Advertisement

Acta Mathematica Sinica, English Series

, Volume 34, Issue 4, pp 787–800 | Cite as

On Characterization of Poisson Integrals of Schrödinger Operators with Morrey Traces

  • Liang Song
  • Xiao Xiao Tian
  • Li Xin Yan
Article
  • 39 Downloads

Abstract

Let L be a Schrödinger operator of the form L = −Δ + V acting on L2(ℝ n ) where the nonnegative potential V belongs to the reverse Hölder class B q for some qn. In this article we will show that a function fL2,λ(ℝ n ), 0 < λ < n, is the trace of the solution of Lu = −u tt + L u = 0, u(x, 0) = f(x), where u satisfies a Carleson type condition
$$\mathop {\sup }\limits_{{x_B},{r_B}} r_B^{ - \lambda }\int_0^{{r_B}} {\int_{B\left( {{x_B},{r_B}} \right)} {t{{\left| {\nabla u\left( {x,t} \right)} \right|}^2}dxdt \leqslant C < \infty .} } $$
Its proof heavily relies on investigate the intrinsic relationship between the classical Morrey spaces and the new Campanato spaces L L 2,λ (ℝ n ) associated to the operator L, i.e.
$$L_L^{2,\lambda }\left( {{\mathbb{R}^n}} \right) = {L^{2,\lambda }}\left( {{\mathbb{R}^n}} \right).$$
Conversely, this Carleson type condition characterizes all the L-harmonic functions whose traces belong to the space L2,λ(ℝ n ) for all 0 < λ < n.

Keywords

Schrödinger operators Dirichlet problem Morrey spaces Campanato spaces Poisson semigroup 

MR(2010) Subject Classification

42B37 42B35 47B38 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Adams, D., Xiao, J.: Morrey spaces in harmonic analysis. Ark. Mat., 50(2), 201–230 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    Campanato, S.: Propriet di una famiglia di spazi funzionali (in Italian). Ann. Scuola Norm. Sup. Pisa, 18, 137–160 (1964)MathSciNetzbMATHGoogle Scholar
  3. [3]
    Deng, D. G., Duong, X. T., Sikora, A., et al.: Comparison of the classical BMO with the BMO spaces associated with operators and applications. Rev. Mat. Iberoam., 24(1), 267–296 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    Duong, X. T., Xiao, J., Yan, L. X.: Old and new Morrey spaces with heat kernel bounds. J. Fourier Anal. Appl., 13(1), 87–111 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    Duong, X. T., Yan, L. X.: New function spaces of BMO type, John-Nirenberg inequality, interpolation and applications. Comm. Pure Appl. Math., 58, 1375–1420 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    Duong, X. T., Yan, L. X.: Duality of Hardy and BMO spaces associated with operators with heat kernel bounds. J. Amer. Math. Soc., 18, 943–973 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    Duong, X. T., Yan, L. X., Zhang, C.: On characterization of Poisson integrals of Schrödinger operators with BMO traces. J. Funct. Anal., 266(4), 2053–2085 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    Dziubanski, J., Garrigos, G., Martinez, T., et al.: BMO spaces related to Schrodinger operators with potentials satisfying a reverse Holder inequality. Math. Z., 249(2), 329–356 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    Fabes, E. B., Johnson, R. L., Neri, U.: Spaces of harmonic functions representable by Poisson integrals of functions in BMO and L p. Indiana Univ. Math. J., 25(2), 159–170 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    Fefferman, C., Stein, E. M.: Hp spaces of several variables. Acta Math., 129, 137–195 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    Janson, S., Taibleson, M. H., Weiss, G.: Elementary characterizations of the Morrey-Campanato spaces. Lecture Notes in Math., 992, 101–114 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    John, F., Nirenberg, L.: On functions of bounded mean oscillation. Comm. Pure Appl. Math., 14, 415–426 (1961)MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    Jiang, R. J., Xiao, J., Yang, D. C.: Towards spaces of harmonic functions with traces in square Campanato space and their scaling invariant. Anal. Appl. (Singap.), 14(5), 679–703 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    Ma, T., Stinga, P., Torrea, J., et al.: Regularity properties of Schrodinger operators. J. Math. Anal. Appl., 388, 817–837 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    Morrey, C. B.: On the solutions of quasi-linear elliptic partial differential equations. Trans. Amer. Math. Soc., 43(1), 126–166 (1938)MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    Nakai, E.: The Campanato, Morrey and Holder spaces on spaces of homogeneous type. Studia Math., 176(1), 1–19 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    Peetre, J.: On the theory of L p spaces. J. Funct. Anal., 4, 71–87 (1969)CrossRefGoogle Scholar
  18. [18]
    Rafeiro, H., Samko, N., Samko, S.: Morrey-Campanato spaces: an overview. Operator theory, pseudo-differential equations, and mathematical physics, 293–323, Oper. Theory Adv. Appl., 228, Birkhäuser/Springer Basel AG, Basel, 2013Google Scholar
  19. [19]
    Shen, Z. W.: L p estimates for Schrodinger operators with certain potentials. Ann. Inst. Fourier (Grenoble), 45, 513–546 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    Shen, Z. W.: On fundamental solutions of generalized Schrodinger operators. J. Funct. Anal., 167(2), 521–564 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    Stein, E. M., Weiss, G.: Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, Princeton, NJ, 1970zbMATHGoogle Scholar
  22. [22]
    Taylor, M. E.: Microlocal Analysis on Morrey Spaces. Singularities and Oscillations (Minneapolis, MN, 1994/1995), 97-135, IMA Vol. Math. Appl., 91, Springer, New York, 1997Google Scholar

Copyright information

© Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsSun Yat-sen UniversityGuangzhouP. R. China

Personalised recommendations