2 results
Hausdorff operators on some classical spaces of analytic functions
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- Journal:
- Canadian Mathematical Bulletin / Volume 67 / Issue 3 / September 2024
- Published online by Cambridge University Press:
- 29 February 2024, pp. 768-780
- Print publication:
- September 2024
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In this note, we start on the study of the sufficient conditions for the boundedness of Hausdorff operators
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,
$$ \begin{align*}(\mathcal{H}_{K,\mu}f)(z):=\int_{\mathbb{D}}K(w)f(\sigma_w(z))d\mu(w)\end{align*} $$
$\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 ESSENTIAL NORMS OF COMPOSITION OPERATORS ON WEIGHTED DIRICHLET SPACES
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 97 / Issue 2 / April 2018
- Published online by Cambridge University Press:
- 31 January 2018, pp. 297-307
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- April 2018
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Let
$\unicode[STIX]{x1D711}$ be an analytic self-map of the unit disc. If 
$\unicode[STIX]{x1D711}$ is analytic in a neighbourhood of the closed unit disc, we give a precise formula for the essential norm of the composition operator 
$C_{\unicode[STIX]{x1D711}}$ on the weighted Dirichlet spaces 
${\mathcal{D}}_{\unicode[STIX]{x1D6FC}}$ for 
$\unicode[STIX]{x1D6FC}>0$. We also show that, for a univalent analytic self-map 
$\unicode[STIX]{x1D711}$ of 
$\mathbb{D}$, if 
$\unicode[STIX]{x1D711}$ has an angular derivative at some point of 
$\unicode[STIX]{x2202}\mathbb{D}$, then the essential norm of 
$C_{\unicode[STIX]{x1D711}}$ on the Dirichlet space is equal to one.
