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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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  • $\begingroup$ 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) $\endgroup$
    – Math604
    Commented Aug 23, 2020 at 16:57
  • $\begingroup$ 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. $\endgroup$
    – Math604
    Commented Aug 23, 2020 at 17:02
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    $\begingroup$ You get $\nabla u \in L^{N/(N-1), \infty}$. $\endgroup$
    – Giorgio Metafune
    Commented Aug 23, 2020 at 18:16
  • $\begingroup$ Giorgio, where can I find such results? It would be great if you can give me more directions to search. $\endgroup$
    – Slm2004
    Commented Aug 24, 2020 at 11:44
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    $\begingroup$ @Slm2004 Sorry for the late answer but I did not get any notification of your comment (you forgot @ before my name). The proof I know uses the heat semigroup and it is written in a paper by A. Lunardi and myself in some generality, but the result for the Laplacian is surely older. The title is "On domains of elliptic operators in $L^1$". If you don't find it, write an email to [email protected] $\endgroup$
    – Giorgio Metafune
    Commented Sep 6, 2020 at 9:53

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You may find useful a paper by Perez in Journal of Functional Analysis in 1995

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  • $\begingroup$ Thank you for your answer $\endgroup$
    – Slm2004
    Commented Aug 24, 2020 at 11:43

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