CAPACITARY INEQUALITIES FOR FRACTIONAL INTEGRALS, WITH APPLICATIONS TO PARTIAL-DIFFERENTIAL EQUATIONS AND SOBOLEV MULTIPLIERS

Some new characterizations of the class of positive measures gamma on R(n) such that H-p(l) subset of Lp(gamma) are given, where H-p(l)(1<p< infinity, 0<1<infinity) is the space of Bessel potentials. This imbedd ing, as well as the corresponding trace inequality parallel to J(l)u p arallel to(Lp(gamma))less than or equal to C parallel to u parallel to (Lp), for Bessel potentials J(l)=(1-Delta)(-1/2), is shown to be equiv alent to one of the following conditions. (a) J(l)(J(l gamma) )(p')les s than or equal to CJ(l gamma) a.e. (b) M(l)(M(l gamma))(p')less than or equal to CM(l gamma) a.e. (c) For all compact subsets E of R(n) int egral(E)(J(l gamma))(p')dx less than or equal to C cap(E,H-p(l)), wher e 1/p+1/p' = 1, M(l) is the fractional maximal operator, and cap(, H-p (l)) is the Bessel capacity. In particular, it is shown that the trace inequality for a positive measure gamma holds if and only if it holds for the measure (J(l gamma))(p')dx. Similar results are proved for th e Riesz potentials I-l gamma=\x\(l-n)gamma. These results are used to get a complete characterization of the positive measures on RR giving rise to bounded pointwise multipliers M(H-p(m) --> H-p(-l)). Some app lications to elliptic partial differential equations are considered, i ncluding coercive estimates for solutions of the Poisson equation, and existence of positive solutions for certain linear and semi-linear eq uations.