Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

The equivalence of two definitions of capacity


Authors: David R. Adams and John C. Polking
Journal: Proc. Amer. Math. Soc. 37 (1973), 529-534
MSC: Primary 31B15
DOI: https://doi.org/10.1090/S0002-9939-1973-0328109-5
MathSciNet review: 0328109
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: It is shown that two definitions for an $ {L_p}$ capacity $ (1 < p < \infty )$ on subsets of Euclidean $ {R^n}$ are equivalent in the sense that as set functions their ratio is bounded above and below by positive finite constants. The classical notions of capacity correspond to the case $ p = 2$.


References [Enhancements On Off] (What's this?)

  • [1] D. R. Adams and N. G. Meyers, Bessel potentials. Inclusion relations among classes of exceptional sets, Indiana Univ. Math. J. (to appear). (An announcement of these results appears in Bull. Amer. Math. Soc. 77 (1971), 968-970.) MR 0284607 (44:1831)
  • [2] A. P. Calderón, Lebesgue spaces of differentiable functions and distributions, Proc. Sympos. Pure Math., vol. 4, Amer. Math. Soc., Providence, R.I., 1961, pp. 33-49. MR 26 #603. MR 0143037 (26:603)
  • [3] B. Fuglede, Extremal length and functional completion. Acta Math. 98 (1957), 171-219. MR 20 #4187. MR 0097720 (20:4187)
  • [4] -, Application du théorème minimax a l'étude diverse capacités, C. R. Acad. Sci. Paris 266 (1968), 921-923.
  • [5] R. Harvey and J. C. Polking, Removable singularities of solutions of linear partial differential equations, Acta Math. 125 (1970), 39-56. MR 43 #5183. MR 0279461 (43:5183)
  • [6] -, A notion of capacity which characterizes removable singularities, Trans. Amer. Math. Soc. 169 (1972), 183-195. MR 0306740 (46:5862)
  • [7] I. I. Hirschman, A convexity theorem for certain groups of transformations, J. Analyse Math. 2 (1952/53), 209-218. MR 15, 295; 1139. MR 0057936 (15:295b)
  • [8] W. Littman, A connection between $ \alpha $-capacity and m-p polarity, Bull. Amer. Math. Soc. 73 (1967), 862-866. MR 36 #2940. MR 0219866 (36:2940)
  • [9] -, Polar sets and removable singularities of partial differential equations, Ark. Mat. 7 (1967), 1-9. MR 37 #559. MR 0224960 (37:559)
  • [10] V. G. Maz'ja and V. P. Havin, A nonlinear analogue of the Newtonian potential and metric properties of $ (p,1)$-capacity, Dokl. Akad. Nauk SSSR 194 (1970), 770-773=Soviet Math. Dokl. 11 (1970), 1294-1298. MR 42 #7926. MR 0273045 (42:7926)
  • [11] V. G. Maz'ja, Imbedding theorems and their applications, Baku Sympos. (1966), ``Nauka", Moscow, 1970, pp. 142-159. (Russian). MR 0313791 (47:2345)
  • [12] -, p-conductance and theorems of imbedding certain function spaces into the space $ \mathfrak{C}$, Dokl. Akad. Nauk SSSR 140 (1961), 299-302=Soviet Math. Dokl. 2 (1961), 1200-1203. MR 28 #460. MR 0157224 (28:460)
  • [13] N. G. Meyers, A theory of capacities for potentials of functions in Lebesgue classes, Math. Scand. 26 (1970), 255-292. MR 43 #3474. MR 0277741 (43:3474)
  • [14] L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa (3) 13 (1959), 115-162. MR 22 #823. MR 0109940 (22:823)
  • [15] J. C. Polking, A Leibniz formula for some differentiation operators of fractional order, Indiana Univ. Math. J. 21 (1972), 1019-1029. MR 0318868 (47:7414)
  • [16] Ju. G. Rešetnjak, The concept of capacity in the theory of functions with generalized derivatives, Sibirsk. Mat. Ž. 10 (1969), 1109-1138=Siberian Math. J. 10 (1969), 818-842. MR 43 #2234. MR 0276487 (43:2234)
  • [17] R. S. Strichartz, Multipliers on fractional Sobolev spaces, J. Math. Mech. 16 (1967), 1031-1060. MR 35 #5927. MR 0215084 (35:5927)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 31B15

Retrieve articles in all journals with MSC: 31B15


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1973-0328109-5
Keywords: Capacity, Bessel potentials of $ {L_p}$ functions, fractional differentiation operators, functions that operate
Article copyright: © Copyright 1973 American Mathematical Society

American Mathematical Society