Daiqing Zhang, Pu Zhang
In the present paper, we study the endpoint Sobolev regularity of the one-sided multilinear maximal operators $\mathfrak{M}_\alpha^{+}$and $\mathfrak{M}_\alpha^{-}$, where $m$ is a positive integer and 0 ≤α＜m. We prove that both the maps $\vec{f} \mapsto\left(\mathfrak{M}_\alpha^{+}(\vec{f})\right)^{\prime}$ and $\vec{f} \mapsto\left(\mathfrak{M}_\alpha^{-}(\vec{f})\right)^{\prime}$ are bounded and continuous from $w^{1,1}(\mathbb{R}) \times \cdots \times w^{1,1}(\mathbb{R})$ to $L^q(\mathbb{R})$ if $q \in\left(\frac{1}{m-\alpha}, \infty\right)$, and bounded and continuous from $W^{1,1}(\mathbb{R}) \times \cdots \times W^{1,1}(\mathbb{R})$ to $L^q(\mathbb{R})$ if $\alpha \in[1, m)$ and $q \in\left(\frac{1}{m-\alpha+1}, \infty\right)$. Here $w^{1,1}(\mathbb{R})$ is the set of all functions $f \in W^{1,1}(\mathbb{R})$ with $\left\|f^{\prime}\right\|_{L^{\infty}(\mathbb{R})}<\infty$. Besides, we show that the boundedness of $\vec{f} \mapsto\left(\mathfrak{M}_\alpha^{+}(\vec{f})\right)^{\prime}$ from $W^{1,1}(\mathbb{R}) \times \cdots \times W^{1,1}(\mathbb{R})$ to $L^q(\mathbb{R})$ with any $q \in\left(\frac{1}{m-\alpha+1}, \infty\right)$ implies its continuity. The above claim also holds for $\mathfrak{M}_\alpha^{-}$. It should be pointed out that all of main results are new even in the linear case $m=1$.
