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 . We show that for such operators which display a zero energy resonance the full range in the Mikhlin theorem cannot be obtained: in the radial, three-dimensional case it shrinks to .

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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:
https://doi.org/10.1090/S0002-9939-06-08621-7

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.