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SHARP CONSTANTS FOR MULTIVARIATE HAUSDORFF $q$-INEQUALITIES

Part of: Difference and functional equations Difference equations Inequalities

Published online by Cambridge University Press:  07 June 2018

DASHAN FAN  and
FAYOU ZHAO
DASHAN FAN
Affiliation:
Department of Mathematical Sciences, University of Wisconsin-Milwaukee, Milwaukee, WI 53201, USA Department of Mathematics, Zhejiang Normal University, Jinhua 321000, PR China email fan@uwm.edu
FAYOU ZHAO*
Affiliation:
Department of Mathematics, Shanghai University, Shanghai 200444, PR China email fyzhao@shu.edu.cn
*
✉fyzhao@shu.edu.cn
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Abstract

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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 $p\in \mathbb{R}\backslash \{0\}$.

Keywords

sharp constantmultivariate Hausdorff operatorq-inequalities

MSC classification

Primary: 26D10: Inequalities involving derivatives and differential and integral operators
Secondary: 26D15: Inequalities for sums, series and integrals 39A13: Difference equations, scaling ($q$-differences)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 106 , Issue 2 , April 2019 , pp. 274 - 286
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

Footnotes

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

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