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

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

Published electronically:
August 4, 2006

MathSciNet review:
2255290

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 .

**[Agm]**Agmon, S.*Spectral properties of Schrödinger operators and scattering theory.*Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 2 (1975), no. 2, 151-218. MR**0397194 (53:1053)****[Aub]**Aubin, T.*Nonlinear analysis on manifolds. Monge-Ampère equations.*Grundlehren der Mathematischen Wissenschaften, 252. Springer-Verlag, New York, 1982. MR**0681859 (85j:58002)****[ErdSch]**Erdogan, M. B., Schlag, W.*Dispersive estimates for Schrödinger operators in the presence of a resonance and/or an eigenvalue at zero energy in dimension three: I*, Dynamics of PDE, vol. 1, no. 4 (2004), 359-379. MR**2127577****[GesZin]**Gesztesy, F., Zinchenko, M.*On spectral theory of Schrödinger operators with strongly singular potentials.*preprint 2005.**[JenKat]**Jensen, A., Kato, T.*Spectral properties of Schrödinger operators and time-decay of the wave functions.*Duke Math. J. 46 (1979), no. 3, 583-611. MR**0544248 (81b:35079)****[JenNak1]**Jensen, A., Nakamura, S.*and Besov estimates for Schrödinger Operators.*Advanced Studies in Pure Math. 23, Spectral and Scattering Theory and Applications (1994), 187-209.**[JenNak2]**Jensen, A., Nakamura, S.*-mapping properties of functions of Schrödinger operators and their applications to scattering theory.*J. Math. Soc. Japan 47 (1995), no. 2, 253-273. MR**1317282 (95m:47087)****[JenNen]**Jensen, A., Nenciu, G.*A unified approach to resolvent expansions at thresholds.*Rev. Math. Phys. 13 (2001), no. 6, 717-754. MR**1841744 (2002e:81031)****[KriSch]**Krieger, J., Schlag, W.*On the focusing critical semi-linear wave equation*, preprint 2005.**[Sog]**Sogge, C.*Lectures on nonlinear wave equations.*Monographs in Analysis, II. International Press, Boston, MA, 1995. MR**1715192 (2000g:35153)****[Ste1]**Stein, E. M.*Topics in harmonic analysis related to the Littlewood-Paley theory.*Annals of Mathematics Studies, No. 63, Princeton University Press, Princeton, N.J., University of Tokyo Press, Tokyo, 1970. MR**0252961 (40:6176)****[Ste2]**Stein, E.*Harmonic analysis*, Princeton University Press, Princeton, 2004.**[Tal]**Talenti, G.*Best constant in Sobolev inequality.*Ann. Mat. Pura Appl. (4)**110**(1976), 353-372. MR**0463908 (57:3846)****[Yaj]**Yajima, K.*The -continuity of wave operators for Schrödinger operators.*J. Math. Soc. Japan 47 (1995), no. 3, 551-581. MR**1331331 (97f:47049)****[Yaj2]**Yajima, K.*Dispersive estimate for Schrödinger equations with threshold resonance and eigenvalue*, preprint 2004, to appear in Comm. Math. Phys.

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
35J10,
42B15,
35P10,
42B25

Retrieve articles in all journals with MSC (2000): 35J10, 42B15, 35P10, 42B25

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.