2018
Том 70
№ 4

All Issues

Wu Jianglong

Articles: 3
Article (English)

Boundedness of Riesz-type potential operators on variable exponent Herz – Morrey spaces

Wu Jianglong

↓ Abstract

Ukr. Mat. Zh. - 2017. - 69, № 9. - pp. 1187-1197

We show the boundedness of the Riesz-type potential operator of variable order $\beta (x)$ from the variable exponent Herz – Morrey spaces $M \dot{K}^{\alpha (\cdot ),\lambda}_{p_1 ,q_1 (\cdot )}(\mathbb{R}^n)$ into the weighted space $M \dot{K}^{\alpha (\cdot ),\lambda}_{p_2 ,q_2 (\cdot )}(\mathbb{R}^n, \omega )$, where $\alpha (x) \in L^{\infty} (\mathbb{R}^n) is log-Holder continuous both at the origin and at infinity, $\omega = (1+| x| ) \gamma (x)$ with some $\gamma (x) > 0$, and $1/q_1 (x) 1/q_2 (x) = \beta (x)/n$ when $q_1 (x)$ is not necessarily constant at infinity. It is assumed that the exponent $q_1 (x)$ satisfies the logarithmic continuity condition both locally and at infinity and $1 < (q_1)_{\infty} \leq q_1(x) \leq (q_1)_+ < \infty, \;x \in \mathbb{R}$.

Article (English)

Weighted Estimates for Multilinear Commutators of Marcinkiewicz Integrals with Bounded Kernel

Liu Qingguo, Wu Jianglong

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2014. - 66, № 4. - pp. 538–550

Let \( {\mu}_{\varOmega, \overrightarrow{b}} \) be a multilinear commutator generalized by the n-dimensional Marcinkiewicz integral with bounded kernel μ Ώ and let \( {b}_j\ \in Os{c_{\exp}}_{L^{r_j}} \) , 1 ≤ jm. We prove the following weighted inequalities for ωA and 0 < p < ∞: $$ {\begin{array}{cc}\hfill {\left\Vert {\mu}_{\varOmega }(f)\right\Vert}_{L^p\left(\omega \right)}\le C{\left\Vert M(f)\right\Vert}_{L^p\left(\omega \right)},\hfill & \hfill \left\Vert {\mu}_{\varOmega, \overrightarrow{b}}(f)\right\Vert \hfill \end{array}}_{L^p\left(\omega \right)}\le C{\left\Vert {M}_{L{\left( \log L\right)}^{1/r}}(f)\right\Vert}_{L^p\left(\omega \right)}. $$

The weighted weak L(log L)1/r -type estimate is also established for p =1 and ωA 1.

Brief Communications (English)

Boundedness of multilinear singular integral operators on the homogeneous Morrey–Herz spaces

Wu Jianglong

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2009. - 61, № 10. - pp. 1434-1440

A boundedness result is established for multilinear singular integral operators on the homogeneous Morrey–Herz spaces. As applications, two corollaries about interesting cases of the boundedness of the considered operators on the homogeneous Morrey–Herz spaces are obtained.