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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A remark on Littlewood-Paley theory for the distorted Fourier transform


Author: W. Schlag
Journal: Proc. Amer. Math. Soc. 135 (2007), 437-451
MSC (2000): Primary 35J10, 42B15; Secondary 35P10, 42B25
Published electronically: August 4, 2006
MathSciNet review: 2255290
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Abstract: We consider the classical theorems of Mikhlin and Littlewood-Paley from Fourier analysis in the context of the distorted Fourier transform. The latter is defined as the analogue of the usual Fourier transform as that transformation which diagonalizes a Schrödinger operator $ -\Delta+V$. We show that for such operators which display a zero energy resonance the full range $ 1<p< \infty$ in the Mikhlin theorem cannot be obtained: in the radial, three-dimensional case it shrinks to $ \frac32<p<3$.


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Additional Information

W. Schlag
Affiliation: Department of Mathematics, University of Chicago, 5734 South University Ave., Chicago, Illinois 60637
Email: schlag@math.uchicago.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-06-08621-7
PII: S 0002-9939(06)08621-7
Keywords: Littlewood-Paley theory, distorted Fourier transform, zero energy resonances of Schr\"odinger operators
Received by editor(s): August 29, 2005
Published electronically: August 4, 2006
Additional Notes: The author was partially supported by NSF grant DMS-0300081 and a Sloan Fellowship.
Communicated by: Andreas Seeger
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.