On Strong Barriers and an Inequality of Hardy for Domains in R ⁿ

Global gradient estimates for degenerate parabolic equations in nonsmooth domains

“…This result was shown by Lewis in [20]. See also Ancona [2] and Mikkonen [23]. Theorem 3.12 Let 1 < p ≤ n. If a set E is uniformly p-thick, then there exists a constant q = q(n, p, µ) such that 1 < q < p for which E is uniformly q-thick.…”

The equivalence between pointwise Hardy inequalities and uniform fatness

“…In R n , for n ≥ 2, Hardytype inequalities first appeared in the paper of Nečas [31] in the context of Lipschitz domains. However, it has been well-known since the works of Ancona [3] (p = 2), Lewis [29], and Wannebo [34], that the regularity of the boundary is not essential for Hardy inequalities, as it was shown in these papers that uniform p-fatness of the complement suffices for a domain to admit the integral p-Hardy inequality (2). Uniform n-fatness of the complement is also necessary for the n-Hardy inequality (in R n ), see [3,29], but this is not true for p < n.…”

Mutual Absolute Continuity of Interior and Exterior Harmonic Measure Implies Rectifiability

“…However, we mention that by a result of Ancona [2] this is equivalent to the more known "capacity density condition" (CDC), and by a deep theorem of Lewis [25], it follows that Ω is ∆-regular if and only if there exists some ε > 0 and some R > 0 such that H n−1+ε ∞ (B(x, r) \ Ω) ≈ r n−1+ε for all x ∈ ∂Ω and all 0 < r ≤ R, where H s ∞ stands for the s-dimensional Hausdorff content. We also remark that, in particular, the nontangentially accessible domains of Jerison and Kenig [20] are examples of ∆-regular domains.…”