Advanced Search
Article Contents
Article Contents

$L^p$ mapping properties for nonlocal Schrödinger operators with certain potentials

  • * Corresponding author: Yong-Cheol Kim

    * Corresponding author: Yong-Cheol Kim
Abstract Full Text(HTML) Figure(1) Related Papers Cited by
  • In this paper, we consider nonlocal Schrödinger equations with certain potentials $V∈{\rm{RH}}^q$($q>\frac{n}{2s}>1$ and $0<s <1$) of the form

    $\begin{equation*}L_K u+V u = f\,\,\text{ in }\; \mathbb{R}^n \end{equation*}$

    where $L_K$ is an integro-differential operator. We denote the solution of the above equation by $\mathcal{S}_V f: = u$, which is called the inverse of the nonlocal Schrödinger operator $L_K+V$ with potential $V$; that is, $\mathcal{S}_V = (L_K+V)^{-1}$. Then we obtain an improved version of the weak Harnack inequality of nonnegative weak subsolutions of the nonlocal equation

    $\begin{equation}\begin{cases}L_K u+V u = 0\,\,&\text{ in }\; \Omega,\\ u = g\,\,&\text{ in }\; \mathbb{R}^n\backslash\Omega, \;\;\;\;\;\;\;\;\; (1)\end{cases}\end{equation}$

    where $g∈ H^s(\mathbb{R}^n)$ and $\Omega$ is a bounded open domain in $\mathbb{R}^n$ with Lipschitz boundary, and also get an improved decay of a fundamental solution $\mathfrak{e}_V$ for $L_K+V$. Moreover, we obtain $L^p$ and $L^p-L^q$ mapping properties of the inverse $\mathcal{S}_V$ of the nonlocal Schrödinger operator $L_K+V$.

    Mathematics Subject Classification: Primary: 7G20, 45K05, 35J60, 35B65, 35D10; Secondary: 60J75.


    \begin{equation} \\ \end{equation}
  • 加载中
  • Figure 1.  The range of $(p,q)$ valid in Theorem 1.4

  •   J. Bourgain , H. Brezis  and  P. Mironescu , Limiting embedding theorems for Ws, p when s↑1 and applications, J. Anal. Math., 87 (2002) , 77-101.  doi: 10.1007/BF02868470.
      W. Choi  and  Y.-C. Kim , The Malgrange-Ehrenpreis theorem for nonlocal Schrödinger operators with certain potential, Comm. Pure and Appl. Anal., 17 (2018) , 1993-2010.  doi: 10.3934/cpaa.2018095.
      A. Di Castro , T. Kuusi  and  G. Palatucci , Nonlocal Harnack inequalities, J. Funct. Anal., 267 (2014) , 1807-1836.  doi: 10.1016/j.jfa.2014.05.023.
      A. Di Castro , T. Kuusi  and  G. Palatucci , Local behavior of fractional $p$-minimizers, Ann. I. H. Poincaré-AN, 33 (2016) , 1279-1299.  doi: 10.1016/j.anihpc.2015.04.003.
      E. Di Nezza , G. Palatucci  and  E. Valdinoci , Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012) , 521-573.  doi: 10.1016/j.bulsci.2011.12.004.
      C. Fefferman , The uncertainty principle, Bull. Amer. Math. Soc.(N.S.), 9 (1983) , 129-206.  doi: 10.1090/S0273-0979-1983-15154-6.
      M. Felsinger  and  M. Kassmann , Local regularity for parabolic nonlocal operators, Comm. Partial Differential Equations, 38 (2013) , 1539-1573.  doi: 10.1080/03605302.2013.808211.
      Q. Han and F. Lin, Elliptic Partial Differential Equations, Courant Lecture Notes in Mathematics, American Mathematical Society, 1997.
      T. Kuusi , G. Mingione  and  Y. Sire , Nonlocal equations with measure data, Comm. Math. phys., 337 (2015) , 1317-1368.  doi: 10.1007/s00220-015-2356-2.
      V. Maz'ya  and  T. Shaposhnikova , On the Bourgain, Brezis and Mironescu theorem concerning limiting embeddings of fractional Sobolev spaces, J. Funct. Anal., 201 (2003) , 298-300.  doi: 10.1016/S0022-1236(03)00002-8.
      R. Servadei  and  E. Valdinoci , Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013) , 2105-2137. 
      Z. Shen , Lp estimates for Schrödinger operators with certain potentials, Annales de l'institut Fourier, 45 (1995) , 513-546.  doi: 10.5802/aif.1463.
      Z. Shen , On Fundamental Solutions of Generalized Schrödinger Operators, J. Funct. Anal., 167 (1999) , 521-564.  doi: 10.1006/jfan.1999.3455.
      E. M. Stein, Harmonic Analysis; Real Variable Methods, orthogonality, and oscillatory integrals, Princeton Univ. Press, 1993.
  • 加载中



Article Metrics

HTML views(364) PDF downloads(280) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint