Hardy spaces and the Szegő projection of the non-smooth worm domain

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• 作者：Alessandro Monguzzi ^{alessandro.monguzzi@unimi.it" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Hardy spaces ; Worm domains ; Szegő projection
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2016
• 出版时间：1 April 2016
• 年：2016
• 卷：436
• 期：1
• 页码：439-466
• 全文大小：582 K

文摘

We define Hardy spaces $H^p(D_{\beta }^{\prime })$, p∈(1,∞)$p\in (1,\infty )$, on the non-smooth worm domain and we prove a series of related results such as the existence of boundary values on the distinguished boundary $\partial D_{\beta }^{\prime }$ of the domain and a Fatou-type theorem (i.e., pointwise convergence to the boundary values). Thus, we study the Szegő projection operator $\tilde{S}$ and the associated Szegő kernel $K_{D_{\beta }^{\prime }}$. More precisely, if $H^p(\partial D_{\beta }^{\prime })$ denotes the space of functions which are boundary values for functions in $H^p(D_{\beta }^{\prime })$, we prove that the operator $\tilde{S}$ extends to a bounded linear operator for every p∈(1,+∞)$p\in (1,+\infty )$ and for every k>0$k>0$. Here W^k,p$W^{k,p}$ denotes the Sobolev space of order k   and underlying L^p$L^p$ norm, p∈(1,∞)$p\in (1,\infty )$. As a consequence of the L^p$L^p$ boundedness of $\tilde{S}$, we prove that $H^p(D_{\beta }^{\prime })\cap \mathcal{C}(\overline{D_{\beta }^{\prime }})$ is a dense subspace of $H^p(D_{\beta }^{\prime })$.