English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Denote by $C^{\alpha}(\mathbb{D})$ the space of the functions $f$ on t}heunit disk $\mathbb{D}$ which are H\"older continuous with the exponent$\alpha$, and denote by $C^{1, \alpha}(\mathbb{D})$ the space which consists ofdifferentiable functions $f$ such that their derivatives are in the space$C^{\alpha}(\mathbb{D})$. Let $\mathcal{C}$ be the Cauchy transform ofDirichlet problem. In this paper, we obtain the norm estimates of$\|\mathcal{C}\|_{L^p\to L^q}$, where $3/2<p<2$ and $q=p/(p-1)$. As anapplication, we show that if $3/2<p<2$, then $u\in C^{\mu}(\mathbb{D})$, where$\mu=2/p-1$. We also show that if $2<p<\infty$, then $u\in C^{1,\nu}(\mathbb{D})$, where $\nu=1-2/p$. Finally, for the case $p=\infty$, we showthat $u$ is not necessarily in $C^{1, 1}(\mathbb{D})$, but its gradient, i.e.,$|\nabla u|$ is Lipschitz continuous with respect to the pseudo-hyperbolicmetric. This paper is inspired by Chapter 4 of [Astala, Iwaniec, Martin:Elliptic partial differential equations and quasiconformal mappings in theplane, Princeton Mathematical Series, Vol. 48, Princeton University Press,Princeton, NJ, 2009, p. xviii+677] and [Kalaj, Cauchy transform and Poisson'sequation, Adv. Math. \textbf{231} (2012), 213--242]

$L^p\to L^q$ norm estimates of Cauchy transforms on the Dirichlet problem and their applications