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Weighted capacity and the Choquet integral


Author: David R. Adams
Journal: Proc. Amer. Math. Soc. 102 (1988), 879-887
MSC: Primary 31B15; Secondary 35J99
DOI: https://doi.org/10.1090/S0002-9939-1988-0934860-7
MathSciNet review: 934860
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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.


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  • [A$ _{1}$] D. R. Adams, Maximal operators and capacity, Proc. Amer. Math. Soc. 34 (1972), 152-156. MR 0350314 (50:2807)
  • [A$ _{2}$] -, Sets and functions of finite $ {L^p}$-capacity, Indiana Univ. Math. J. 27 (1978), 611-627. MR 0486575 (58:6297)
  • [A$ _{3}$] -, A note on differentiation with respect to $ {L^p}$-capacity, preprint, Univ. of Kentucky, 1978, 8 pp.
  • [A$ _{4}$] -, Lectures on $ {L^p}$-potential theory, Umeå Univ. Reports, No. 2, 1981.
  • [A$ _{5}$] -, Weighted nonlinear potential theory, Trans. Amer. Math. Soc. 297 (1986), 73-94. MR 849468 (88m:31011)
  • [An] B. Anger, Representation of capacities, Math. Ann. 229 (1977), 245-258. MR 0466588 (57:6465)
  • [BZ] T. Bagby and W. Ziemer, Pointwise differentiability and absolute continuity, Trans. Amer. Math. Soc. 191 (1974), 129-148. MR 0344390 (49:9129)
  • [C$ _{1}$] G. Choquet, Theory of capacities, Ann. Inst. Fourier 5 (1953), 131-295. MR 0080760 (18:295g)
  • [C$ _{2}$] -, Forme abstraite du theoreme de capacitabilitie, Ann. Inst. Fourier 9 (1959), 83-89. MR 0112844 (22:3692b)
  • [CF] R. Coifman and C. Fefferman, Weighted norm inequalities for maximal functions and singular integrals, Studia Math. 60 (1974), 241-250. MR 0358205 (50:10670)
  • [EP] D. E. Edmunds and L. A. Peletier, A Harnack inequality for weak solutions of degenerate quasilinear elliptic equations, J. London Math. Soc. 5 (1972), 21-31. MR 0298217 (45:7269)
  • [FKS] E. Fabes, C. Kenig, and R. Serapioni, The local regularity of solutions of degenerate elliptic equations, Comm. Partial Differential Equations 7 (1982), 77-116. MR 643158 (84i:35070)
  • [FJK] E. Fabes, D. Jerison, and C. Kenig, The Wiener test for degenerate elliptic equations, Ann. Inst. Fourier 32 (1982), 151-182. MR 688024 (84g:35067)
  • [Ha] K. Hansson, Imbedding theorems of Sobolev type in potential theory, Math. Scand. 45 (1979), 77-102. MR 567435 (81j:31007)
  • [H] L. I. Hedberg, On certain convolution inequalities, Proc. Amer. Math. Soc. 36 (1972), 505-510. MR 0312232 (47:794)
  • [LSW] W. Littman, G. Stampacchia, and H. Weinberger, Regular points for elliptic equations with discontinuous coefficients, Ann. Scuola Norm. Sup. Pisa 17 (1963), 45-79. MR 0161019 (28:4228)
  • [M] N. Meyers, A theory of capacities for potentials of functions in Lebesgue classes, Math. Scand. 26 (1970), 255-292. MR 0277741 (43:3474)
  • [MS] M. K. Murthy and G. Stampacchia, Boundary value problems for some degenerate elliptic operators, Ann. Mat. Pura Appl. 80 (1968), 1-122. MR 0249828 (40:3069)
  • [S] G. Stampacchia, Le probleme de Dirichlet pour les equations elliptique du second order á coefficients discontinuous, Ann. Inst. Fourier (Grenoble) 15 (1965), 189-258. MR 0192177 (33:404)
  • [St] E. Stein, Maximal functions: Spherical means, Proc. Nat. Acad. Sci. U.S.A. 73 (1976), 2174-2175. MR 0420116 (54:8133a)

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DOI: https://doi.org/10.1090/S0002-9939-1988-0934860-7
Article copyright: © Copyright 1988 American Mathematical Society

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