Conductor and Capacitary Inequalities with Applications to Sobolev-Type Embeddings

Abstract

Let Ω be an open set in $\Bbb{R}^{n}$ and let μ and ν be locally finite, nonzero Borel measures on Ω. We also use the following notation: l is a positive integer, 1≤p≤∞, q>0, dx is an element of the Lebesgue measure m _n on $\Bbb{R}^{n}$, and f is an arbitrary function in $C^{\infty}_{0}(\varOmega)$, i.e., an infinitely differentiable function with compact support in Ω. By $\mathcal{L}_{t}$ we mean the set {x∈Ω:|f(x)|>t}, where t>0. We shall use the equivalence relation a∼b to denote that the ratio a/b admits upper and lower bounds by positive constants depending only on n, l, p, and q.