978-0-8218-2772-7
List Price: US$53
Individual Members: US$31.80
Institutional Members: US$42.40
Order Code: MEMO/156/742
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Under minimal assumptions on a function \(\psi\) we obtain wavelet-type frames of the form \[\psi_{j,k}(x) = r^{(1/2)n j} \psi(r^j x - sk), \qquad j \in \mathbb{Z}, k \in \mathbb{Z}^n,\] for some \(r > 1\) and \(s > 0\). This collection is shown to be a frame for a scale of Triebel-Lizorkin spaces (which includes Lebesgue, Sobolev and Hardy spaces) and the reproducing formula converges in norm as well as pointwise a.e. The construction follows from a characterization of those operators which are bounded on a space of smooth molecules. This characterization also allows us to decompose a broad range of singular integral operators in terms of smooth molecules.

Readership

Graduate students and research mathematicians interested in functional analysis, Calderón-Zygmund theory, singular integral operators, and wavelets.

Table of Contents

  • Main results
  • Molecular decompositions of operators
  • Frames
  • Maximal theorems and equi-convergence
  • Appendix. Proof of basic estimates
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