Journal of the
Korean Mathematical Society JKMS

In this paper, two weight conditions are introduced and the multiple weighted strong and weak characterizations of the multilinear fractional new maximal operator $\mathcal{M}_{\varphi,\beta}$ are established. Meanwhile, we introduce the $S_{(\vec{p},q),\beta}(\varphi)$ and $B_{(\vec{p},q),\beta}(\varphi)$ conditions and obtain the characterization of two-weighted inequalities for $\mathcal{M}_{\varphi,\beta}$. Finally, the relationships of the conditions $S_{(\vec{p},q),\beta}(\varphi)$, $\mathcal{A}_{(\vec{p},q),\beta}(\varphi)$ and $B_{(\vec{p},q),\beta}(\varphi)$ and the characterization of the one-weight $A_{(\vec{p},q),\beta}(\varphi)$ are given.

Keywords: Multilinear fractional new maximal operator, $\mathcal{A}_{(\vec{p},q),\beta}(\varphi)$ condition, $S_{(\vec{p},q),\beta}(\varphi)$ condition, $B_{(\vec{p},q),\beta}(\varphi)$ condition, new multiple weights

MSC numbers: Primary 42B25, 47G10

Supported by: This work was financially supported by the National Natural Science Foundation of China(Grant No. 11561062).