## Nonlocal Harnack inequalities for nonlocal Schrödinger operators with $$A_1$$-Muckenhoupt potentials.(English)Zbl 1478.35057

In this paper, by applying the De Giorgi-Nash-Moser theory, the author obtains nonlocal Harnack inequalities for (locally nonnegative in $$\Omega$$) weak solutions of the nonlocal Schrödinger equations \begin{gather*} L_Ku + V u = 0 \;\; \text{in} \,\, \Omega, \\ u = g \;\; \text{in} \,\, \mathbb{R}^n \setminus \Omega \end{gather*} where $$V = V_+ -V_-$$ with $$V_- \in L^1_{\mathrm{loc}}(\mathbb{R}^n)$$ and $$V_+ \in L^q_{\mathrm{loc}}(\mathbb{R}^n)$$, with $$q>n$$ and such that the potentials $$V, V_+, V_-$$ belong to suitable $$A_1$$-Muckenhoupt classes. Noteworthy, this result implies the classical Harnack inequalities for globally nonnegative weak solutions of the equations. Nonlocal weak Harnack inequalities for weak supersolutions are a straightforwad consequence of the above results that are still working for any nonnegative potential in $$L^q_{\mathrm{loc}}(\mathbb{R}^n)$$.

### MSC:

 35B45 A priori estimates in context of PDEs 35B65 Smoothness and regularity of solutions to PDEs 35J10 Schrödinger operator, Schrödinger equation 35J15 Second-order elliptic equations 35R09 Integro-partial differential equations
Full Text:

### References:

 [1] Aizenman, M.; Simon, B., Brownian motions and Harnack inequality for Schrödinger operators, Commun. Pure Appl. Math., 35, 209-273 (1982) · Zbl 0459.60069 [2] Bourgain, J.; Brezis, H.; Mironescu, P., Limiting embedding theorems for $$W^{s , p}$$ when $$s \uparrow 1$$ and applications, J. Anal. Math., 87, 77-101 (2002) · Zbl 1029.46030 [3] Chiarenza, F.; Fabes, E.; Garofalo, N., Harnack’s inequality for Schrödinger operators and the continuity of solutions, Proc. Am. Math. Soc., 98, 3, 415-425 (1986) · Zbl 0626.35022 [4] Choi, W.; Kim, Y.-C., The Malgrange-Ehrenpreis theorem for nonlocal Schrödinger operators with certain potentials, Commun. Pure Appl. Anal., 17, 1993-2010 (2018) · Zbl 06923931 [5] Choi, W.; Kim, Y.-C., $$L^p$$-mapping properties for nonlocal Schrödinger operators with certain potentials, Discrete Contin. Dyn. Syst., Ser. A, 38, 5811-5834 (2018) · Zbl 1400.45011 [6] Di Castro, A.; Kuusi, T.; Palatucci, G., Nonlocal Harnack inequalities, J. Funct. Anal., 267, 1807-1836 (2014) · Zbl 1302.35082 [7] Di Castro, A.; Kuusi, T.; Palatucci, G., Local behavior of fractional p-minimizers, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 33, 5, 1279-1299 (2016) · Zbl 1355.35192 [8] Di Nezza, E.; Palatucci, G.; Valdinoci, E., Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math., 136, 521-573 (2012) · Zbl 1252.46023 [9] Fefferman, C., The uncertainty principle, Bull. Am. Math. Soc. (N.S.), 9, 129-206 (1983) · Zbl 0526.35080 [10] Felsinger, M.; Kassmann, M., Local regularity for parabolic nonlocal operators, Commun. Partial Differ. Equ., 38, 1539-1573 (2013) · Zbl 1277.35090 [11] Garcia-Cuerva, J.; Rubio De Francia, J. L., Weighted Norm Inequalities and Related Topics (1985), North-Holland · Zbl 0578.46046 [12] Giaquinta, M.; Giusti, E., On the regularity of the minima of variational integrals, Acta Math., 148, 31-46 (1982) · Zbl 0494.49031 [13] Giusti, E., Direct Methods in the Calculus of Variations (2003), World Scientific Publishing Co., Inc.: World Scientific Publishing Co., Inc. River Edge · Zbl 1028.49001 [14] Grafakos, L., Classical and Modern Fourier Analysis (2004), Prentice Hall, Pearson Education Inc. · Zbl 1148.42001 [15] Hinz, A. M.; Kalf, H., Subsolution estimates and Harnack’s inequality for Schrödinger operators, J. Reine Angew. Math., 404, 118-134 (1990) · Zbl 0779.35026 [16] Y.-C. Kim, Hölder continuity of weak solutions to nonlocal Schrödinger equations with $$A_1$$-Muckenhoupt potentials, submitted for publication. [17] Kim, Y.-C., Nonlocal Harnack inequalities for nonlocal heat equations, J. Differ. Equ., 267, 6691-6757 (2019) · Zbl 07104401 [18] Kim, Y.-C., Local properties for weak solutions of nonlocal heat equations, Nonlinear Anal. TMA, 192, Article 111689 pp. (2020) · Zbl 1439.35183 [19] Y.-C. Kim, The existence of a fundamental solution for nonlocal Schrödinger operators with $$A_1$$-Muckenhoupt potentials, submitted for publication. [20] Kinnunen, J.; Shanmugalingam, N., Regularity of quasi-minimizers on metric spaces, Manuscr. Math., 105, 401-423 (2001) · Zbl 1006.49027 [21] Krylov, N. V.; Safonov, M. V., Certain properties of solutions of parabolic equations with measurable coefficients, Izv. Akad. Nauk SSSR, 40, 161-175 (1980), (in Russian) [22] Laskin, N., Fractional Schrödinger equation, Phys. Rev. E, 66, Article 056108 pp. (2002) [23] Maz’ya, V.; Shaposhnikova, T., On the Bourgain, Brezis and Mironescu theorem concerning limiting embeddings of fractional Sobolev spaces, J. Funct. Anal., 201, 298-300 (2003) [24] Murata, M., Structure of positive solutions to $$(- \operatorname{\Delta} + V) u = 0$$ in $$\mathbb{R}^n$$, Duke Math. J., 53, 4, 869-943 (1986) · Zbl 0624.35023 [25] Servadei, R.; Valdinoci, E., Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33, 2105-2137 (2013) · Zbl 1303.35121 [26] Shen, Z., On the Neumann problem for Schrödinger operators in Lipschitz domains, Indiana Univ. Math. J., 43, 1, 143-176 (1994) · Zbl 0798.35044 [27] Shen, Z., $$L^p$$ estimates for Schrödinger operators with certain potentials, Ann. Inst. Fourier, 45, 2, 513-546 (1995) · Zbl 0818.35021 [28] Simon, B., Schrödinger semigroups, Bull. Am. Math. Soc., 7, 3, 447-526 (1982) · Zbl 0524.35002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.