Suppose $u$ is a strong solution of $$\begin{cases}\Delta u =f &\quad \text{in} \quad B_1(0)\\ u=0 &\quad \text{on}\quad \partial B_1\end{cases}$$ The well known $W^{2,p}$ estimates says if $f\in L^p(B_1)$, $1<p<\infty$, then $$||u||_{W^{2,p}(B_{\frac12})}\leq C||f||_{L^p(B_1)}$$ It is obtained through Calderon-Zygmund theory.

Does anyone know what will happen for $p=1$? What is the best we can say about $u$ and $\nabla u$? I saw a paper that says $\nabla u\in L^{2,\infty}$ (the weak $L^2$ space) through the knowledge of singular integral. I did not know how they get this. Can anyone give me more reference? Thank you.

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