In the present paper, we study the endpoint Sobolev regularity of the one-sided multilinear maximal operators \(\frak{M}_{\alpha}^{+}\) and \(\frak{M}_{\alpha}^{-}\) , where m is a positive integer and 0 ≤ a ≤ m. We prove that both the maps \(\overrightarrow{f}\mapsto({\frak{M}}_{\alpha}^{+}(\overrightarrow{f}))^{\prime}\) and \(\overrightarrow{f}\mapsto({\frak{M}}_{\alpha}^{-}(\overrightarrow{f}))^{\prime}\) are bounded and continuous from w1,1(ℝ) × ⋯ × w1,1(ℝ) to Lq(ℝ) if \(q \in ({{1 \over {m - \alpha}},\infty})\) , and bounded and continuous from W1,1 (ℝ) × ⋯ × W1,1 (ℝ) to Lq(ℝ) if α ∈ [1, m) and \(q \in ({{1 \over {m - \alpha +1}},\infty})\) . Here w1,1(ℝ) is the set of all functions f ∈ W1,1(ℝ) with \(\Vert f^{\prime}\Vert_{L^{\infty}(\mathbb{R})}<\infty\) . Besides, we show that the boundedness of \(\overrightarrow{f}\mapsto({\frak{M}}_{\alpha}^{+}(\overrightarrow{f}))^{\prime}\) from W1,1(ℝ) × ⋯ × W1,1(ℝ) to Lq(ℝ) with any \(q \in ({{1 \over {m - \alpha +1}},\infty})\) implies its continuity. The above claim also holds for \(\frak{M}_{\alpha}^{-}\) . It should be pointed out that all of main results are new even in the linear case m = 1.
