Global subrepresentation formulas in chain domains with irregular boundaries

Abstract

For a large class of chain domains with rough boundaries, we derive global subrepresentation formulas (i.e., pointwise inequalities reminiscent of part of the Fundamental Theorem of Calculus). The results are new even for John and Boman domains. We also show how they lead to global first order Poincaré–Sobolev estimates in a large collection of Φ-John domains. The restriction on Φ is expressed as an integral condition on the number of chaining balls of various specific sizes.