关键词: Multilinear local maximal operators, $A_{(\vec{p},q)}^{\beta}$ weights, two-weight inequalities

Abstract: Let $ 0<\beta <1$ and $\Omega$ be a proper open and non-empty subset of $\mathbf{R}^n$. In this paper, the object of our investigation is the multilinear local maximal operator $\mathcal{M}_{\beta}$, defined by $$\mathcal{M}_{\beta}(\vec{f})(x)= \sup_{\substack{Q \ni x \\ Q\in{\mathcal{F}_{\beta}}}} \prod_{i=1}^m \frac{1}{|Q|} \int_Q |f_i(y_i)|{\rm d}y_i,$$ where $\mathcal{F}_{\beta}=\{Q(x,l):x \in \Omega, l< \beta {\rm d}(x, \Omega^c)\}$, $Q=Q(x,l)$ is denoted as a cube with sides parallel to the axes, and $x$ and $l$ denote its center and half its side length. Two-weight characterizations for the multilinear local maximal operator $\mathcal{M}_{\beta}$ are obtained. A formulation of the Carleson embedding theorem in the multilinear setting is proved.

Key words: Multilinear local maximal operators, $A_{(\vec{p},q)}^{\beta}$ weights, two-weight inequalities