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.

[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.

[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.

[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.

[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.

[6]

C. Fefferman, The uncertainty principle, Bull. Amer. Math. Soc.(N.S.), 9 (1983), 129-206. doi: 10.1090/S0273-0979-1983-15154-6.

[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.

[8]

Q. Han and F. Lin, Elliptic Partial Differential Equations, Courant Lecture Notes in Mathematics, American Mathematical Society, 1997.

[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.

[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.

[11]

R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137.

[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.

[13]

Z. Shen, On Fundamental Solutions of Generalized Schrödinger Operators, J. Funct. Anal., 167 (1999), 521-564. doi: 10.1006/jfan.1999.3455.

[14]

E. M. Stein, Harmonic Analysis; Real Variable Methods, orthogonality, and oscillatory integrals, Princeton Univ. Press, 1993.

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.

[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.

[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.

[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.

[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.

[6]

C. Fefferman, The uncertainty principle, Bull. Amer. Math. Soc.(N.S.), 9 (1983), 129-206. doi: 10.1090/S0273-0979-1983-15154-6.

[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.

[8]

Q. Han and F. Lin, Elliptic Partial Differential Equations, Courant Lecture Notes in Mathematics, American Mathematical Society, 1997.

[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.

[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.

[11]

R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137.

[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.

[13]

Z. Shen, On Fundamental Solutions of Generalized Schrödinger Operators, J. Funct. Anal., 167 (1999), 521-564. doi: 10.1006/jfan.1999.3455.

[14]

E. M. Stein, Harmonic Analysis; Real Variable Methods, orthogonality, and oscillatory integrals, Princeton Univ. Press, 1993.

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