Harnack inequalities for Schrödinger operators

Wolfhard Hansen

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (1999)

  • Volume: 28, Issue: 3, page 413-470
  • ISSN: 0391-173X

How to cite

top

Hansen, Wolfhard. "Harnack inequalities for Schrödinger operators." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 28.3 (1999): 413-470. <http://eudml.org/doc/84384>.

@article{Hansen1999,
author = {Hansen, Wolfhard},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
keywords = {Harnack inequalities; Schrödinger operators},
language = {eng},
number = {3},
pages = {413-470},
publisher = {Scuola normale superiore},
title = {Harnack inequalities for Schrödinger operators},
url = {http://eudml.org/doc/84384},
volume = {28},
year = {1999},
}

TY - JOUR
AU - Hansen, Wolfhard
TI - Harnack inequalities for Schrödinger operators
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 1999
PB - Scuola normale superiore
VL - 28
IS - 3
SP - 413
EP - 470
LA - eng
KW - Harnack inequalities; Schrödinger operators
UR - http://eudml.org/doc/84384
ER -

References

top
  1. [AS82] M. Aizenman - B. Simon, Brownian motion and Harnack's inequality for Schrödinger operators, Comm. Pure Appl. Math.35 (1982), 209-271. Zbl0459.60069MR644024
  2. [BH86] J. Bliedtner - W. Hansen, "Potential Theory - An Analytic and Probabilistic Approach to Balayage", Universitext. Springer, Berlin-Heidelberg-New York- Tokyo, 1986. Zbl0706.31001MR850715
  3. [BH91]] N. Bouleau - F. Hirsch, "Dirichlet Forms and Analysis on Wiener Space", vol. 14 of Studies in Mathematics. de Gruyter, Berlin - New York, 1991. Zbl0748.60046MR1133391
  4. [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. Zbl0566.31005MR887788
  5. [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. Zbl0659.35025MR887788
  6. [BM68] N. Boboc - P. Mustaţă, "Espaces harmoniques associés aux opérateurs différentiels linéaires du second ordre de type elliptique" , Lecture Notes in Mathematics68, Springer, Berlin - New York, 1968. Zbl0167.40301MR241681
  7. [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. Zbl0176.09703MR262881
  8. [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, Roma1970, pp. 69-119. Zbl0205.10704MR277744
  9. [Bou79] A. Boukricha, Das Picard-Prinzip und verwandte Fragen bei Störung von harmonischen Räumen, Math. Ann.239 (1979), 247-270. Zbl0377.31011MR522783
  10. [CC72] C. Constantinescu - A. Cornea, "Potential Theory on Harmonic Spaces", Grundlehren d. math. Wiss. Springer, Berlin - Heidelberg - New York, 1972. Zbl0248.31011MR419799
  11. [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. Zbl0626.35022MR857933
  12. [CFZ88] M. Cranston - E. Fabes - Z. Zhao, Conditional gauge and potential theory for the Schrödinger operator, Trans. Amer. Math. Soc.307 (1988), 171-194. Zbl0652.60076MR936811
  13. [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. Zbl0795.35018MR1221840
  14. [DL54] J. Deny - J.L. Lions, Les espaces du type Beppo-Levi, Ann. Inst. Fourier (Grenoble) 5 (1953/54), 305-370. Zbl0065.09903MR74787
  15. [dLP90] A. De La Pradelle, Sur les perturbations d'espaces harmoniques, Acad. Roy. Belg. Bull. Cl. Sci. (6) 1 (1990), 201-212. Zbl0770.31010MR1146274
  16. [dLP99] A. De La Pradelle, "Sur les perturbations d'espaces harmoniques" Erratum, Acad. Roy. Belg. Bull. Cl. Sci. (6) (1999). 
  17. [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. Zbl0645.35018
  18. [FOT94] M. Fukushima - Y. Oshima - M. Takeda, "Dirichlet Forms and Symmetric Markov Processes", Studies in Mathematics. de Gruyter, Berlin - New York, 1994. Zbl0838.31001MR1303354
  19. [FS82] G.B. Folland - E.M. Stein, "Hardy Spaces on Homogeneous Groups", Mathematical Notes28. Princeton University Press, Princeton, New Jersey, 1982. Zbl0508.42025MR657581
  20. [Han81 ] W. Hansen, Semi-polar sets and quasi-balayage, Math. Ann.257 (1981), 495-517. Zbl0458.31008MR1513285
  21. [Han93] W. Hansen, A note on continuous solutions of the Schrödinger equation, Proc. Amer. Math. Soc. (2) 117 (1993), 381-384. Zbl0770.31008MR1107921
  22. [Han99] W. Hansen, Potentials with given oscillations, Exposition. Math. (1999), to appear. Zbl0937.31007MR1706216
  23. [Her62] R.M. Hervé., Recherches axiomatiques sur la théorie des fonctions surharmoniques et du potentiel, Ann. Inst. Fourier (Grenoble) 12 (1962). 415-517. Zbl0101.08103MR139756
  24. [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. Zbl0176.09801MR261027
  25. [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. Zbl0224.31014MR377092
  26. [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 Paris75 (1985). Zbl0608.31006
  27. [Her87] R.-M. Hervé, Inégalité de Harnack pour un faisceau perturbé et théorie adjointe, Publication de l'Université de ParisVI81 (1987). 
  28. [HH87] W. Hansen - H. Hueber, The Dirichlet problem for sublaplacians on nilpotent Lie groups - geometric criteria for regularity, Math. Ann.276 (1987), 537-547. Zbl0601.31007MR879533
  29. [HK90] A.M. Hinz - H. Kalf, Subsolution estimates and Harnack's inequality for Schrödinger operators, J. Reine Angew. Math.404 (1990), 118-134. Zbl0779.35026MR1037432
  30. [HM90] W. Hansen - Z.M. Ma, Perturbations by differences of unbounded potentials, Math. Ann.287 (1990), 553-569. Zbl0685.31005MR1066814
  31. [HS82] H. Hueber - M. Sieveking, Uniform bounds for Green functions on C1,1-domains, Ann. Inst. Fourier (Grenoble) 32 (1982), 105-117. Zbl0465.35028MR658944
  32. [HS84] H. Hueber - M. Sieveking, Quotients of Green functions on R n, Math. Ann.269 (1984), 263-278. Zbl0535.31004MR759112
  33. [Hue88] H. Hueber, The domination principle for the sum of squares of vector fields, Exposition. Math.6 (1988), 183-184. Zbl0655.31008MR938182
  34. [Keu90] J.-M. Keuntje, "Störung harmonischer Räume mit Differenzen beschränkter Potentiale", PhD thesis, Universität Bielefeld, 1990. Zbl0744.31006
  35. [Kro88] P. Kroeger, Harmonic spaces associated with parabolic and elliptic differential operators, Math. Ann.285 (1988), 393-403. Zbl0664.31011MR1019709
  36. [KS93] P. Kröger - K.-Th. Sturm, Hölder continuity of normalized solutions of the Schrödinger equation, Math. Ann.297 (1993), 663-670. Zbl0822.35033MR1245411
  37. [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. Zbl0805.35017MR1291523
  38. [Kuw96] H. Kuwano, On Harnack's inequality for some degenerate elliptic equations, Bull. Fukuoka Univ. Ed. III45 (1996), 1-7. Zbl0887.35010MR1435785
  39. [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. Zbl0890.35003MR1476279
  40. [Moh98] A. Mohammed, Harnack's principle for some non-divergence structure elliptic operators, Comm. Partial Differential Equations23 (1998), 287-306. Zbl0897.35022MR1608528
  41. [Nak96] M Nakai, Brelot spaces of Schrödinger equations, J. Math. Soc. Japan48 (1996), 275-298. Zbl0864.31007MR1376082
  42. [Net75] I. Netuka, Continuity and maximum principle for potentials of signed measures, Czechoslovak Math. J.25 (1975), 309-316. Zbl0309.31019MR382690
  43. [NSW85] A. Nagel - E.M. Stein - S. Wainger, Balls and metrics defined by vector fields I: Basic properties, Acta Math.155 (1985), 103-147. Zbl0578.32044MR793239
  44. [SC84] A. Sanchez-Calle, Fundamental solutions and geometry of the sum of squares of vector fields, Invent. Math.78 (1984), 143-160. Zbl0582.58004MR762360
  45. [Sim90] C.G. Simader, An elementary proof of Harnack's inequality for Schrödinger operators and related topics, Math. Z.203 (1990), 129-152. Zbl0697.35017MR1030712
  46. [Zah] M. Zahid, Inégalité de Harnack et perturbation d'espaces harmoniques de Brelot par des mesures quelconques, Extracta Math., to appear. 
  47. [Zah96] M. Zahid, Fonctionelle additive et ellipticité, Extracta Math.11 (1996), 288-300. Zbl0888.31006MR1437453

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.