Equivalence of analytic and Sobolev Poincaré inequalities for planar domains

Abstract

For a finitely connected planar domain Ω it is shown that the analytic-Poincaré inequalityholds uniformly for all holomorphic functions f on Ω (z 0 ∈ Ω fixed, K a p (Ω) an absolute constant) if and only if the Sobolev-holds for an absolute constant K p (Ω) and for all u ∈ C 1 (Ω) whose integral over Ω is zero. This paper extends a result of Hamilton (1986) who established this equivalence when 1 < p < ∞.

Stable Phase Retrieval in Infinite Dimensions

“…for all F ∈ O(D), and 1 ≤ p ≤ ∞. Such inequalities are studied in [33]. Although (2.2) features the point z 0 ∈ D, it turns out that whether or not the domain D is an analytic Poincaré domain is independent of z 0 .…”

“…Although (2.2) features the point z 0 ∈ D, it turns out that whether or not the domain D is an analytic Poincaré domain is independent of z 0 . However [33], the best-possible constant C a poinc (p, D, z 0 ) depends on the choice of z 0 . Denote by C poinc (p, D) the usual Poincaré constant of the domain D, i.e.…”

Dirichlet problem and Sokhotski-Plemelj jump formula on Weil-Petersson class quasidisks

“…Now it is well-known, see for example [16] and [17], that for any quasidisk Ω (which is a so called John domain), for any arbitrary z 0 in Ω, and for F holomorphic in Ω, one has the analytic Poincaré inequality…”

“…In particular we require some results of of Smith and Stegenga [16] and Stanoyevitch and Stegenga [17] concerning Poincaré inequalities on John domains. Later in the investigation of the regularity of the WP-class quasicircles we utilize a result from geometric measure theory due to Falconer and Marsh [7] characterizing bi-Lipschitz equivalence of quasidisks.…”

Well-posedness of a Riemann–Hilbert problem on d-regular quasidisks

“…It is well-known, using a combination of the results in [8] and [9], that for any finitely connected Hölder domain Ω with finite area and any z 0 in Ω, one has the analytic Poincaré inequality…”