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
$0<s <1$
) 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
$L_K+V$
.
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. BourgainH. 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 CastroT. 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 CastroT. 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 NezzaG. 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. KuusiG. 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

show all references

References:
[1]

J. BourgainH. 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 CastroT. 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 CastroT. 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 NezzaG. 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. KuusiG. 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

Figure 1.  The range of $(p,q)$ valid in Theorem 1.4
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