The classical Schauder estimate says that if $u$ is a solution of \begin{equation} \Delta u = f \end{equation} where $f \in C^{\alpha}(B_1)$, then $u \in C^{2, \alpha}(B_{1/2})$. Moreover, we have \begin{equation} \|u\|_{C^{2, \alpha}(B_{1/2)}} \le C \{ \|u\|_{L^\infty(B_1)} + \|f\|_{C^\alpha(B_1)} \}. \end{equation} There are other "Schauder estimates" and I'm looking for one where $f \in L^n$ where $n$ is the dimension of the space, in order to obtain the Hölder regularity of the solution. I will be very grateful.
2026-03-28 11:53:34.1774698814
Schauder estimate with right hand side in $L^n$.
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