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 $1<p<\infty$ and $\alpha >{\text{dim}}_A(\partial \Omega )-2$, we establish a norm equivalence between ${\int }_{\Omega }|g{|}^p{\delta }^{\alpha }dA$ and ${\int }_{\Omega }|g^{\prime }{|}^p{\delta }^{\alpha +p}dA$ 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 Poincaré inequality on John domains. Taking $\alpha =mp-2$ yields the corresponding comparison between the $m$-th and $(m+1)$-st derivatives. For $m\ge 2$ the boundary-dimension condition is automatic, so the only dimension-sensitive case is the comparison between ${\int }_{\Omega }|f^{\prime }{|}^p{\delta }^{p-2}dA$ and ${\int }_{\Omega }|f^{\prime \prime }{|}^p{\delta }^{2p-2}dA$ when $1<p<2$. We show that this restriction is sharp within the class of quasidisks by using self-similar Rohde snowflakes. We also construct, for every $s>1$, an inward-cusp $s$-John domain on which the comparison fails, showing that the ordinary John condition cannot in general be weakened.

15 pages, 2 figures