# Boardline case of $W^{2,p}$ estimates on elliptic equations

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 $$f\in L^p(B_1)$$, $$1, then $$||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.

• I have no clue about these $L^{2,\infty}$ spaces or best results. But I think a duality argument probably gives optimal $L^p$ estimates for $u$ and $\nabla u$ (and is also a great example to apply this duality stuff) Aug 23, 2020 at 16:57
• also you can apply a duality argument to get results even if you weaken $L^1$ condition on $f$ to $\int_{B_1} |f(x)| \delta(x) dx \le C$ where $\delta(x)$ is the distance to the boundary function. Aug 23, 2020 at 17:02
• You get $\nabla u \in L^{N/(N-1), \infty}$. Aug 23, 2020 at 18:16
• Giorgio, where can I find such results? It would be great if you can give me more directions to search. Aug 24, 2020 at 11:44
• I know how to show $\nabla u\in L^{2,\infty}$. It just follows from Riesz potential argument. when $p=1$. Sep 4, 2020 at 13:48