Absolute continuity for elliptic-caloric measures

Caroline Sweezy

Studia Mathematica (1996)

  • Volume: 120, Issue: 2, page 95-112
  • ISSN: 0039-3223

Abstract

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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].

How to cite

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Sweezy, Caroline. "Absolute continuity for elliptic-caloric measures." Studia Mathematica 120.2 (1996): 95-112. <http://eudml.org/doc/216330>.

@article{Sweezy1996,
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].},
author = {Sweezy, Caroline},
journal = {Studia Mathematica},
keywords = {Carleson condition},
language = {eng},
number = {2},
pages = {95-112},
title = {Absolute continuity for elliptic-caloric measures},
url = {http://eudml.org/doc/216330},
volume = {120},
year = {1996},
}

TY - JOUR
AU - Sweezy, Caroline
TI - Absolute continuity for elliptic-caloric measures
JO - Studia Mathematica
PY - 1996
VL - 120
IS - 2
SP - 95
EP - 112
AB - 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].
LA - eng
KW - Carleson condition
UR - http://eudml.org/doc/216330
ER -

References

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  1. [1] D. G. Aronson, Non-negative solutions of linear parabolic equations, Ann. Scuola Norm. Sup. Pisa 22 (1968), 607-694. Zbl0182.13802
  2. [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. Zbl0063.00353
  3. [3] R. Coifman and C. Fefferman, Weighted norm inequalities for maximal functions and singular integrals, Studia Math. 51 (1974), 241-250. Zbl0291.44007
  4. [4] B. E. J. Dahlberg, On the absolute continuity of elliptic measures, Amer. J. Math. 108 (1986), 1119-1138. Zbl0644.35032
  5. [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. Zbl0537.35025
  6. [6] J. Doob, Classical Potential Theory and its Probabilistic Counterpart, Springer, 1984. 
  7. [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. Zbl0625.35006
  8. [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. Zbl0770.35014
  9. [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. [10] C. Kenig, Harmonic analysis techniques for second order elliptic boundary value problems, preprint. 
  11. [11] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, 1970. Zbl0207.13501
  12. [12] C. Sweezy, Fatou theorems for parabolic equations, Proc. Amer. Math. Soc., to appear. Zbl0857.35052

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