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SHARP CONSTANTS FOR MULTIVARIATE HAUSDORFF
$q$-INEQUALITIES
- Part of
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- Journal:
- Journal of the Australian Mathematical Society / Volume 106 / Issue 2 / April 2019
- Published online by Cambridge University Press:
- 07 June 2018, pp. 274-286
- Print publication:
- April 2019
-
- Article
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In this paper, we focus on the multivariate Hausdorff operator of the form
where
$$\begin{eqnarray}\mathbf{H}_{\unicode[STIX]{x1D6F7}}(f)(x)=\int _{(0,+\infty )^{n}}{\displaystyle \frac{\unicode[STIX]{x1D6F7}\big(\frac{x_{1}}{t_{1}},\frac{x_{2}}{t_{2}},\ldots ,\frac{x_{n}}{t_{n}}\big)}{t_{1}t_{2}\cdots t_{n}}}f(t_{1},t_{2},\ldots ,t_{n})\,\mathbf{dt},\end{eqnarray}$$
$\mathbf{dt}=dt_{1}\,dt_{2}\cdots \,dt_{n}$ or 
$\mathbf{dt}=d_{q}t_{1}\,d_{q}t_{2}\cdots d_{q}t_{n}$ is the discrete measure in 
$q$-analysis. The sharp bounds for the multivariate Hausdorff operator on spaces 
$L^{p}$ with power weights are calculated, where 
$p\in \mathbb{R}\backslash \{0\}$.
