Equivalence of analytic and Sobolev Poincaré inequalities for planar domains

Alexander Stanoyevitch and David A. Stegenga

Vol. 178 (1997), No. 2, 363–375

DOI: 10.2140/pjm.1997.178.363

Abstract

For a finitely connected planar domain Ω it is shown that the analytic-Poincaré inequality

holds uniformly for all holomorphic functions f on (z₀ ∈ fixed, K_p^a(Ω) an absolute constant) if and only if the Sobolev-Poincaré inequality ∥u(z)∥_L^p() ≤ K_p(Ω)∥ν(z)∥_L^p() holds for an absolute constant K_p(Ω) and for all u ∈𝒞¹(Ω) whose integral over is zero. This paper extends a result of Hamilton (1986) who established this equivalence when 1 < p < ∞.

Milestones

Received: 1 August 1995

Published: 1 April 1997