Abstract
Let H be a Schrödinger operator on ℝ n. Under a polynomial decay condition for the kernel of its spectral operator, we show that the Besov spaces and Triebel-Lizorkin spaces associated with H are well defined. We further give a Littlewood-Paley characterization of Lp spaces in terms of dyadic functions of H. This generalizes and strengthens the previous result when the heat kernel of H satisfies certain upper Gaussian bound.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Benedetto, J. and Zheng, S., Besov Spaces for the Schrödinger Operator with Barrier Potential (Submitted). http://lanl.arXiv.org/math.CA/0411348, (2005).
D’Ancona, P. and Pierfelice, V., On the Wave Equation with a Large Rough Potential, J. Funct. Anal., 227:1(2005), 30–77.
Davies, E., Heat Kernels and Spectral Theory, Cambridge University Press, Cambridge, 1989.
—, Pointwise Bounds on the Space and Time Derivatives of Heat Kernels, J. Operator Theory, 21(1989), 367–378.
Duong, X. and Yan, L., Duality of Hardy and BMO Spaces Associated with Operators with Heat Kernel Bounds, J. Amer. Math. Soc., 18(2005), 943–973.
Dziubański, J., Triebel-Lizorkin Spaces Associated with Laguerre and Hermite Expansions, Proc. Amer. Math. Soc., 125(1997), 3547–3554.
Dziubański, J., and Zienkiewicz, J., Hardy Spaces H 1 for Schrödinger Operators with Compactly Supported Potentials, Ann. Mat. Pura Appl., 184:3(2005), 315–326.
Dziubański, J. and Zienkiewicz, J., H p Spaces for Schrödinger Operators, in: Fourier Analysis and Related Topics, Banach Center Publ., 56, Inst. Math., Polish Acad. Sci., 2002, 45–53.
Epperson, J., Triebel-Lizorkin Spaces for Hermite Expansions, Studia Math., 114(1995), 87–103.
—, Hermite and Laguerre, Wave Packet Expansions, Studia Math., 126:3(1997), 199–217.
Hebisch, W., A Multiplier Theorem for Schrödinger Operators, Colloq. Math., 60/61:2(1990), 659–664.
Jensen, A. and Nakamura, S., Mapping Properties of Functions of Schrödinger Operators Between L p Spaces and Besov Spaces, in: Spectral and Scattering Theory and Applications, Advanced Studies in Pure Math., 23, 1994.
Keel, M. and Tao, T., Endpoint Strichartz Estimates, Amer. J. Math., 120(1998), 955–980.
Ólafsson, G. and Zheng, S., Function Spaces Associated with Schrödinger Operators: the Pöschl-Teller Potential (Submitted).
Rodnianski, I. and Tao, T., Long-Time Decay Estimates for the Schrödinger Equation on Manifolds, Preprint.
Schlag, W., A Remark on Littlewood-Paley Theory for the Distorted Fourier Transform, http://lanl.arXiv.org/math.AP/0508577, (2005).
Simon, B., Schrödinger Semigroups, Bull. Amer. Math. Soc., 7:3(1982), 447–526.
Thangavelu, S., Lectures on Hermite and Laguerre Expansions, Princeton Univ. Press, 1993.
—, Hermite and Laguerre Semigroups, Some Recent Developments, Preprint.
Triebel, H., Theory of Function Spaces, Birkhäuser Verlag, 1983.
—, Theory of Function Spaces II, Monographs Math., 84, Birkhäuser, Basel, 1992.
Zheng, S., A Representation Formula Related to Schrödinger Operators, Anal. Theo. Appl., 20:3(2004), 294–296.
—, Spectral Multipliers, Function spaces and dispersive estimates for Schrödinger Operators, Preprint.
Author information
Authors and Affiliations
Additional information
Supported by DARPA grant HM1582-05-2-0001. The author gratefully thanks the hospitality and support of Department of Mathematics, University of South Carolina, during his visiting at the Industrial Mathematics Institute.
Rights and permissions
About this article
Cite this article
Zheng, S. Littlewood-Paley theorem for Schrödinger operators. Analys in Theo Applic 22, 353–361 (2006). https://doi.org/10.1007/s10496-006-0353-1
Received:
Issue Date:
DOI: https://doi.org/10.1007/s10496-006-0353-1