Sharp Constants for Multivariate Hausdorff -Inequalities

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

Estimates for the Commutators of p-Adic Hausdorff Operator on Herz-Morrey Spaces

“…If the kernel function Φ is chosen wisely then the Hausdorff operator reduces to some classical operators like the Hardy operator, the adjoint Hardy operator, the Hardy-Littlewood averaging operator and the Cesàro operator. Here we would like to mention some important publications including [11][12][13][14][15][16][17][18][19][20][21][22][23][24] which discussed the boundedness of Hausdorff operator on function spaces. On the other hand, the p-adic matrix Hausdorff operator was introduced by Volosivets [25], which is given by…”

Weak And Strong Boundedness For P-adic Fractional Hausdorff Operator And Its Commutator

“…|y| n−β f (y)dy, 0 ≤ β < n. Later, the weak and strong estimates of two kinds of multilinear fractional Hausdorff operator on Lebesgue space were studied by Fan and Zhao, see [7]. Gao and Zhao [9] obtained the sharp weak bounds for Hausdorff operators.…”

Sharp bounds for Hardy operators on product spaces

“…Fu et al [16,45] investigated sharp constants for Hardy-type operators on p-adic field. Maligranda et al [34], Guo and Zhao [20], Fan and Zhao [9] studied Hardy q-type integral inequalities. In addition, there are many extensions of the Hardy-type operators, such as weighted Hardy-Littlewood averages [14,15,46] and Hausdorff operators [3,5,27,43].…”