A question about convolution.

Question

Let $U\subset\mathbb{R}^n$ be open and bounded (with lipschitz boundary if necessary). Let $u\in W^{1,1}(U)$. Can I have the following convolution equality: $\frac{\partial ^2}{\partial x_i^2}u*v=\frac{\partial }{\partial x_i}u*\frac{\partial}{\partial x_i}v.$ For $v\in C_c^\infty(U)$ and $i=1,...,n$. I think is this is true if $u\in W^{2,1}(U)$ but I want this for $W^{1,1}$. And I think this is true, since $\frac{\partial^2}{\partial x_i^2}(u*v)=\frac{\partial^2}{\partial x_i^2}u*v=\frac{\partial}{\partial x_i}u*\frac{\partial}{\partial x_i}v$.