Note on Gradient Estimate of Heat Kernel for Schrödinger Operators

ABSTRACT

Let be a Schrödinger operator on . We show that gradient estimates for the heat kernel of with upper Gaussian bounds imply polynomial decay for the kernels of certain smooth dyadic spectral operators. The latter decay property has been known to play an important role in the Littlewood-Paley theory for and Sobolev spaces. We are able to establish the result by modifying Hebisch and the author’s recent proofs. We give a counterexample in one dimension to show that there exists in the Schwartz class such that the long time gradient heat kernel estimate fails.

Let be a Schrödinger operator on . We show that gradient estimates for the heat kernel of with upper Gaussian bounds imply polynomial decay for the kernels of certain smooth dyadic spectral operators. The latter decay property has been known to play an important role in the Littlewood-Paley theory for and Sobolev spaces. We are able to establish the result by modifying Hebisch and the author’s recent proofs. We give a counterexample in one dimension to show that there exists in the Schwartz class such that the long time gradient heat kernel estimate fails.

Cite this paper

nullS. Zheng, "Note on Gradient Estimate of Heat Kernel for Schrödinger Operators,"*Applied Mathematics*, Vol. 1 No. 5, 2010, pp. 425-430. doi: 10.4236/am.2010.15056.

nullS. Zheng, "Note on Gradient Estimate of Heat Kernel for Schrödinger Operators,"

References

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[2] S. Zheng, “Littlewood-Paley Theorem for Schr?dinger Operators,” Analysis in Theory and Applications, Vol. 22, No. 4, 2006, pp. 353-361.

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[6] J. Benedetto and S. Zheng, “Besov Spaces for the Schr?dinger Operator with Barrier Potential,” to Appear in Complex Analysis and Operator Theory, Birkh?user.

[7] W. Hebisch, “A Multiplier Theorem for Schr?dinger Operators,” Colloquium Mathematicum, Vol. 60-61, No. 2, 1990, pp. 659-664.

[8] W. Hebisch, “Almost Everywhere Summability of Eigenfunction Expansions Associated to Elliptic Operators,” Studia Mathematica, Vol. 96, No. 3, 1990, pp. 263-275.

[9] E. Ouhabaz, “Sharp Gaussian Bounds and Growth of Semigroups Associated with Elliptic and Schr?dinger Operators,” Proceedings of the American Mathematical Society, Vol. 134, No. 12, 2006, pp. 3567-3575.

[10] G. Furioli, C. Melzi and A. Veneruso, “Littlewood-Paley Decompositions and Besov Spaces on Lie Groups of Polynomial Growth,” Mathematische Nachrichten, Vol. 279, No. 9-10, 2006, pp. 1028-1040.

[11] A. Grigor’yan, “Upper Bounds of Derivatives of the Heat Kernel on an Arbitrary Complete Manifold,” Journal of Functional Analysis, Vol. 127, No. 2, 1995, pp. 363-389.

[12] Q. Zhang, “Global Bounds of Schr?dinger Heat Kernels with Negative Potentials,” Journal of Functional Analysis, Vol. 182, No. 2, 2001, pp. 344-370.

[13] B. Simon, “Schr?dinger Semigroups,” Bulletin of the American Mathematical Society, Vol. 7, No. 3, 1982, pp. 447-526.

[14] H.-J. Schmeisser and H. Triebel, “Topics in Fourier Analysis and Function Spaces,” Wiley-Interscience, Chichester, 1987.

[15] P. D’Ancona and V. Pierfelice, “On the Wave Equation with a Large Rough Potential,” Journal of Functional Analysis, Vol. 227, No. 1, 2005, pp. 30-77.

[16] S. Zheng, “A Representation Formula Related to Schr?dinger Operators,” Analysis in Theory and Applications, Vol. 20, No. 3, 2004, pp. 294-296.

[1] G. ólafsson and S. Zheng, “Harmonic Analysis Related to Schr?dinger Operators,” Contemporary Mathematics, Vol. 464, 2008, pp. 213-230.

[2] S. Zheng, “Littlewood-Paley Theorem for Schr?dinger Operators,” Analysis in Theory and Applications, Vol. 22, No. 4, 2006, pp. 353-361.

[3] J. Epperson, “Triebel-Lizorkin Spaces for Hermite Expansions,” Studia Mathematica, Vol. 114, No. 1, 1995, pp. 87-103.

[4] J. Dziubański, “Triebel-Lizorkin Spaces Associated with Laguerre and Hermite Expansions,” Proceedings of the American Mathematical Society, Vol. 125, No. 12, 1997, pp. 3547-3554.

[5] G. ólafsson and S. Zheng, “Function Spaces Associated with Schr?dinger Operators: The P?schl-Teller Potential,” Journal of Fourier Analysis and Application, Vol. 12, No. 6, 2006, pp. 653-674.

[6] J. Benedetto and S. Zheng, “Besov Spaces for the Schr?dinger Operator with Barrier Potential,” to Appear in Complex Analysis and Operator Theory, Birkh?user.

[7] W. Hebisch, “A Multiplier Theorem for Schr?dinger Operators,” Colloquium Mathematicum, Vol. 60-61, No. 2, 1990, pp. 659-664.

[8] W. Hebisch, “Almost Everywhere Summability of Eigenfunction Expansions Associated to Elliptic Operators,” Studia Mathematica, Vol. 96, No. 3, 1990, pp. 263-275.

[9] E. Ouhabaz, “Sharp Gaussian Bounds and Growth of Semigroups Associated with Elliptic and Schr?dinger Operators,” Proceedings of the American Mathematical Society, Vol. 134, No. 12, 2006, pp. 3567-3575.

[10] G. Furioli, C. Melzi and A. Veneruso, “Littlewood-Paley Decompositions and Besov Spaces on Lie Groups of Polynomial Growth,” Mathematische Nachrichten, Vol. 279, No. 9-10, 2006, pp. 1028-1040.

[11] A. Grigor’yan, “Upper Bounds of Derivatives of the Heat Kernel on an Arbitrary Complete Manifold,” Journal of Functional Analysis, Vol. 127, No. 2, 1995, pp. 363-389.

[12] Q. Zhang, “Global Bounds of Schr?dinger Heat Kernels with Negative Potentials,” Journal of Functional Analysis, Vol. 182, No. 2, 2001, pp. 344-370.

[13] B. Simon, “Schr?dinger Semigroups,” Bulletin of the American Mathematical Society, Vol. 7, No. 3, 1982, pp. 447-526.

[14] H.-J. Schmeisser and H. Triebel, “Topics in Fourier Analysis and Function Spaces,” Wiley-Interscience, Chichester, 1987.

[15] P. D’Ancona and V. Pierfelice, “On the Wave Equation with a Large Rough Potential,” Journal of Functional Analysis, Vol. 227, No. 1, 2005, pp. 30-77.

[16] S. Zheng, “A Representation Formula Related to Schr?dinger Operators,” Analysis in Theory and Applications, Vol. 20, No. 3, 2004, pp. 294-296.