On the theory of potentials in locally compact spaces

Le théorème du minimax et la théorie fine du potentiel

“…On peut en fait construire un noyau symétrique k (satisfaisant à tous les principes habituels de la théorie du potentiel) sur un espace localement compact, mais non dénombrable à l'infini, tel qu'il existe des ensembles fermés non capacitables (voir [14], ex. 10, p. 213).…”

Potential Theory of Geometric Stable Processes

“…Since the kernel G satisfies the maximum principle (see, for example, Theorem 5.2.2 in [6]), it follows from ( [7], page 159) that for any compact subset…”

Polar sets and capacitary potentials in homogeneous spaces

“…Our starting point is to combine the potential theory in locally compact spaces developed by Fuglede [10] in the 1960's with the homogeneous spaces of Coifman and Weiss [5] from the early 1970's. A homogeneous space (X, d, µ) consists of a quasimetric space (X, d) equipped with a nonnegative doubling measure µ (see Section 2 for the details).…”

“…The C k -capacity of a compact set K is defined by C k (K) −1 = inf I k (ν), where infimum is over all nonnegative measures ν supported in K with total mass ∥ν∥ 1 = 1 and I k (ν) denotes the energy integral I k (ν) =´´k(x, y) dν(x) dν(y). A complete homogeneous space is locally compact and we can apply the basic existence theorem for capacitary measures and capacitary potentials for compact sets in [10,Theorem 2.4]. The corresponding result for more general sets in a locally compact space is harder and requires stronger assumptions on the kernel, see [10,Chap.…”

“…A complete homogeneous space is locally compact and we can apply the basic existence theorem for capacitary measures and capacitary potentials for compact sets in [10,Theorem 2.4]. The corresponding result for more general sets in a locally compact space is harder and requires stronger assumptions on the kernel, see [10,Chap. II].…”