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Let $\Omega$ be a bounded smooth domain and $u\in W^{2,p}(\Omega)\cap H^1_0(\Omega)$. By the classical $L^p$ theory of second order elliptic equation, we have $$ \|\nabla^2u\|_{L^p(\Omega)}\leq C\|\Delta u\|_{L^p(\Omega)}, $$ where $p\geq 1$, $C$ is a constant dependent of $\Omega$ and independent of $u$.

Now let $g$ be a smooth function. Define a weighted norm $$ \|u\|_{L^p(\Omega, |g|)}=\left(\int_\Omega|u|^p|g|dx\right)^{\frac{1}{p}}. $$ What I want to ask is, whether we have $$ \|\nabla^2u\|_{L^p(\Omega, |g|)}\leq C\|\Delta u\|_{L^p(\Omega, |g|)}. $$ If it does not hold, under what conditions on $g$ we can make this become true?

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    $\begingroup$ If $\Omega=\mathbb R^n$ this is true when the weight $|g|$ is in the Muckhenaupt class $A_p$. It should be the same in $\Omega$. $\endgroup$ Mar 17, 2022 at 23:02

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