Harnack inequalities for Schrödinger operators
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 4, Tome 28 (1999) no. 3, pp. 413-470. 
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     author = {Hansen, Wolfhard},
     title = {Harnack inequalities for {Schr\"odinger} operators},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     pages = {413--470},
     publisher = {Scuola normale superiore},
     volume = {Ser. 4, 28},
     number = {3},
     year = {1999},
     mrnumber = {1736524},
     zbl = {0940.35063},
     language = {en},
     url = {http://archive.numdam.org/item/ASNSP_1999_4_28_3_413_0/}
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Hansen, Wolfhard. Harnack inequalities for Schrödinger operators. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 4, Tome 28 (1999) no. 3, pp. 413-470. http://archive.numdam.org/item/ASNSP_1999_4_28_3_413_0/

[AS82] M. Aizenman - B. Simon, Brownian motion and Harnack's inequality for Schrödinger operators, Comm. Pure Appl. Math. 35 (1982), 209-271. | MR | Zbl

[BH86] J. Bliedtner - W. Hansen, "Potential Theory - An Analytic and Probabilistic Approach to Balayage", Universitext. Springer, Berlin-Heidelberg-New York- Tokyo, 1986. | MR | Zbl

[BH91]] N. Bouleau - F. Hirsch, "Dirichlet Forms and Analysis on Wiener Space", vol. 14 of Studies in Mathematics. de Gruyter, Berlin - New York, 1991. | MR | Zbl

[BHH85] 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 | Zbl

[BHH87] 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 | Zbl

[BM68] 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 | Zbl

[Bon69] 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 | Zbl

[Bon70] J.M. Bony, Opérateurs elliptiques dégénérés associés aux axiomatiques de la théorie du potentiel, In: "Potential Theory" (CIME, 1° Ciclo, Stresa 1969) Edizioni Cremonese, Roma 1970, pp. 69-119. | MR | Zbl

[Bou79] A. Boukricha, Das Picard-Prinzip und verwandte Fragen bei Störung von harmonischen Räumen, Math. Ann. 239 (1979), 247-270. | MR | Zbl

[CC72] C. Constantinescu - A. Cornea, "Potential Theory on Harmonic Spaces", Grundlehren d. math. Wiss. Springer, Berlin - Heidelberg - New York, 1972. | MR | Zbl

[CFG86] 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 | Zbl

[CFZ88] M. Cranston - E. Fabes - Z. Zhao, Conditional gauge and potential theory for the Schrödinger operator, Trans. Amer. Math. Soc. 307 (1988), 171-194. | MR | Zbl

[CGL93] 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 | Zbl

[DL54] J. Deny - J.L. Lions, Les espaces du type Beppo-Levi, Ann. Inst. Fourier (Grenoble) 5 (1953/54), 305-370. | Numdam | MR | Zbl

[dLP90] A. De La Pradelle, Sur les perturbations d'espaces harmoniques, Acad. Roy. Belg. Bull. Cl. Sci. (6) 1 (1990), 201-212. | MR | Zbl

[dLP99] A. De La Pradelle, "Sur les perturbations d'espaces harmoniques" Erratum, Acad. Roy. Belg. Bull. Cl. Sci. (6) (1999).

[FdLP88] D. Feyel - A. De La Pradelle, Étude de l'équation 1/2Δu - uμ = 0 où μ est une mesure positive, Ann. Inst. Fourier (Grenoble) 38 (1988), 199-218. | Numdam | Zbl

[FOT94] M. Fukushima - Y. Oshima - M. Takeda, "Dirichlet Forms and Symmetric Markov Processes", Studies in Mathematics. de Gruyter, Berlin - New York, 1994. | MR | Zbl

[FS82] G.B. Folland - E.M. Stein, "Hardy Spaces on Homogeneous Groups", Mathematical Notes 28. Princeton University Press, Princeton, New Jersey, 1982. | MR | Zbl

[Han81 ] W. Hansen, Semi-polar sets and quasi-balayage, Math. Ann. 257 (1981), 495-517. | MR | Zbl

[Han93] W. Hansen, A note on continuous solutions of the Schrödinger equation, Proc. Amer. Math. Soc. (2) 117 (1993), 381-384. | MR | Zbl

[Han99] W. Hansen, Potentials with given oscillations, Exposition. Math. (1999), to appear. | MR | Zbl

[Her62] R.M. Hervé., Recherches axiomatiques sur la théorie des fonctions surharmoniques et du potentiel, Ann. Inst. Fourier (Grenoble) 12 (1962). 415-517. | Numdam | MR | Zbl

[Her68] 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 | Zbl

[Her72] 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 | Zbl

[Her85] 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

[Her87] R.-M. Hervé, Inégalité de Harnack pour un faisceau perturbé et théorie adjointe, Publication de l'Université de Paris VI 81 (1987).

[HH87] W. Hansen - H. Hueber, The Dirichlet problem for sublaplacians on nilpotent Lie groups - geometric criteria for regularity, Math. Ann. 276 (1987), 537-547. | MR | Zbl

[HK90] A.M. Hinz - H. Kalf, Subsolution estimates and Harnack's inequality for Schrödinger operators, J. Reine Angew. Math. 404 (1990), 118-134. | MR | Zbl

[HM90] W. Hansen - Z.M. Ma, Perturbations by differences of unbounded potentials, Math. Ann. 287 (1990), 553-569. | MR | Zbl

[HS82] H. Hueber - M. Sieveking, Uniform bounds for Green functions on C1,1-domains, Ann. Inst. Fourier (Grenoble) 32 (1982), 105-117. | Numdam | MR | Zbl

[HS84] H. Hueber - M. Sieveking, Quotients of Green functions on R n, Math. Ann. 269 (1984), 263-278. | MR | Zbl

[Hue88] H. Hueber, The domination principle for the sum of squares of vector fields, Exposition. Math. 6 (1988), 183-184. | MR | Zbl

[Keu90] J.-M. Keuntje, "Störung harmonischer Räume mit Differenzen beschränkter Potentiale", PhD thesis, Universität Bielefeld, 1990. | Zbl

[Kro88] P. Kroeger, Harmonic spaces associated with parabolic and elliptic differential operators, Math. Ann. 285 (1988), 393-403. | MR | Zbl

[KS93] P. Kröger - K.-Th. Sturm, Hölder continuity of normalized solutions of the Schrödinger equation, Math. Ann. 297 (1993), 663-670. | MR | Zbl

[Kur94] 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 | Zbl

[Kuw96] H. Kuwano, On Harnack's inequality for some degenerate elliptic equations, Bull. Fukuoka Univ. Ed. III 45 (1996), 1-7. | MR | Zbl

[LM97] 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 | Zbl

[Moh98] A. Mohammed, Harnack's principle for some non-divergence structure elliptic operators, Comm. Partial Differential Equations 23 (1998), 287-306. | MR | Zbl

[Nak96] M Nakai, Brelot spaces of Schrödinger equations, J. Math. Soc. Japan 48 (1996), 275-298. | MR | Zbl

[Net75] I. Netuka, Continuity and maximum principle for potentials of signed measures, Czechoslovak Math. J. 25 (1975), 309-316. | MR | Zbl

[NSW85] A. Nagel - E.M. Stein - S. Wainger, Balls and metrics defined by vector fields I: Basic properties, Acta Math. 155 (1985), 103-147. | MR | Zbl

[SC84] A. Sanchez-Calle, Fundamental solutions and geometry of the sum of squares of vector fields, Invent. Math. 78 (1984), 143-160. | MR | Zbl

[Sim90] C.G. Simader, An elementary proof of Harnack's inequality for Schrödinger operators and related topics, Math. Z. 203 (1990), 129-152. | MR | Zbl

[Zah] M. Zahid, Inégalité de Harnack et perturbation d'espaces harmoniques de Brelot par des mesures quelconques, Extracta Math., to appear.

[Zah96] M. Zahid, Fonctionelle additive et ellipticité, Extracta Math. 11 (1996), 288-300. | MR | Zbl