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In $\mathbb{R}^d$ there is estimate (see 1.3, Chapter III of E.M.Stein' book Singular Integrals and Differentiability Properties of Functions) $$\left\|\frac{\partial^2 f}{\partial x_i \partial x_j} \right\|_{L^p(\mathbb{R}^d)} \leq A_p \|\Delta f\|_{L^p(\mathbb{R}^d)}. $$ Similar estimate for bounded domain $\Omega \subset \mathbb{R}^d$ and $p=2$ may be found in the P.Grisvard' book Elliptic problems in nonsmooth domains, Theorem 3.1.2.1.

I'm looking for a reference for similar estimate in bounded domain with $p\ne 2$.

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    $\begingroup$ Without some assumption on $f$, this is clearly false, just take some $f$ with $\Delta f=0$. $\endgroup$
    – Michael Renardy
    Commented Nov 3 at 2:38

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For your estimate to hold on a bounded domain we clearly need $u$ to have zero boundary value, otherwise we can just take a (nonlinear) harmonic function to see that it cannot be true. So assume $u \in W_{0}^{2,p}(\Omega)$, where $1<p<\infty $ . Approximate $u$ in $ W^{2,p}$ norm by $ C^{\infty}_{c}(\Omega)$ functions $u_m.$ Apply the inequality you found in Stein's book and take limits.

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  • $\begingroup$ The desired estimate cannot hold on arbitrary domains, you need some boundary regularity. However, your answer does not need boundary regularity and, consequently, it should be invalid. $\endgroup$
    – gerw
    Commented Nov 5 at 13:50

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