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A criterion for the absolute continuity of the harmonic measure associated with an elliptic operator. (English) Zbl 0694.35050

The author proves a criterion on a difference of the two linear elliptic operators in divergence form which implies the absolute continuity of the harmonic measure associated with one of the operators with respect to the surface measure if the measure associated with the other operator is absolutely continuous.
Reviewer: B.Nowak

MSC:

35J25 Boundary value problems for second-order elliptic equations
35C15 Integral representations of solutions to PDEs
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[1] W. Littman, G. Stampacchia, and H. F. Weinberger, Regular points for elliptic equations with discontinuous coefficients, Ann. Scuola Norm. Sup. Pisa (3) 17 (1963), 43 – 77. · Zbl 0116.30302
[2] Luis A. Caffarelli, Eugene B. Fabes, and Carlos E. Kenig, Completely singular elliptic-harmonic measures, Indiana Univ. Math. J. 30 (1981), no. 6, 917 – 924. · Zbl 0482.35020
[3] Eugene B. Fabes, David S. Jerison, and Carlos E. Kenig, Necessary and sufficient conditions for absolute continuity of elliptic-harmonic measure, Ann. of Math. (2) 119 (1984), no. 1, 121 – 141. · Zbl 0551.35024
[4] Björn E. J. Dahlberg, On the absolute continuity of elliptic measures, Amer. J. Math. 108 (1986), no. 5, 1119 – 1138. · Zbl 0644.35032
[5] R. R. Coifman and C. Fefferman, Weighted norm inequalities for maximal functions and singular integrals, Studia Math. 51 (1974), 241 – 250. · Zbl 0291.44007
[6] E. M. Stein, Note on the class \? \?\?\? \?, Studia Math. 32 (1969), 305 – 310. · Zbl 0182.47803
[7] L. Caffarelli, E. Fabes, S. Mortola, and S. Salsa, Boundary behavior of nonnegative solutions of elliptic operators in divergence form, Indiana Univ. Math. J. 30 (1981), no. 4, 621 – 640. · Zbl 0512.35038
[8] Björn E. J. Dahlberg, David S. Jerison, and Carlos E. Kenig, Area integral estimates for elliptic differential operators with nonsmooth coefficients, Ark. Mat. 22 (1984), no. 1, 97 – 108. · Zbl 0537.35025
[9] F. John and L. Nirenberg, On functions of bounded mean oscillation, Comm. Pure Appl. Math. 14 (1961), 415 – 426. · Zbl 0102.04302
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