Capacities and embeddings of Besov spaces via general convolution kernels

Abstract

This note is denoted to establish equivalent characterizations of Carleson embeddings of fractional Besov spaces Λp,q β (Rn) into the Lorentz spaces Lq0,pμ (R1+n+) induced by general convolution kernels Φt(·). When (p, q) ∈ (1, n/β) x (1,∞), the embeddings will be characterized in terms of capacitary type inequalities for open subsets of Rn. When p = q ∈ (0, 1], the embeddings will be characterized in terms of fractional Besov capacities or the associated variational functional of a nonnegative Radon measure μ. Especially, when p = q = 1 and β ∈ (0, 1), the characterization can be also established in terms of fractional perimeters.