Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
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 $ -\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$.


References [Enhancements On Off] (What's this?)

  • [Agm] Shmuel Agmon, Spectral properties of Schrödinger operators and scattering theory, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 2 (1975), no. 2, 151–218. MR 0397194
  • [Aub] Thierry Aubin, Nonlinear analysis on manifolds. Monge-Ampère equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 252, Springer-Verlag, New York, 1982. MR 681859
  • [ErdSch] M. Burak Erdoğan and Wilhelm Schlag, Dispersive estimates for Schrödinger operators in the presence of a resonance and/or an eigenvalue at zero energy in dimension three. I, Dyn. Partial Differ. Equ. 1 (2004), no. 4, 359–379. MR 2127577, 10.4310/DPDE.2004.v1.n4.a1
  • [GesZin] Gesztesy, F., Zinchenko, M. On spectral theory of Schrödinger operators with strongly singular potentials. preprint 2005.
  • [JenKat] Arne Jensen and Tosio Kato, Spectral properties of Schrödinger operators and time-decay of the wave functions, Duke Math. J. 46 (1979), no. 3, 583–611. MR 544248
  • [JenNak1] Jensen, A., Nakamura, S. $ L^p$ and Besov estimates for Schrödinger Operators. Advanced Studies in Pure Math. 23, Spectral and Scattering Theory and Applications (1994), 187-209.
  • [JenNak2] Arne Jensen and Shu Nakamura, 𝐿^{𝑝}-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, 10.2969/jmsj/04720253
  • [JenNen] Arne Jensen and Gheorghe Nenciu, A unified approach to resolvent expansions at thresholds, Rev. Math. Phys. 13 (2001), no. 6, 717–754. MR 1841744, 10.1142/S0129055X01000843
  • [KriSch] Krieger, J., Schlag, W. On the focusing critical semi-linear wave equation, preprint 2005.
  • [Sog] Christopher D. Sogge, Lectures on nonlinear wave equations, Monographs in Analysis, II, International Press, Boston, MA, 1995. MR 1715192
  • [Ste1] Elias M. Stein, 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
  • [Ste2] Stein, E. Harmonic analysis, Princeton University Press, Princeton, 2004.
  • [Tal] Giorgio Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl. (4) 110 (1976), 353–372. MR 0463908
  • [Yaj] Kenji Yajima, The 𝑊^{𝑘,𝑝}-continuity of wave operators for Schrödinger operators, J. Math. Soc. Japan 47 (1995), no. 3, 551–581. MR 1331331, 10.2969/jmsj/04730551
  • [Yaj2] Yajima, K. Dispersive estimate for Schrödinger equations with threshold resonance and eigenvalue, preprint 2004, to appear in Comm. Math. Phys.

Similar Articles

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.