Capacitary inequalities for fractional integrals, with applications to partial differential equations and Sobolev multipliers

Abstract

Some new characterizations of the class of positive measures $\gamma$ on $\mathbf R^n$ such that $H{_{p}^{l}} \subset L_p (\gamma)$ are given, where $H{_{p}^{l}} (1\lt p \lt ∞, 0 \lt l \lt ∞)$ is the space of Bessel potentials. This imbedding, as well as the corresponding trace inequality $$||J_l u||_{L_p (\gamma )} \leq C||u||_{L_p },$$ for Bessel potentials $J_l = (1-Δ)^{-1/2}$, is shown to be equivalent to one of the following conditions.

1. $J_l (J_{l\gamma})^{p^\prime} \leq C J_{l\gamma}$ a.e.

2. $M_l (M_{l\gamma})^{p^\prime} \leq C M_{l\gamma}$ a.e.

3. For all compact subsets $E$ of $\mathbf R^n$

$$\int_E (J_{l\gamma)})^{p^\prime} dx \leq C \text{ cap} (E, H{_{p}^{l}}),$$ where $1/p+1/p'=1$, $M_l$ is the fractional maximal operator, and cap$(\cdot, H{_{p}^{l}})$ 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 $(J_{l\gamma})^{p^\prime} dx$. Similar results are proved for the Riesz potentials $I_{l\gamma} = |x|^{l-n}*\gamma$.

These results are used to get a complete characterization of the positive measures on $\mathbf R^n$ giving rise to bounded pointwise multipliers $M (H{_{p}^{m}} → H {_{p}^{−l}})$. Some applications to elliptic partial differential equations are considered, including coercive estimates for solutions of the Poisson equation, and existence of positive solutions for certain linear and semi-linear equations.