A notion of capacity which characterizes removable singularities

Harvey, Reese; Polking, John C.

doi:10.1090/S0002-9947-1972-0306740-4

A notion of capacity which characterizes removable singularities
HTML articles powered by AMS MathViewer

by Reese Harvey and John C. Polking

Trans. Amer. Math. Soc. 169 (1972), 183-195

DOI: https://doi.org/10.1090/S0002-9947-1972-0306740-4

PDF | Request permission

Abstract:

In this paper the authors define a capacity for a given linear partial differential operator acting on a Banach space of distributions. This notion has as special cases Newtonian capacity, analytic capacity, and AC capacity. It is shown that the sets of capacity zero are precisely those sets which are removable sets for the corresponding homogeneous equation. Simple properties of the capacity are derived and special cases examined.

References

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
Lars V. Ahlfors, Bounded analytic functions, Duke Math. J. 14 (1947), 1–11. MR 21108
D. G. Aronson, Removable singularities for linear parabolic equations, Arch. Rational Mech. Anal. 17 (1964), 79–84. MR 177206, DOI 10.1007/BF00283868
Lennart Carleson, Selected problems on exceptional sets, Van Nostrand Mathematical Studies, No. 13, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1967. MR 0225986
E. B. Fabes and N. M. Rivière, Singular integrals with mixed homogeneity, Studia Math. 27 (1966), 19–38. MR 209787, DOI 10.4064/sm-27-1-19-38
Bent Fuglede, Extremal length and functional completion, Acta Math. 98 (1957), 171–219. MR 97720, DOI 10.1007/BF02404474

Applications du théorème minimax à l’étude diverse capacités

266

P. R. Garabedian, Schwarz’s lemma and the Szegö kernel function, Trans. Amer. Math. Soc. 67 (1949), 1–35. MR 32747, DOI 10.1090/S0002-9947-1949-0032747-8
Reese Harvey and John Polking, Removable singularities of solutions of linear partial differential equations, Acta Math. 125 (1970), 39–56. MR 279461, DOI 10.1007/BF02838327

Linear partial differential operators

28

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

Uniform approximation of nowhere dense sets by solutions of elliptic equations

Ju. G. Rešetnjak, The concept of capacity in the theory of functions with generalized derivatives, Sibirsk. Mat. Ž. 10 (1969), 1109–1138 (Russian). MR 0276487