arXiv subjects
Publications and source records attributed to ZhuoRan Hu.
This paper provides a study of problems related to Hardy spaces left by G.\,Weiss in \cite{We}. First, We will prove that the Hardy spaces $H^p(\mathbb{R}^n)$ can be characterized by a fixed Lipschitz function.
For $p>p_0=\frac{2λ}{2λ+1}$ with $λ>0$, the Hardy space $H_λ^p(\mathbb R_+^2)$ associated with the Dunkl transform $\mathcal{F}_λ$ and the Dunkl operator $D$ on the real line $\mathbb R$, where $D_xf(x)=f'(x)+\fracλ{x}[f(x)-f(-x)]$, is the set of functions $F=u+iv$ on the upper half plane $\mathbb R^2_+=\left\{(x, y): x\in\mathbb R, y>0\right\}$, satisfying $λ$-Cauchy-Riemann equations: $ D_xu-\partial_y v=0$, $\partial_y u +D_xv=0$, and $\sup\limits_{y>0}\int_{\mathbb R}|F(x+iy)|^p|x|^{2λ}dx<\infty$. In this paper, we will give a further characterization of $H_λ^p(\mathbb R_+^2)$ in \cite{ZhongKai Li 3}. We prove the inequality $\|F\|_{H_λ^p(\mathbb R_+^2)}\leq c\|u_{\nabla}^*\|_{L^p_λ}$, which gives a Real Characterization of the class $H_λ^p(\mathbb R_+^2)$ for $\frac{2λ}{2λ+1}<p\leq1$ as a main result.
For $p>\frac{2λ}{2λ+1}$ with $λ>0$, the Hardy spaces $H_λ^{p}(\mathbb{R}^{2}_+)$ associated with the Dunkl transform $\mathscr{F}_λ$ and the Dunkl operator $D_x$ on the line, where $D_xf(x)=f'(x)+\fracλ{x}[f(x)-f(-x)]$, is the set of function $F=u+iv$ on the upper half plane $\mathbb{R}_+^2=\big\{(x, y): y>0\big\}$, satisfying the $λ$-Cauchy-Riemann equations: $D_xu-\partial_y v=0, \partial_y u +D_xv=0$, and $\sup_{y>0}\int_{\mathbb{R}}|F(x, y)||x|^{2λ}dx<0$. In this paper, we will study the boundedness of Cesàro operator on $H_λ^{p}(\mathbb{R}^{2}_+)$. We will prove the following inequality $$ \|C_αf\|_{H_λ^p(\mathbb{R}_+^2)}\leq C\|f\|_{H_λ^p(v_+^2)},$$ for $\frac{2λ}{2λ+1}< p<\infty$, where C is dependent on $α$, $p$, $λ$, and the average function for the Cesàro operator $C_α$ is $ϕ_α(t)=α(1-t)^{α-1}$ with $α>0$.
For $p>p_0=\frac{2λ}{2λ+1}$ with $λ>0$, the Hardy space $H_λ^p(\mathbb{R}_+^2)$ associated with the Dunkl transform $\mathcal{F}_λ$ and the Dunkl operator $D$ on the real line $\mathbb{R}$, where $D_xf(x)=f'(x)+\fracλ{x}[f(x)-f(-x)]$, is the set of functions $F=u+iv$ on the upper half plane $\mathbb{R}^2_+=\left\{(x, y): x\in\mathbb{R}, y>0\right\}$, satisfying $λ$-Cauchy-Riemann equations: $ D_xu-\partial_y v=0$, $\partial_y u +D_xv=0$, and $\sup\limits_{y>0}\int_{\mathbb{R}}|F(x+iy)|^p|x|^{2λ}dx<\infty$ in [7]. Then it is proved in [11] that the real Dunkl-Hardy Spaces $H_λ^p(\mathbb{R})$ for $\frac{1}{1+γ_λ}<p\leq1$ are Homogeneous Hardy Spaces. In this paper, we will continue to investigate $λ$-Hilbert transform on the real Dunkl-Hardy Spaces $H_λ^p(\mathbb{R})$ for $\frac{1}{1+γ_λ}<p\leq1$ with $γ_λ=1/(4λ+2)$ and extend the results of $λ$-Hilbert transform in [7].
We investigate $λ$-Hilbert transform, $λ$-Possion integral and conjugate $λ$-Poisson integral on the atomic Hardy space in the Dunkl setting and establish a new version of Paley type inequality which extends the results in \cite{F} and \cite{ZhongKai Li 3}.