Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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Lp ESTIMATES FOR SCHRÖDINGER TYPE OPERATORS ON THE HEISENBERG GROUP

  • Yu, Liu (Department of Mathematics and Mechanics, University of Science and Technology Beijing)
  • Published : 2010.03.01
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Abstract

We investigate the Schr$\ddot{o}$dinger type operator $H_2\;=\;(-\Delta_{\mathbb{H}^n})^2+V^2$ on the Heisenberg group $\mathbb{H}^n$, where $\Delta_{\mathbb{H}^n}$ is the sublaplacian and the nonnegative potential V belongs to the reverse H$\ddot{o}$lder class $B_q$ for $q\geq\frac{Q}{2}$, where Q is the homogeneous dimension of $\mathbb{H}^n$. We shall establish the estimates of the fundamental solution for the operator $H_2$ and obtain the $L^p$ estimates for the operator $\nabla^4_{\mathbb{H}^n}H^{-1}_2$, where $\nabla_{\mathbb{H}^n}$ is the gradient operator on $\mathbb{H}^n$.

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References

  1. C. Benson, A. H. Dooley, and G. Ratcliff, Fundamental solutions for powers of the Heisenberg sub-Laplacian, Illinois J. Math. 37 (1993), no. 3, 455–476.
  2. C. Berenstein, D. C. Chang, and J. Z. Tie, Laguerre Calculus And its Applications on The Heisenberg Group, American Mathematical Society, International press, 2001.
  3. T. Coulhon, D. Muller, and J. Zienkiewicz, About Riesz transforms on the Heisenberg groups, Math. Ann. 305 (1996), no. 2, 369–379. https://doi.org/10.1007/BF01444227
  4. C. Fefferman, The uncertainty principle, Bull. Amer. Math. Soc. (N.S.) 9 (1983), no. 2, 129–206. https://doi.org/10.1090/S0273-0979-1983-15154-6
  5. G. B. Folland and E. M. Stein, Hardy Spaces on Homogeneous Groups, Mathematical Notes, 28. Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1982.
  6. F. Gehring, The $L^p$-integrability of the partial derivatives of a quasiconformal mapping, Acta Math. 130 (1973), 265–277. https://doi.org/10.1007/BF02392268
  7. D. Jerison and A. Sanchez-Calle, Estimates for the heat kernel for a sum of squares of vector fields, Indiana Univ. Math. J. 35 (1986), no. 4, 835–854. https://doi.org/10.1512/iumj.1986.35.35043
  8. H. Q. Li, Estimations $L^p$ des operateurs de Schr¨odinger sur les groupes nilpotents, J. Funct. Anal. 161 (1999), no. 1, 152–218. https://doi.org/10.1006/jfan.1998.3347
  9. N. Lohoue, Estimees de type Hardy-Sobolev sur certaines varietes riemanniennes et les groupes de Lie, J. Funct. Anal. 112 (1993), no. 1, 121–158. https://doi.org/10.1006/jfan.1993.1028
  10. G. Z. Lu, A Fefferman-Phong type inequality for degenerate vector fields and applications, Panamer. Math. J. 6 (1996), no. 4, 37–57.
  11. Z. W. Shen, $L^p$ estimates for Schrodinger operators with certain potentials, Ann. Inst. Fourier (Grenoble) 45 (1995), no. 2, 513–546. https://doi.org/10.5802/aif.1463
  12. Z. W. Shen, On the Neumann problem for Schrodinger operators in Lipschitz domains, Indiana Univ. Math. J. 43 (1994), no. 1, 143–176. https://doi.org/10.1512/iumj.1994.43.43007
  13. S. Sugano, $L^p$ estimates for some Schrodinger operators and a Calderon-Zygmund operator of Schrodinger type, Tokyo J. Math. 30 (2007), no. 1, 179–197. https://doi.org/10.3836/tjm/1184963655
  14. J. P. Zhong, Harmonic analysis for some Schr¨odinger type operators, Ph. D. Princeton University, 1993.

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