## Weighted capacity and the Choquet integral

HTML articles powered by AMS MathViewer

- by David R. Adams
- Proc. Amer. Math. Soc.
**102**(1988), 879-887 - DOI: https://doi.org/10.1090/S0002-9939-1988-0934860-7
- PDF | 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
—, - 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

*A note on differentiation with respect to*${L^p}$

*-capacity*, preprint, Univ. of Kentucky, 1978, 8 pp. —,

*Lectures on*${L^p}$

*-potential theory*, UmeåUniv. Reports, No. 2, 1981.

## Bibliographic Information

- © Copyright 1988 American Mathematical Society
- 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