Let $u\in L^2(\mathbb R)$ and $w \in W^{1,1}(\mathbb R)$, we consider the convolution
$$u*v$$
Is it true that $w*u \in W^{1,2}(\mathbb R)$?
What regularity can we put on $w$ for this to be true?
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4$\begingroup$ Sure, $(u*w)' = u*w'$, and use Young's convolution inequality, but I don't think this is a good question for MathOverflow. $\endgroup$– Aleksei KulikovCommented Aug 22 at 18:57
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