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SHARP CONSTANTS FOR MULTIVARIATE HAUSDORFF $q$ -INEQUALITIES

  • DASHAN FAN (a1) (a2) and FAYOU ZHAO (a3)

Abstract

In this paper, we focus on the multivariate Hausdorff operator of the form

$$\begin{eqnarray}\mathbf{H}_{\unicode[STIX]{x1D6F7}}(f)(x)=\int _{(0,+\infty )^{n}}{\displaystyle \frac{\unicode[STIX]{x1D6F7}\big(\frac{x_{1}}{t_{1}},\frac{x_{2}}{t_{2}},\ldots ,\frac{x_{n}}{t_{n}}\big)}{t_{1}t_{2}\cdots t_{n}}}f(t_{1},t_{2},\ldots ,t_{n})\,\mathbf{dt},\end{eqnarray}$$
where $\mathbf{dt}=dt_{1}\,dt_{2}\cdots \,dt_{n}$ 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 $p\in \mathbb{R}\backslash \{0\}$ .

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The research was supported by National Natural Science Foundation of China (Grant Nos. 11471288, 11601456).

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SHARP CONSTANTS FOR MULTIVARIATE HAUSDORFF $q$ -INEQUALITIES

  • DASHAN FAN (a1) (a2) and FAYOU ZHAO (a3)

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