Uniformly fat sets
HTML articles powered by AMS MathViewer
- by John L. Lewis PDF
- Trans. Amer. Math. Soc. 308 (1988), 177-196 Request permission
Abstract:
In this paper we study closed sets $E$ which are "locally uniformly fat" with respect to a certain nonlinear Riesz capacity. We show that $E$ is actually "locally uniformly fat" with respect to a weaker Riesz capacity. Two applications of this result are given. The first application is concerned with proving Sobolev-type inequalities in domains whose complements are uniformly fat. The second application is concerned with the Fekete points of $E$.References
- David R. Adams, Traces of potentials. II, Indiana Univ. Math. J. 22 (1972/73), 907–918. MR 313783, DOI 10.1512/iumj.1973.22.22075
- David R. Adams and Norman G. Meyers, Thinness and Wiener criteria for non-linear potentials, Indiana Univ. Math. J. 22 (1972/73), 169–197. MR 316724, DOI 10.1512/iumj.1972.22.22015
- David R. Adams and Lars Inge Hedberg, Inclusion relations among fine topologies in nonlinear potential theory, Indiana Univ. Math. J. 33 (1984), no. 1, 117–126. MR 726108, DOI 10.1512/iumj.1984.33.33005
- David R. Adams and John C. Polking, The equivalence of two definitions of capacity, Proc. Amer. Math. Soc. 37 (1973), 529–534. MR 328109, DOI 10.1090/S0002-9939-1973-0328109-5
- David R. Adams and Norman G. Meyers, Thinness and Wiener criteria for non-linear potentials, Indiana Univ. Math. J. 22 (1972/73), 169–197. MR 316724, DOI 10.1512/iumj.1972.22.22015
- Alano Ancona, On strong barriers and an inequality of Hardy for domains in $\textbf {R}^n$, J. London Math. Soc. (2) 34 (1986), no. 2, 274–290. MR 856511, DOI 10.1112/jlms/s2-34.2.274
- Lennart Carleson, An interpolation problem for bounded analytic functions, Amer. J. Math. 80 (1958), 921–930. MR 117349, DOI 10.2307/2372840
- Björn E. J. Dahlberg, On the distribution of Fekete points, Duke Math. J. 45 (1978), no. 3, 537–542. MR 507457
- V. G. Maz′ja and V. P. Havin, A nonlinear potential theory, Uspehi Mat. Nauk 27 (1972), no. 6, 67–138. MR 0409858
- W. K. Hayman and Ch. Pommerenke, On analytic functions of bounded mean oscillation, Bull. London Math. Soc. 10 (1978), no. 2, 219–224. MR 500932, DOI 10.1112/blms/10.2.219
- Lars Hörmander, $L^{p}$ estimates for (pluri-) subharmonic functions, Math. Scand. 20 (1967), 65–78. MR 234002, DOI 10.7146/math.scand.a-10821
- N. S. Landkof, Foundations of modern potential theory, Die Grundlehren der mathematischen Wissenschaften, Band 180, Springer-Verlag, New York-Heidelberg, 1972. Translated from the Russian by A. P. Doohovskoy. MR 0350027
- John L. Lewis, Some applications of Riesz capacities, Complex Variables Theory Appl. 12 (1989), no. 1-4, 237–244. MR 1040923, DOI 10.1080/17476938908814368
- 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
- Charles B. Morrey Jr., Multiple integrals in the calculus of variations, Die Grundlehren der mathematischen Wissenschaften, Band 130, Springer-Verlag New York, Inc., New York, 1966. MR 0202511
- Ch. Pommerenke, Über die Verteilung der Fekete-Punkte. II, Math. Ann. 179 (1969), 212–218 (German). MR 247108, DOI 10.1007/BF01358488 P. Sjögren, On the regularity of the distribution of the Fekete points of a compact surface in ${{\mathbf {R}}^n}$, Ark. Mat. 11 (1973), 148-151.
- David A. Stegenga, A geometric condition which implies BMOA, Harmonic analysis in Euclidean spaces (Proc. Sympos. Pure Math., Williams Coll., Williamstown, Mass., 1978) Proc. Sympos. Pure Math., XXXV, Part, Amer. Math. Soc., Providence, R.I., 1979, pp. 427–430. MR 545283
- Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
Additional Information
- © Copyright 1988 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 308 (1988), 177-196
- MSC: Primary 31B99
- DOI: https://doi.org/10.1090/S0002-9947-1988-0946438-4
- MathSciNet review: 946438