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Harnack inequalities for Schrödinger operators. (English) Zbl 0940.35063
The paper deals with Harnack inequalities for uniformly elliptic operators and sums of squares of smooth vector fields. Precisely, let \(\mu\) be a signed Radon measure on a domain \(X\subset {\mathbb R}^d,\) \(d\geq 1,\) with Green function \(G_X\) and assume that the potential \(G_X^{1_B|\mu|}\) is bounded for every ball \(B\subset X.\) Set \(d_{\mu^\pm}(x)=\lim\sup_{y\to x} G_X^{1_B\mu^\pm}(y)-G_X^{1_B\mu^\pm}(x),\) \(x\in B,\) \(\overline B\subset X.\)
Then, if \(U\) is a domain in \(X\) admitting a positive \(\mu\)-harmonic function which is locally bounded and not identically zero, then Harnack inequalities hold for positive \(\mu\)-harmonic functions on \(U\) and every \(\mu\)-harmonic function on \(U\) is locally bounded. In particular, Harnack inequalities always hold as long as \(d_{\mu^-}\leq \gamma<1.\)

MSC:
35J10 Schrödinger operator, Schrödinger equation
35B45 A priori estimates in context of PDEs
31C05 Harmonic, subharmonic, superharmonic functions on other spaces
35J15 Second-order elliptic equations
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References:
[1] M. Aizenman - B. Simon , Brownian motion and Harnack’s inequality for Schrödinger operators , Comm. Pure Appl. Math. 35 ( 1982 ), 209 - 271 . MR 644024 | Zbl 0459.60069 · Zbl 0459.60069
[2] J. Bliedtner - W. Hansen , ” Potential Theory - An Analytic and Probabilistic Approach to Balayage ”, Universitext. Springer , Berlin - Heidelberg - New York - Tokyo , 1986 . MR 850715 | Zbl 0706.31001 · Zbl 0706.31001
[3] ] N. Bouleau - F. Hirsch , ”Dirichlet Forms and Analysis on Wiener Space”, vol. 14 of Studies in Mathematics . de Gruyter , Berlin - New York , 1991 . MR 1133391 | Zbl 0748.60046 · Zbl 0748.60046
[4] A. Boukricha - W. Hansen - H. Hueber , Continuous solutions of the generalized Schrödinger equation and perturbation of harmonic spaces , BiBoS (Research Center Bielefeld-Bochum-Stochastics, Bielefeld University) . Preprint 54 ( 1985 ), 1 - 70 . MR 887788 | Zbl 0566.31005 · Zbl 0566.31005
[5] A. Boukricha - W. Hansen - H. Hueber , Continuous solutions of the generalized Schrödinger equation and perturbation of harmonic spaces , Exposition. Math. 5 ( 1987 ), 97 - 135 . MR 887788 | Zbl 0659.35025 · Zbl 0659.35025
[6] N. Boboc - P. Mustaţă , ”Espaces harmoniques associés aux opérateurs différentiels linéaires du second ordre de type elliptique” , Lecture Notes in Mathematics 68 , Springer , Berlin - New York , 1968 . MR 241681 | Zbl 0167.40301 · Zbl 0167.40301
[7] J.M. Bony , Principe du maximum, inégalité de Harnack et unicité du problème de Cauchy pour les opérateurs elliptiques dégénérés , Ann. Inst. Fourier ( Grenoble ) 19 ( 1969 ), 277 - 304 . Numdam | MR 262881 | Zbl 0176.09703 · Zbl 0176.09703
[8] J.M. Bony , Opérateurs elliptiques dégénérés associés aux axiomatiques de la théorie du potentiel , In: ”Potential Theory” (CIME, 1^\circ Ciclo, Stresa 1969) Edizioni Cremonese , Roma 1970 , pp. 69 - 119 . MR 277744 | Zbl 0205.10704 · Zbl 0205.10704
[9] A. Boukricha , Das Picard-Prinzip und verwandte Fragen bei Störung von harmonischen Räumen , Math. Ann. 239 ( 1979 ), 247 - 270 . MR 522783 | Zbl 0377.31011 · Zbl 0377.31011
[10] C. Constantinescu - A. Cornea , ”Potential Theory on Harmonic Spaces”, Grundlehren d. math. Wiss . Springer , Berlin - Heidelberg - New York , 1972 . MR 419799 | Zbl 0248.31011 · Zbl 0248.31011
[11] F. Chiarenza - E. Fabes - N. Garofalo , Harnack’s inequality for Schrödinger operators and the continuity of solutions , Proc. Amer. Math. Soc. 98 ( 1986 ), 415 - 425 . MR 857933 | Zbl 0626.35022 · Zbl 0626.35022
[12] M. Cranston - E. Fabes - Z. Zhao , Conditional gauge and potential theory for the Schrödinger operator , Trans. Amer. Math. Soc. 307 ( 1988 ), 171 - 194 . MR 936811 | Zbl 0652.60076 · Zbl 0652.60076
[13] G. Citti - N. Garofalo - E. Lanconelli , Harnack’s inequality for sum of squares of vector fields plus a potential , Amer. J. Math. 115 ( 1993 ), 699 - 734 . MR 1221840 | Zbl 0795.35018 · Zbl 0795.35018
[14] J. Deny - J.L. Lions , Les espaces du type Beppo-Levi , Ann. Inst. Fourier ( Grenoble ) 5 ( 1953 /54), 305 - 370 . Numdam | MR 74787 | Zbl 0065.09903 · Zbl 0065.09903
[15] A. De La Pradelle , Sur les perturbations d’espaces harmoniques , Acad. Roy. Belg. Bull. Cl. Sci. ( 6 ) 1 ( 1990 ), 201 - 212 . MR 1146274 | Zbl 0770.31010 · Zbl 0770.31010
[16] A. De La Pradelle , ”Sur les perturbations d’espaces harmoniques” Erratum , Acad. Roy. Belg. Bull. Cl. Sci. ( 6 ) ( 1999 ). · Zbl 1194.31011
[17] D. Feyel - A. De La Pradelle , Étude de l’équation 1/2\Delta u - u\mu = 0 où \mu est une mesure positive , Ann. Inst. Fourier ( Grenoble ) 38 ( 1988 ), 199 - 218 . Numdam | Zbl 0645.35018 · Zbl 0645.35018
[18] M. Fukushima - Y. Oshima - M. Takeda , ” Dirichlet Forms and Symmetric Markov Processes ”, Studies in Mathematics . de Gruyter , Berlin - New York , 1994 . MR 1303354 | Zbl 0838.31001 · Zbl 0838.31001
[19] G.B. Folland - E.M. Stein , ” Hardy Spaces on Homogeneous Groups ”, Mathematical Notes 28 . Princeton University Press , Princeton, New Jersey , 1982 . MR 657581 | Zbl 0508.42025 · Zbl 0508.42025
[20] W. Hansen , Semi-polar sets and quasi-balayage , Math. Ann. 257 ( 1981 ), 495 - 517 . MR 1513285 | Zbl 0458.31008 · Zbl 0458.31008
[21] W. Hansen , A note on continuous solutions of the Schrödinger equation , Proc. Amer. Math. Soc . ( 2 ) 117 ( 1993 ), 381 - 384 . MR 1107921 | Zbl 0770.31008 · Zbl 0770.31008
[22] W. Hansen , Potentials with given oscillations , Exposition. Math. ( 1999 ), to appear. MR 1706216 | Zbl 0937.31007 · Zbl 0937.31007
[23] R.M. Hervé . , Recherches axiomatiques sur la théorie des fonctions surharmoniques et du potentiel , Ann. Inst. Fourier ( Grenoble ) 12 ( 1962 ). 415 - 517 . Numdam | MR 139756 | Zbl 0101.08103 · Zbl 0101.08103
[24] R.-M. Hervé - M. Hervé , Les fonctions surharmoniques associées à un opérateur elliptique du second ordre à coefficients discontinus , Ann. Inst. Fourier ( Grenoble ) 19 ( 1968 ), 305 - 359 . Numdam | MR 261027 | Zbl 0176.09801 · Zbl 0176.09801
[25] R.-M. Hervé - M. Hervé , Les fonctions surharmoniques dans l’axiomatique de M. Brelot associées à un opérateur elliptique dégénéré , Ann. Inst. Fourier ( Grenoble ) 22 ( 1972 ), 131 - 145 . Numdam | MR 377092 | Zbl 0224.31014 · Zbl 0224.31014
[26] R.-M. Hervé , Perturbation d’ un espace harmonique de M. Brelot. Recherche d’ une bijection entre potentiels et potentiels perturbés , Publication de l’Université de Paris 75 ( 1985 ). Zbl 0608.31006 · Zbl 0608.31006
[27] R.-M. Hervé , Inégalité de Harnack pour un faisceau perturbé et théorie adjointe , Publication de l’Université de Paris VI 81 ( 1987 ).
[28] W. Hansen - H. Hueber , The Dirichlet problem for sublaplacians on nilpotent Lie groups - geometric criteria for regularity , Math. Ann. 276 ( 1987 ), 537 - 547 . MR 879533 | Zbl 0601.31007 · Zbl 0601.31007
[29] A.M. Hinz - H. Kalf , Subsolution estimates and Harnack’s inequality for Schrödinger operators , J. Reine Angew. Math. 404 ( 1990 ), 118 - 134 . Article | MR 1037432 | Zbl 0779.35026 · Zbl 0779.35026
[30] W. Hansen - Z.M. Ma , Perturbations by differences of unbounded potentials , Math. Ann. 287 ( 1990 ), 553 - 569 . Article | MR 1066814 | Zbl 0685.31005 · Zbl 0685.31005
[31] H. Hueber - M. Sieveking , Uniform bounds for Green functions on C1,1-domains , Ann. Inst. Fourier ( Grenoble ) 32 ( 1982 ), 105 - 117 . Numdam | MR 658944 | Zbl 0465.35028 · Zbl 0465.35028
[32] H. Hueber - M. Sieveking , Quotients of Green functions on R n , Math. Ann. 269 ( 1984 ), 263 - 278 . MR 759112 | Zbl 0535.31004 · Zbl 0535.31004
[33] H. Hueber , The domination principle for the sum of squares of vector fields , Exposition. Math. 6 ( 1988 ), 183 - 184 . MR 938182 | Zbl 0655.31008 · Zbl 0655.31008
[34] J.-M. Keuntje , ” Störung harmonischer Räume mit Differenzen beschränkter Potentiale ”, PhD thesis, Universität Bielefeld , 1990 . Zbl 0744.31006 · Zbl 0744.31006
[35] P. Kroeger , Harmonic spaces associated with parabolic and elliptic differential operators , Math. Ann. 285 ( 1988 ), 393 - 403 . MR 1019709 | Zbl 0664.31011 · Zbl 0664.31011
[36] P. Kröger - K.-Th. Sturm , Hölder continuity of normalized solutions of the Schrödinger equation , Math. Ann. 297 ( 1993 ), 663 - 670 . MR 1245411 | Zbl 0822.35033 · Zbl 0822.35033
[37] K. Kurata , Continuity and Harnack’s inequality for solutions of elliptic partial differential equations of second order , Indiana Univ. Math. J. 43 ( 1994 ), 411 - 440 . MR 1291523 | Zbl 0805.35017 · Zbl 0805.35017
[38] H. Kuwano , On Harnack’s inequality for some degenerate elliptic equations , Bull. Fukuoka Univ. Ed. III 45 ( 1996 ), 1 - 7 . MR 1435785 | Zbl 0887.35010 · Zbl 0887.35010
[39] M.R. Lancia - M.V. Marchi , Harnack inequalities for nonsymmetric operators of Hörmander type with discontinuous coefficients , Adv. Math. Sci. Appl. 7 ( 1997 ), 833 - 857 . MR 1476279 | Zbl 0890.35003 · Zbl 0890.35003
[40] A. Mohammed , Harnack’s principle for some non-divergence structure elliptic operators , Comm. Partial Differential Equations 23 ( 1998 ), 287 - 306 . MR 1608528 | Zbl 0897.35022 · Zbl 0897.35022
[41] M Nakai , Brelot spaces of Schrödinger equations , J. Math. Soc. Japan 48 ( 1996 ), 275 - 298 . Article | MR 1376082 | Zbl 0864.31007 · Zbl 0864.31007
[42] I. Netuka , Continuity and maximum principle for potentials of signed measures , Czechoslovak Math. J. 25 ( 1975 ), 309 - 316 . Article | MR 382690 | Zbl 0309.31019 · Zbl 0309.31019
[43] A. Nagel - E.M. Stein - S. Wainger , Balls and metrics defined by vector fields I: Basic properties , Acta Math. 155 ( 1985 ), 103 - 147 . MR 793239 | Zbl 0578.32044 · Zbl 0578.32044
[44] A. Sanchez-Calle , Fundamental solutions and geometry of the sum of squares of vector fields , Invent. Math. 78 ( 1984 ), 143 - 160 . MR 762360 | Zbl 0582.58004 · Zbl 0582.58004
[45] C.G. Simader , An elementary proof of Harnack’s inequality for Schrödinger operators and related topics , Math. Z. 203 ( 1990 ), 129 - 152 . MR 1030712 | Zbl 0697.35017 · Zbl 0697.35017
[46] M. Zahid , Inégalité de Harnack et perturbation d’espaces harmoniques de Brelot par des mesures quelconques , Extracta Math. , to appear.
[47] M. Zahid , Fonctionelle additive et ellipticité , Extracta Math. 11 ( 1996 ), 288 - 300 . MR 1437453 | Zbl 0888.31006 · Zbl 0888.31006
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