Weighted capacity and the Choquet integral

Adams, David R.

doi:10.1090/S0002-9939-1988-0934860-7

Weighted capacity and the Choquet integral
HTML articles powered by AMS MathViewer

by David R. Adams PDF

Proc. Amer. Math. Soc. 102 (1988), 879-887 Request permission

Abstract:

The capacity set function that is naturally associated with a linear second-order elliptic partial differential operator in divergence form is related to the concept of the Choquet integral of a weight function with respect to Newtonian capacity. The weight function comes from the coefficients of the differential operator. This idea is reminiscent of the Radon-Nikodym Theorem, but now for capacities instead of measures.

References

David R. Adams, Maximal operators and capacity, Proc. Amer. Math. Soc. 34 (1972), 152–156. MR 350314, DOI 10.1090/S0002-9939-1972-0350314-1
David R. Adams, Sets and functions of finite $L^{p}$-capacity, Indiana Univ. Math. J. 27 (1978), no. 4, 611–627. MR 486575, DOI 10.1512/iumj.1978.27.27040

A note on differentiation with respect to

-capacity

Lectures on

-potential theory

David R. Adams, Weighted nonlinear potential theory, Trans. Amer. Math. Soc. 297 (1986), no. 1, 73–94. MR 849468, DOI 10.1090/S0002-9947-1986-0849468-4
Bernd Anger, Representation of capacities, Math. Ann. 229 (1977), no. 3, 245–258. MR 466588, DOI 10.1007/BF01391470
Thomas Bagby and William P. Ziemer, Pointwise differentiability and absolute continuity, Trans. Amer. Math. Soc. 191 (1974), 129–148. MR 344390, DOI 10.1090/S0002-9947-1974-0344390-6
Gustave Choquet, Theory of capacities, Ann. Inst. Fourier (Grenoble) 5 (1953/54), 131–295 (1955). MR 80760
Gustave Choquet, Ensembles ${\cal K}$-analytiques et ${\cal K}$-sousliniens. Cas général et cas métrique, Ann. Inst. Fourier (Grenoble) 9 (1959), 75–81 (French). MR 112843
R. R. Coifman and C. Fefferman, Weighted norm inequalities for maximal functions and singular integrals, Studia Math. 51 (1974), 241–250. MR 358205, DOI 10.4064/sm-51-3-241-250
D. E. Edmunds and L. A. Peletier, A Harnack inequality for weak solutions of degenerate quasilinear elliptic equations, J. London Math. Soc. (2) 5 (1972), 21–31. MR 298217, DOI 10.1112/jlms/s2-5.1.21
Eugene B. Fabes, Carlos E. Kenig, and Raul P. Serapioni, The local regularity of solutions of degenerate elliptic equations, Comm. Partial Differential Equations 7 (1982), no. 1, 77–116. MR 643158, DOI 10.1080/03605308208820218
E. Fabes, D. Jerison, and C. Kenig, The Wiener test for degenerate elliptic equations, Ann. Inst. Fourier (Grenoble) 32 (1982), no. 3, vi, 151–182 (English, with French summary). MR 688024
Kurt Hansson, Imbedding theorems of Sobolev type in potential theory, Math. Scand. 45 (1979), no. 1, 77–102. MR 567435, DOI 10.7146/math.scand.a-11827
Lars Inge Hedberg, On certain convolution inequalities, Proc. Amer. Math. Soc. 36 (1972), 505–510. MR 312232, DOI 10.1090/S0002-9939-1972-0312232-4
W. Littman, G. Stampacchia, and H. F. Weinberger, Regular points for elliptic equations with discontinuous coefficients, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 17 (1963), 43–77. MR 161019
Norman G. Meyers, A theory of capacities for potentials of functions in Lebesgue classes, Math. Scand. 26 (1970), 255–292 (1971). MR 277741, DOI 10.7146/math.scand.a-10981
M. K. V. Murthy and G. Stampacchia, Boundary value problems for some degenerate-elliptic operators, Ann. Mat. Pura Appl. (4) 80 (1968), 1–122 (English, with Italian summary). MR 249828, DOI 10.1007/BF02413623
Guido Stampacchia, Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus, Ann. Inst. Fourier (Grenoble) 15 (1965), no. fasc. 1, 189–258 (French). MR 192177
Elias M. Stein, Maximal functions. I. Spherical means, Proc. Nat. Acad. Sci. U.S.A. 73 (1976), no. 7, 2174–2175. MR 420116, DOI 10.1073/pnas.73.7.2174