Download PDF - Absolute continuity for elliptic-caloric measuresAll rights reserved. Copyright exceptions and limitations apply

ArticleOriginal scientific text

Title

Absolute continuity for elliptic-caloric measures

Authors 1

Affiliations

  1. Department of Mathematical Sciences, College of Arts and Sciences, Box 30001, Dept. 3MB, Las Cruces, New Mexico 88003-8001 U.S.A.

Abstract

A Carleson condition on the difference function for the coefficients of two elliptic-caloric operators is shown to give absolute continuity of one measure with respect to the other on the lateral boundary. The elliptic operators can have time dependent coefficients and only one of them is assumed to have a measure which is doubling. This theorem is an extension of a result of B. Dahlberg [4] on absolute continuity for elliptic measures to the case of the heat equation. The method of proof is an adaptation of Fefferman, Kenig and Pipher's proof of Dahlberg's result [8].

Bibliography

  1. D. G. Aronson, Non-negative solutions of linear parabolic equations, Ann. Scuola Norm. Sup. Pisa 22 (1968), 607-694.
  2. A. S. Besicovitch, A general form of the covering principle and relative differentiation of additive functions, II, Proc. Cambridge Philos. Soc. 42 (1946), 1-10.
  3. R. Coifman and C. Fefferman, Weighted norm inequalities for maximal functions and singular integrals, Studia Math. 51 (1974), 241-250.
  4. B. E. J. Dahlberg, On the absolute continuity of elliptic measures, Amer. J. Math. 108 (1986), 1119-1138.
  5. B. E. J. Dahlberg, D. S. Jerison, and C. E. Kenig, Area integral estimates for elliptic differential operators with non-smooth coefficients, Ark. Mat. 22 (1984), 97-108.
  6. J. Doob, Classical Potential Theory and its Probabilistic Counterpart, Springer, 1984.
  7. E. Fabes, N. Garofalo, and S. Salsa, A backward Harnack inequality and Fatou theorem for nonnegative solutions of parabolic equations, Illinois J. Math. 20 (1986), 536-565.
  8. R. Fefferman, C. Kenig, and J. Pipher, The theory of weights and the Dirichlet problem for elliptic equations, Ann. of Math. 134 (1991), 65-124.
  9. Y. Heurteaux, Inégalités de Harnack à la frontière pour des opérateurs paraboliques, C. R. Acad. Sci. Paris Sér. I 308 (1989), 401-404, 441-444.
  10. C. Kenig, Harmonic analysis techniques for second order elliptic boundary value problems, preprint.
  11. E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, 1970.
  12. C. Sweezy, Fatou theorems for parabolic equations, Proc. Amer. Math. Soc., to appear.
Studia Mathematica
1996/120/2
Export to PBN, Opens in new tab
Pages:
95-112
Main language of publication
English
Received
1994-03-15
Accepted
1996-05-06
Published
1996
Exact and natural sciences