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  • Research Article
  • Open Access

Harnack Inequality for the Schrödinger Problem Relative to Strongly Local Riemannian -Homogeneous Forms with a Potential in the Kato Class

Boundary Value Problems20072007:024806

  • Received: 17 May 2006
  • Accepted: 21 September 2006
  • Published:


We define a notion of Kato class of measures relative to a Riemannian strongly local -homogeneous Dirichlet form and we prove a Harnack inequality (on balls that are small enough) for the positive solutions to a Schrödinger-type problem relative to the form with a potential in the Kato class.


  • Differential Equation
  • Partial Differential Equation
  • Ordinary Differential Equation
  • Functional Equation
  • Kato


Authors’ Affiliations

Dipartimento di Matematica "Francesco Brioschi", Politecnico di Milano, Piazza Leonardo Da Vinci 32, 20133 Milano, Italy
Accademia Nazionale delle Scienze detta dei XL, Via L. Spallanzani 7, 00161 Roma, Italy
Dipartimento di Matematica, Università di Parma, Viale Usberti 53/A, 43100 Parma, Italy


  1. Aizenman M, Simon B: Brownian motion and Harnack inequality for Schrödinger operators. Communications on Pure and Applied Mathematics 1982,35(2):209–273. 10.1002/cpa.3160350206MATHMathSciNetView ArticleGoogle Scholar
  2. Chiarenza F, Fabes E, Garofalo N: Harnack's inequality for Schrödinger operators and the continuity of solutions. Proceedings of the American Mathematical Society 1986,98(3):415–425.MATHMathSciNetGoogle Scholar
  3. Citti G, Garofalo N, Lanconelli E: Harnack's inequality for sum of squares of vector fields plus a potential. American Journal of Mathematics 1993,115(3):699–734. 10.2307/2375077MATHMathSciNetView ArticleGoogle Scholar
  4. Biroli M: Weak Kato measures and Schrödinger problems for a Dirichlet form. Rendiconti della Accademia Nazionale delle Scienze detta dei XL. Memorie di Matematica e Applicazioni. Serie V. Parte I 2000, 24: 197–217.MathSciNetGoogle Scholar
  5. Biroli M, Mosco U: Sobolev inequalities on homogeneous spaces. Potential Analysis 1995,4(4):311–324. 10.1007/BF01053449MATHMathSciNetView ArticleGoogle Scholar
  6. Biroli M, Mosco U: A Saint-Venant type principle for Dirichlet forms on discontinuous media. Annali di Matematica Pura ed Applicata. Serie Quarta 1995,169(1):125–181. 10.1007/BF01759352MATHMathSciNetView ArticleGoogle Scholar
  7. Biroli M: Nonlinear Kato measures and nonlinear subelliptic Schrödinger problems. Rendiconti della Accademia Nazionale delle Scienze detta dei XL. Memorie di Matematica e Applicazioni. Serie V. Parte I 1997, 21: 235–252.MathSciNetGoogle Scholar
  8. Malý J: Pointwise estimates of nonnegative subsolutions of quasilinear elliptic equations at irregular boundary points. Commentationes Mathematicae Universitatis Carolinae 1996,37(1):23–42.MATHMathSciNetGoogle Scholar
  9. Malý J, Ziemer WP: Fine Regularity of Solutions of Elliptic Partial Differential Equations, Mathematical Surveys and Monographs. Volume 51. American Mathematical Society, Rhode Island; 1997:xiv+291.View ArticleGoogle Scholar
  10. Biroli M, Marchi S: Oscillation estimates relative to-homogeneous forms and Kato measures data. to appear in Le MatematicheGoogle Scholar
  11. Biroli M: Strongly local nonlinear Dirichlet functionals and forms. to appear in Rendiconti della Accademia Nazionale delle Scienze detta dei XL. Memorie di Matematica e ApplicazioniGoogle Scholar
  12. Biroli M, Vernole PG: Strongly local nonlinear Dirichlet functionals and forms. Advances in Mathematical Sciences and Applications 2005,15(2):655–682.MATHMathSciNetGoogle Scholar
  13. Fukushima M, Ōshima Y, Takeda M: Dirichlet Forms and Symmetric Markov Processes, de Gruyter Studies in Mathematics. Volume 19. Walter de Gruyter, Berlin; 1994:x+392.View ArticleGoogle Scholar
  14. Coifman RR, Weiss G: Analyse Harmonique Non-Commutative sur Certains Espaces Homogènes, Lecture Notes in Mathematics. Volume 242. Springer, Berlin; 1971:v+160.Google Scholar
  15. Malý J, Mosco U: Remarks on measure-valued Lagrangians on homogeneous spaces. Ricerche di Matematica 1999,48(suppl.):217–231.MATHMathSciNetGoogle Scholar
  16. Kato T: Schrödinger operators with singular potentials. Israel Journal of Mathematics 1972, 13: 135–148 (1973). 10.1007/BF02760233MathSciNetView ArticleGoogle Scholar
  17. Biroli M, Mosco U: Kato space for Dirichlet forms. Potential Analysis 1999,10(4):327–345. 10.1023/A:1008684104029MATHMathSciNetView ArticleGoogle Scholar
  18. Biroli M: Schrödinger type and relaxed Dirichlet problems for the subelliptic -Laplacian. Potential Analysis 2001,15(1–2):1–16.MATHMathSciNetView ArticleGoogle Scholar
  19. Biroli M, Tchou NA: Nonlinear subelliptic problems with measure data. Rendiconti della Accademia Nazionale delle Scienze detta dei XL. Memorie di Matematica e Applicazioni. Serie V. Parte I 1999, 23: 57–82.MathSciNetGoogle Scholar
  20. Biroli M, Vernole P: Harnack inequality for harmonic functions relative to a nonlinear -homogeneous Riemannian Dirichlet form. Nonlinear Analysis 2006,64(1):51–68. 10.1016/ ArticleGoogle Scholar


© M. Biroli and S. Marchi 2007

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