Weighted Trudinger-type inequalities

Abstract

Abstract. We establish sharp inequalities of Trudinger-type (with non-standard Young functions) on general domains Ω with respect to measures ν, µ. Our results are new even when Ω is a Euclidean ball, and ν, µ are defined in terms of powers of distance to the boundary. In the Euclidean case, sharpness results are proved for the Young functions involved and the class of domains considered. New characterizations of QHBC domains are given.

Criteria for optimal global integrability of Hajłasz–Sobolev functions

“…We point out that, as proved in [4,5], every simply connected domain in R 2 or every domain in R n with n ≥ 3 that is quasiconformally equivalent to a uniform domain satisfies a slice property and a separation property. Every John domain satisfies both a separation and a slice property; see [6].…”

Weak Slice Conditions and Hölder Imbeddings

“…The basic Euclidean 0-wSlice condition defined below is essentially taken from [5], where it is assumed uniformly for all x and a fixed y, but the α > 0 case and non-Euclidean variants have not been considered before. We also prove some basic properties of these weak slice conditions in this subsection.…”

“…In [3], the strong geometric condition (John) is equivalent to the combination of the weak geometric condition (separation) and a Sobolev-Poincaré imbedding. However in [4], the weak geometric condition (slice) is not implied by the strong geometric condition (mean cigar) for any value of p > n. For p = n, Buckley and O'Shea [5] overcame this deficiency by showing that the strong geometric condition is equivalent to the combination of a so-called weak slice condition (which is implied by a slice condition) and the Trudinger imbedding. Here we prove the following analogous result for p > n; the terminology is explained in Sections 1 and 2.…”

On weighted critical imbeddings of Sobolev spaces

“…There are far fewer results on the weighted limiting imbeddings; they are the subject e.g., of [3,7]. The paper [7] gives necessary and sufficient conditions for critical imbeddings with weights of the special type |x| α log(1/|x|) β ; it is based on estimates for the Riesz transform, and one of the estimates established there also gave some motivation for our considerations here concerning more general weights.…”