Distance-Weighted Norm Equivalences for Analytic Functions on John Domains

Abstract

Let $\Omega\subset\mathbb C$ be a bounded John domain and set $\delta(z)=\text{dist}(z,\partial\Omega)$. For $1 \text{dim}_A(\partial\Omega)-2$, we establish a norm equivalence between $\int_\Omega |g|^p\delta^\alpha\,dA$ and $\int_\Omega |g'|^p\delta^{\alpha+p}\,dA$ for analytic functions $g$ on $\Omega$, with a point-evaluation term fixing the additive constant. The estimate of the derivative term is local and holds on every proper planar domain, whereas the converse follows from a distance-weighted Poincar\'e inequality on John domains. Taking $\alpha=mp-2$ yields the corresponding comparison between the $m$-th and $(m+1)$-st derivatives. For $m\ge2$ the boundary-dimension condition is automatic, so the only dimension-sensitive case is the comparison between $\int_\Omega |f'|^p\delta^{p-2}\,dA$ and $\int_\Omega |f''|^p\delta^{2p-2}\,dA$ when $1 1$, an inward-cusp $s$-John domain on which the comparison fails, showing that the ordinary John condition cannot in general be weakened.