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In this note, we start on the study of the sufficient conditions for the boundedness of Hausdorff operators

$$ \begin{align*}(\mathcal{H}_{K,\mu}f)(z):=\int_{\mathbb{D}}K(w)f(\sigma_w(z))d\mu(w)\end{align*} $$

on three important function spaces (i.e., derivative Hardy spaces, weighted Dirichlet spaces, and Bloch type spaces), which is a continuation of the previous works of Mirotin et al. Here,

$\mu $ is a positive Radon measure, K is a

$\mu $-measurable function on the open unit disk

$\mathbb {D}$, and

$\sigma _w(z)$ is the classical Möbius transform of

$\mathbb {D}$.

The aim of this paper is to characterize the non-negative functions φ defined on (0,∞) for which the Hausdorff operator

$${\rm {\cal H}}_\varphi f(z) = \int_0^\infty f \left( {\displaystyle{z \over t}} \right)\displaystyle{{\varphi (t)} \over t}{\rm d}t$$

is bounded on the Hardy spaces of the upper half-plane

${\rm {\cal H}}_a^p ({\open C}_ + )$,

$p\in [1,\infty ]$. The corresponding operator norms and their applications are also given.

For a wide family of multivariate Hausdorff operators, the boundedness of an operator from this family i s proved on the real Hardy space. By this we extend and strengthen previous results due to Andersen and Móricz.

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Hausdorff operators on some classical spaces of analytic functions

Hausdorff operators on holomorphic Hardy spaces and applications

Multidimensional Hausdorff operators on the real Hardy space

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3 results

Hausdorff operators on some classical spaces of analytic functions

Hausdorff operators on holomorphic Hardy spaces and applications

Multidimensional Hausdorff operators on the real Hardy space