论文标题
$ l^p \ to l^q $ Quart估算了Dirichlet问题及其申请
$L^p\to L^q$ norm estimates of Cauchy transforms on the Dirichlet problem and their applications
论文作者
论文摘要
用$ c^α（\ mathbb {d}）$表示函数$ f $的空间t} he单位磁盘$ \ mathbb {d} $，它们与指数$α$连续，并用$ c^{1，α}（1，α}（\ mathbb {d d}）$不同的空间，该功能是$ cons $ n. $ c^α（\ mathbb {d}）$。令$ \ Mathcal {C} $为Dirichlet问题的Cauchy变换。在本文中，我们获得了$ \ | \ Mathcal {c} \ | _ {l^p \ to l^q} $的规范估计值，其中$ 3/2 <p <2 $和$ q = p/（p/p/（p-1）$。作为一个应用程序，我们表明，如果$ 3/2 <p <2 $，则$ u \ in c^μ（\ mathbb {d}）$，其中$μ= 2/p-1 $。我们还表明，如果$ 2 <p <\ infty $，则$ u \ in c^{1，ν}（\ Mathbb {d}）$，其中$ν= 1-2/p $。最后，对于$ p = \ infty $的情况，我们表明$ u $不一定在$ c^{1，1}（\ mathbb {d}）$中，而是它的梯度，即$ | \ nabla u | $ | $是Lipschitz，lipslipschitz与Pseudo-Hyperbolic Metric有关。本文的灵感来自[Astala，Iwaniec，Martin的第4章：椭圆形的部分微分方程和飞机上的准形式映射，普林斯顿数学系列，第1卷。 48，普林斯顿大学出版社，新泽西州普林斯顿，2009年，第1页。 XVIII+677]和[Kalaj，Cauchy Transform and Poisson's方程，Adv。数学。 \ textbf {231}（2012），213--242]
Denote by $C^α(\mathbb{D})$ the space of the functions $f$ on t}he unit disk $\mathbb{D}$ which are Hölder continuous with the exponent $α$, and denote by $C^{1, α}(\mathbb{D})$ the space which consists of differentiable functions $f$ such that their derivatives are in the space $C^α(\mathbb{D})$. Let $\mathcal{C}$ be the Cauchy transform of Dirichlet 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 an application, we show that if $3/2<p<2$, then $u\in C^μ(\mathbb{D})$, where $μ=2/p-1$. We also show that if $2<p<\infty$, then $u\in C^{1, ν}(\mathbb{D})$, where $ν=1-2/p$. Finally, for the case $p=\infty$, we show that $u$ is not necessarily in $C^{1, 1}(\mathbb{D})$, but its gradient, i.e., $|\nabla u|$ is Lipschitz continuous with respect to the pseudo-hyperbolic metric. This paper is inspired by Chapter 4 of [Astala, Iwaniec, Martin: Elliptic partial differential equations and quasiconformal mappings in the plane, Princeton Mathematical Series, Vol. 48, Princeton University Press, Princeton, NJ, 2009, p. xviii+677] and [Kalaj, Cauchy transform and Poisson's equation, Adv. Math. \textbf{231} (2012), 213--242]
