A remark on LittlewoodPaley theory for the distorted Fourier transform
Author:
W. Schlag
Journal:
Proc. Amer. Math. Soc. 135 (2007), 437451
MSC (2000):
Primary 35J10, 42B15; Secondary 35P10, 42B25
Published electronically:
August 4, 2006
MathSciNet review:
2255290
Fulltext PDF Free Access
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Abstract: We consider the classical theorems of Mikhlin and LittlewoodPaley 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, threedimensional case it shrinks to .
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 Agmon, S. Spectral properties of Schrödinger operators and scattering theory. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 2 (1975), no. 2, 151218. MR 0397194 (53:1053)
 [Aub]
 Aubin, T. Nonlinear analysis on manifolds. MongeAmpère equations. Grundlehren der Mathematischen Wissenschaften, 252. SpringerVerlag, 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), 359379. 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 timedecay of the wave functions. Duke Math. J. 46 (1979), no. 3, 583611. 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), 187209.
 [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, 253273. MR 1317282 (95m:47087)
 [JenNen]
 Jensen, A., Nenciu, G. A unified approach to resolvent expansions at thresholds. Rev. Math. Phys. 13 (2001), no. 6, 717754. MR 1841744 (2002e:81031)
 [KriSch]
 Krieger, J., Schlag, W. On the focusing critical semilinear 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 LittlewoodPaley 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), 353372. MR 0463908 (57:3846)
 [Yaj]
 Yajima, K. The continuity of wave operators for Schrödinger operators. J. Math. Soc. Japan 47 (1995), no. 3, 551581. 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.
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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/S0002993906086217
PII:
S 00029939(06)086217
Keywords:
LittlewoodPaley 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 DMS0300081 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.
