arXiv subjects
Publications and source records attributed to Guantie Deng.
In this paper, we investigate a class of domains $Ω^{n+1}_γ=\{(z,w)\in \mathbb{C}^n\times \mathbb{C}: |z|^γ< |w| < 1\}$ for $γ>0$ that generalizes the Hartogs triangle. We obtain a sharp range of $p$ for the boundedness of the Bergman projection on the domain considered here. It generalizes the results by Edholm and McNeal \cite{LD1} for n = 1 to any dimension n.
In this paper, we develop the theory of weighted Bergman space and obtain a general representation formula of the Bergman kernel function for the spaces on the Reinhardt domain containing the origin. As applications, we calculate the concrete forms of the Bergman kernels for some special weights on the Reinhardt domains $\mathbb{C}^n,$ $D_{n,m}:= \{(z, w)\in \mathbb{C}^n \times \mathbb{C}^m : \|w\|^2 <{e}^{-μ_1\|z\|^{μ_2}}\}$ and $V_η:=\{(z, z', w) \in \mathbb{C}^{n} \times \mathbb{C}^{m} \times \mathbb{C} : \sum_{j=1}^{n} e^{η_{j}|w|^{2}}|z_{j}|^{2}+\|z'\|^{2}<1\}$.
This paper aims to obtain decompositions of higher dimensional $L^p(\mathbb{R}^n)$ functions into sums of non-tangential boundary limits of the corresponding Hardy space functions on tubes for the index range $0<p<1$. In the one-dimensional case, Deng and Qian \cite{DQ} recently obtained such Hardy space decomposition result: for any function $f\in L^p(\mathbb{R}),\ 0<p<1$, there exist functions $f_1$ and $f_2$ such that $f=f_1+f_2$, where $f_1$ and $f_2$ are, respectively, the non-tangential boundary limits of some Hardy space functions in the upper-half and lower-half planes. In the present paper, we generalize the one-dimensional Hardy space decomposition result to the higher dimensions, and discuss the uniqueness issue of such decomposition.
We define Hardy spaces $H^p(Ω_\pm)$ on half-strip domain~$Ω_+$ and $Ω_-= \mathbb{C}\setminus\overline{Ω_+}$, where $0<p<\infty$, and prove that functions in $H^p(Ω_\pm)$ has non-tangential boundary limit a.e. on $Γ$, the common boundary of $Ω_\pm$. We then prove that Cauchy integral of functions in $L^p(Γ)$ are in $H^p(Ω_\pm)$, where $1<p<\infty$, that is, Cauchy transform is bounded. Besides, if $1\leqslant p<\infty$, then $H^p(Ω_\pm)$ functions are the Cauchy integral of their non-tangential boundary limits. We also establish an isomorphism between $H^p(Ω_\pm)$ and $H^p(\mathbb{C}_\pm)$, the classical Hardy spaces over upper and lower half complex planes.
We provide the details of the first proof in~\cite{CJS89}, which proved that Cauchy transform of $L^2$~functions on Lipschitz curves is bounded. We then prove that every $L^2$~function on Lipschitz curves is the sum of non-tangential boundary limit of functions in $H^2(Ω_\pm)$, the Hardy spaces on domains over and under the Lipschitz curve. We also obtain a more accurate boundary of Cauchy transform under the condition that the Lipschitz curve is the real axis.
Let $Γ$ be a Lipschitz curve on the complex plane $\mathbb{C}$ and $Ω_+$ is the domain above $Γ$, we define Hardy space $H^p(Ω_+)$ as the set of holomorphic functions $F$ satisfying $\sup_{τ>0}(\int_Γ |F(ζ+\mathrm{i}τ)|^p |\,\mathrm{d}ζ|)^{\frac1p}< \infty$. We mainly focus on the case of $1<p<\infty$ in this paper, and prove that if $F(w)\in H^p(Ω_+)$, then $F(w)$ has non-tangential boundary limit $F(ζ)$ a.e. on $Γ$, and $F(w)$ is the Cauchy integral of $F(ζ)$. We denote the conformal mapping from $\mathbb{C}_+$ onto $Ω_+$ as $Φ$, and then prove that, $ H^p(Ω_+)$ is isomorphic to $H^p(\mathbb{C}_+)$, the classical Hardy space on upper half plane, under the mapping $T\colon F\to F(Φ(z))\cdot (Φ'(z))^\frac{1}{p}$, where $F\in H^p(Ω_+)$.
Let $0 0}(\int_Γ |F(ζ+\mathrm{i}τ)|^p |\,\mathrm{d}ζ|)^{\frac1p}< \infty$. We denote the conformal mapping from $\mathbb{C}_+$ onto $Ω_+$ as $Φ$, and prove that, $H^p(Ω_+)$ is isomorphic to $H^p(\mathbb{C}_+)$, the classical Hardy space on the upper half plane~$\mathbb{C}_+$, under the mapping $T\colon F\to F(Φ)\cdot (Φ')^{\frac1p}$. Besides, $T$ and $T^{-1}$ are both bounded. We also prove that if $F(w)\in H^p(Ω_+)$, then $F(w)$ has non-tangential boundary limit $F(ζ)$ a.e. on $Γ$, and, if $1\leqslant p< \infty$, $F(w)$ is the Cauchy integral on $Γ$ of $F(ζ)$.
Subsequent to our recent work on Fourier spectrum characterization of Hardy spaces $H^p(\mathbb{R})$ for the index range $1\leq p\leq \infty,$ in this paper we prove further results on rational Approximation, integral representation and Fourier spectrum characterization of functions in the Hardy spaces $H^p(\mathbb{R}), 0 < p\leq \infty,$ with particular interest in the index range $ 0< p \leq 1.$ We show that the set of rational functions in $ H^p(\mathbb{C}_{+1}) $ with the single pole $-i$ is dense in $ H^p(\mathbb{C}_{+1}) $ for $0 0\}$. We give Laplace integral representation formulas for functions in the Hardy spaces $H^p,$ $0<p\leq2.$ Besides one in the integral representation formula we give an alternative version of Fourier spectrum characterization for functions in the boundary Hardy spaces $H^p$ for $0<p\leq 1.$
In the paper we give some criterions for a-minimally thin sets and a-rarefied sets associated with the stationary Schrödinger operator at a fixed Martin boundary point or {\infty} with respect to a cone. Moreover, we show that a positive superfunction on a cone behaves regularly outside a-rarefied set. Finally we illustrate the relation between a-minimally thin set and a-rarefied set in a cone.