American Institute of Mathematical Sciences

November  2018, 38(11): 5811-5834. doi: 10.3934/dcds.2018253

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

 1 Department of Mathematics Education, Incheon National University, Incheon 22012, Republic of Korea 2 Department of Mathematics Education, Korea University, Seoul 02841, Republic of Korea

* Corresponding author: Yong-Cheol Kim

Received  February 2018 Revised  June 2018 Published  August 2018

In this paper, we consider nonlocal Schrödinger equations with certain potentials
 $V∈{\rm{RH}}^q$
(
 $q>\frac{n}{2s}>1$
and
 $0 ) 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 $$$\begin{cases}L_K u+V u = 0\,\,&\text{ in }\; \Omega,\\ u = g\,\,&\text{ in }\; \mathbb{R}^n\backslash\Omega, \;\;\;\;\;\;\;\;\; (1)\end{cases}$$$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 $L_K+V$. Citation: Woocheol Choi, Yong-Cheol Kim.$L^p$mapping properties for nonlocal Schrödinger operators with certain potentials. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5811-5834. doi: 10.3934/dcds.2018253 References:  [1] 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. Google Scholar [2] 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. Google Scholar [3] 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. Google Scholar [4] 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. Google Scholar [5] E. Di Nezza, G. 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Anal., 201 (2003), 298-300. doi: 10.1016/S0022-1236(03)00002-8. Google Scholar [11] R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137. Google Scholar [12] Z. Shen, Lp estimates for Schrödinger operators with certain potentials, Annales de l'institut Fourier, 45 (1995), 513-546. doi: 10.5802/aif.1463. Google Scholar [13] Z. Shen, On Fundamental Solutions of Generalized Schrödinger Operators, J. Funct. Anal., 167 (1999), 521-564. doi: 10.1006/jfan.1999.3455. Google Scholar [14] E. M. Stein, Harmonic Analysis; Real Variable Methods, orthogonality, and oscillatory integrals, Princeton Univ. Press, 1993. Google Scholar show all references References:  [1] 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. Google Scholar [2] 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. Google Scholar [3] 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. Google Scholar [4] 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. Google Scholar [5] 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. Google Scholar [6] C. Fefferman, The uncertainty principle, Bull. Amer. Math. Soc.(N.S.), 9 (1983), 129-206. doi: 10.1090/S0273-0979-1983-15154-6. Google Scholar [7] 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. Google Scholar [8] Q. Han and F. Lin, Elliptic Partial Differential Equations, Courant Lecture Notes in Mathematics, American Mathematical Society, 1997. Google Scholar [9] 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. Google Scholar [10] 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. Google Scholar [11] R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137. Google Scholar [12] Z. Shen, Lp estimates for Schrödinger operators with certain potentials, Annales de l'institut Fourier, 45 (1995), 513-546. doi: 10.5802/aif.1463. Google Scholar [13] Z. Shen, On Fundamental Solutions of Generalized Schrödinger Operators, J. Funct. Anal., 167 (1999), 521-564. doi: 10.1006/jfan.1999.3455. Google Scholar [14] E. M. Stein, Harmonic Analysis; Real Variable Methods, orthogonality, and oscillatory integrals, Princeton Univ. Press, 1993. Google Scholar The range of$(p,q)\$ valid in Theorem 1.4
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