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

Some new characterizations of the class of positive measures γ on Rⁿ such that Hₚˡ ⊂ Lₚ (γ) are given, where Hₚˡ (1< p < ∞, 0 < l < ∞) is the space of Bessel potentials. This imbedding, as well as the corresponding trace inequality ||Jₗ u||Lₚ (γ ) ≤ C||u||Lₚ, for Bessel potentials Jₗ = (1-Δ)-1/2, is shown to be equivalent to one of the following conditions. Jₗ (Jlγ)p^ ≤ C Jlγ a.e.Mₗ (Mlγ)p^ ≤ C Mlγ a.e.For all compact subsets E of Rⁿ ∫E (Jlγ))p^ dx ≤ C cap (E, Hₚˡ), where $1/p+1/p'=1$, Mₗ is the fractional maximal operator, and cap(·, Hₚˡ) is the Bessel capacity. In particular it is shown that the trace inequality for a positive measure γ holds if and only if it holds for the measure (Jlγ)p^ dx. Similar results are proved for the Riesz potentials Ilγ = |x|ˡ⁻ⁿ*γ. These results are used to get a complete characterization of the positive measures on Rⁿ giving rise to bounded pointwise multipliers M (Hₚᵐ → 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.