Regularity of local multilinear fractional maximal and strong maximal operators

Abstract

We prove that the local multilinear fractional maximal operator is bounded from $L^{p_1}(\mathrm{\Omega })\times \cdots \times L^{p_m}(\mathrm{\Omega })$ to $W^{1,q}(\mathrm{\Omega })$ when and $1∕q=1∕p_1+\cdots +1∕p_m$ with , which was previously unknown for the case . In addition, we introduce and investigate the Sobolev regularity properties of the local multilinear strong maximal operator, as well as its fractional variant. Several new pointwise estimates for the weak gradients of the above maximal functions will be established when $\overrightarrow{f}=(f_1,\dots ,f_m)$, with each $f_j\in W^{1,p_j}(\mathrm{\Omega })$ for some $p_j\in (1,\infty )$. As applications, we obtain certain boundedness for the above operators in the first-order Sobolev spaces and the Sobolev spaces with zero boundary values.