THE RELATIVE p-AFFINE CAPACITY.

In this paper, the relative p-affine capacities are introduced, developed, and subsequently applied to the trace theory of affine Sobolev spaces. In particular, we geometrically characterize such a nonnegative Radon measure μ given on an open set O ⊆ Rⁿ that naturally induces an embedding of the p-affine Sobolev class W^1,p_0,d (O) into the Lebesgue space L^q(O, μ) (under 1 ⩽ p ⩽ q < ∝) and the exponentially-integrable Lebesgue space exp ((nω^1/n_n ∣f∣)ⁿ/^(n-1)) ∊ L¹(O, μ) (under p = n) as well as the Lebesgue space L^∝(O, μ) (under n < p < ∝) with μ(O) < ∝. The results discovered here are new and nontrivial. [ABSTRACT FROM AUTHOR]