대한수학회지 (Journal of the Korean Mathematical Society)

  • 제59권1호
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  • Pages.129-150
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  • 2022
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  • 0304-9914(pISSN)
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  • 2234-3008(eISSN)
대한수학회 (Korean Mathematical Society)
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REGULARITY OF THE GENERALIZED POISSON OPERATOR

  • Li, Pengtao (School of Mathematics and Statistics Qingdao University) ;
  • Wang, Zhiyong (School of Mathematics and Statistics Qingdao University) ;
  • Zhao, Kai (School of Mathematics and Statistics Qingdao University)
  • 투고 : 2021.04.03
  • 심사 : 2021.09.24
  • 발행 : 2022.01.01
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초록

Let L = -∆ + V be a Schrödinger operator, where the potential V belongs to the reverse Hölder class. In this paper, by the subordinative formula, we investigate the generalized Poisson operator PLt,σ, 0 < σ < 1, associated with L. We estimate the gradient and the time-fractional derivatives of the kernel of PLt,σ, respectively. As an application, we establish a Carleson measure characterization of the Campanato type space 𝒞𝛄L (ℝn) via PLt,σ.

키워드

과제정보

This work was financially supported by the National Natural Science Foundation of China (No. 12071272) and Shandong Natural Science Foundation of China (Nos. ZR2020MA004, ZR2017JL008).

참고문헌

  1. L. A. Caffarelli, S. Salsa, and L. Silvestre, Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian, Invent. Math. 171 (2008), no. 2, 425-461. https://doi.org/10.1007/s00222-007-0086-6
  2. L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations 32 (2007), no. 7-9, 1245-1260. https://doi.org/10.1080/03605300600987306
  3. R. R. Coifman, Y. Meyer, and E. M. Stein, Some new function spaces and their applications to harmonic analysis, J. Funct. Anal. 62 (1985), no. 2, 304-335. https://doi.org/10.1016/0022-1236(85)90007-2
  4. D. Deng, X. T. Duong, L. Song, C. Tan, and L. Yan, Functions of vanishing mean oscillation associated with operators and applications, Michigan Math. J. 56 (2008), no. 3, 529-550. https://doi.org/10.1307/mmj/1231770358
  5. X. T. Duong, L. Yan, and C. Zhang, On characterization of Poisson integrals of Schrodinger operators with BMO traces, J. Funct. Anal. 266 (2014), no. 4, 2053-2085. https://doi.org/10.1016/j.jfa.2013.09.008
  6. J. Dziubanski, G. Garrigos, T. Martinez, J. L. Torrea, and J. Zienkiewicz, BMO spaces related to Schrodinger operators with potentials satisfying a reverse Holder inequality, Math. Z. 249 (2005), no. 2, 329-356. https://doi.org/10.1007/s00209-004-0701-9
  7. J. Dziubanski and J. Zienkiewicz, Hp spaces for Schrodinger operators, in Fourier analysis and related topics, 45-53, Banach Center Publ., 56, Polish Acad. Sci. Inst. Math., Warsaw, 2000. https://doi.org/10.4064/bc56-0-4
  8. C. Fefferman and E. M. Stein, Hp spaces of several variables, Acta Math. 129 (1972), no. 3-4, 137-193. https://doi.org/10.1007/BF02392215
  9. J. Huang, M. Duan, Y. Wang, and W. Li, Fractional Carleson measure associated with Hermite operator, Anal. Math. Phys. 9 (2019), no. 4, 2075-2097. https://doi.org/10.1007/s13324-019-00300-2
  10. J. Huang, P. Li, and Y. Liu, Regularity properties of the heat kernel and area integral characterization of Hardy space H1𝓛 related to degenerate Schrodinger operators, J. Math. Anal. Appl. 466 (2018), no. 1, 447-470. https://doi.org/10.1016/j.jmaa.2018.06.008
  11. S. Hofmann, G. Lu, D. Mitrea, M. Mitrea, and L. Yan, Hardy spaces associated to non-negative self-adjoint operators satisfying Davies-Gaffney estimates, Mem. Amer. Math. Soc. 214 (2011), no. 1007, vi+78 pp. https://doi.org/10.1090/S0065-9266-2011-00624-6
  12. P. Li, Riesz potentials of Hardy-Hausdorff spaces and Q-type spaces, Sci. China Math. 63 (2020), no. 10, 2017-2036. https://doi.org/10.1007/s11425-018-9443-7
  13. P. Li, Z. Wang, T. Qian, and C. Zhang, Regularity of fractional heat semigroup associated with Schrodinger operators, preprint available at arXiv:2012.07234.
  14. C.-C. Lin and H. Liu, BMOL(ℍn) spaces and Carleson measures for Schrodinger operators, Adv. Math. 228 (2011), no. 3, 1631-1688. https://doi.org/10.1016/j.aim.2011.06.024
  15. T. Ma, P. R. Stinga, J. L. Torrea, and C. Zhang, Regularity properties of Schrodinger operators, J. Math. Anal. Appl. 388 (2012), no. 2, 817-837. https://doi.org/10.1016/j.jmaa.2011.10.006
  16. Z. W. Shen, Lp estimates for Schrodinger operators with certain potentials, Ann. Inst. Fourier (Grenoble) 45 (1995), no. 2, 513-546. https://doi.org/10.5802/aif.1463
  17. L. Song, X. X. Tian, and L. X. Yan, On characterization of Poisson integrals of Schrodinger operators with Morrey traces, Acta Math. Sin. (Engl. Ser.) 34 (2018), no. 4, 787-800. https://doi.org/10.1007/s10114-018-7368-3
  18. P. R. Stinga and J. L. Torrea, Extension problem and Harnack's inequality for some fractional operators, Comm. Partial Differential Equations 35 (2010), no. 11, 2092-2122. https://doi.org/10.1080/03605301003735680
  19. Y. Wang, Y. Liu, C. Sun, and P. Li, Carleson measure characterizations of the Campanato type space associated with Schrodinger operators on stratified Lie groups, Forum Math. 32 (2020), no. 5, 1337-1373. https://doi.org/10.1515/forum-2019-0224
  20. D. Yang, D. Yang, and Y. Zhou, Localized Morrey-Campanato spaces on metric measure spaces and applications to Schrodinger operators, Nagoya Math. J. 198 (2010), 77-119. https://doi.org/10.1215/00277630-2009-008
  21. D. Yang, D. Yang, and Y. Zhou, Localized BMO and BLO spaces on RD-spaces and applications to Schrodinger operators, Commun. Pure Appl. Anal. 9 (2010), no. 3, 779-812. https://doi.org/10.3934/cpaa.2010.9.779