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The aim of this paper is to characterize the nonnegative functions $\varphi$ defined on $(0,\infty)$ for which the Hausdorff operator $$\mathscr H_\varphi f(z)= \int_0^\infty f\left(\frac{z}{t}\right)\frac{\varphi(t)}{t}dt$$ is bounded on the Hardy spaces of the upper half-plane $\mathcal H_a^p(\mathbb C_+)$, $p\in[1,\infty]$. The corresponding operator norms and their applications are also given., Comment: Proc. Roy. Soc. Edinburgh Sect. A (to appear)

Let \(\varphi \) be a nonnegative integrable function on \((0,\infty )\). It is well-known that the Hausdorff operator \({{\mathcal {H}}}_\varphi \) generated by \(\varphi \) is bounded on the real Hardy space \(H^1({{\mathbb {R}}})\). The aim of this paper is to give the exact norm of \({{\mathcal {H}}}_\varphi \). More precisely, we prove that $$\begin{aligned} \Vert {{\mathcal {H}}}_\varphi \Vert _{H^1({{\mathbb {R}}})\rightarrow H^1({{\mathbb {R}}})}= {\mathop {\int }\limits _{0}^{\infty }} \varphi (t)dt. \end{aligned}$$

This paper deals with a general class of weighted multilinear Hardy-Cesaro operators that acts on the product of Lebesgue spaces and central Morrey spaces. Their sharp bounds are also obtained. In addition, we obtain sufficient and necessary conditions on weight functions so that the commutators of these weighted multilinear Hardy-Cesaro operators (with symbols in central BMO spaces) are bounded on the product of central Morrey spaces. These results extends known results on multilinear Hardy operators.

This paper aims to investigate the norms of the weighted Hardy–Cesaro operator on weighted Lebesgue and BMO spaces. Under certain conditions on s(t) and on ω, we characterize the weight function ψ so that Uψ, s is bounded on Lp(ω), BMO(ω). The corresponding operator norms are worked out too. We also give a necessary condition on the weight function ψ, for the boundedness of the commutators of operator Uψ, s on Lp(ω) with symbols in BMO(ω).

We give a very simple proof of a result by Dafni that states that the weak $^{*}$ -convergence is true in the local Hardy space $h^{1}(\mathbb{R}^{d})$ .

Let $(\mathcal{X}, d, \mu)$ be a complete RD-space. Let $\rho$ be an admissible function on $\mathcal{X}$, which means that $\rho$ is a positive function on $\mathcal{X}$ and there exist positive constants $C_0$ and $k_0$ such that, for any $x, y \in \mathcal{X}$,\[ \rho(y) \leq C_0 [\rho(x)]^{1/(1+k_0)} [\rho(x) + d(x, y)]^{k_0/(1+k_0)}.\]In this paper, we define a space $\operatorname{VMO}_\rho(\mathcal{X})$ and show that it is the predual of the localized Hardy space $H^1_\rho(\mathcal{X})$ introduced by Yang and Zhou [14]. Then we prove a version of the classical theorem of Jones and Journé [7] on weak$^*$-convergence in $H^1_\rho(\mathcal{X})$. As an application, we give an atomic characterization of $H^1_\rho(\mathcal{X})$.

Let $T$ be a pseudo-differential operator whose symbol belongs to the H\"ormander class $S^m_{\rho,\delta}$ with $0\leq \delta, Comment: Taiwanese J. Math. (to appear)