Global $W^{2,p}$ estimate for the Laplacian on Euclidean space

Fourier identification. Let $u\in C_c^{\mathrm{\infty }}({\mathbb{R}}^n)$ and put $f:=-\mathrm{\Delta }u\in C_c^{\mathrm{\infty }}({\mathbb{R}}^n)$. Since $u$ is smooth, the classical identity $\widehat{{\mathrm{\partial }}_i{\mathrm{\partial }}_{j}u}=(2\pi i{\xi }_i)(2\pi i{\xi }_j)\widehat{u}$ and the distributional Fourier calculus of [F2] give, as tempered distributions, $\mathcal{F}({\mathrm{\partial }}_i{\mathrm{\partial }}_{j}u)=(2\pi i{\xi }_i)(2\pi i{\xi }_j)\mathcal{F}u=-4{\pi }^2{\xi }_i{\xi }_j\mathcal{F}u$ and $\mathcal{F}(-\mathrm{\Delta }u)=4{\pi }^2\mathrm{\mid }\xi {\mathrm{\mid }}^2\mathcal{F}u$; on $L^2$ classes these equal ${\mathcal{F}}_2({\mathrm{\partial }}_i{\mathrm{\partial }}_{j}u)$ and ${\mathcal{F}}_{2}f$ respectively by the agreement statement of [F2]. Since $f\in L^2$ and $R_j$ is the $L^2$ multiplier by $m_j$, the composition satisfies $R_i{R}_{j}f={\mathcal{F}}_2^{-1}\left(m_i{m}_j{\mathcal{F}}_{2}f\right)$ and, on $\{\xi \mathrm{\ne }0\}$, $m_i{m}_j\cdot 4{\pi }^2\mathrm{\mid }\xi {\mathrm{\mid }}^2=(-i{\xi }_i\mathrm{/}\mathrm{\mid }\xi \mathrm{\mid })(-i{\xi }_j\mathrm{/}\mathrm{\mid }\xi \mathrm{\mid })4{\pi }^2\mathrm{\mid }\xi {\mathrm{\mid }}^2=-4{\pi }^2{\xi }_i{\xi }_j$. Plancherel injectivity [F2] therefore gives the $L^2$ identity ${\mathrm{\partial }}_i{\mathrm{\partial }}_{j}u=R_i{R}_{j}f=R_i{R}_j(-\mathrm{\Delta }u)$.

$L^p$ bound for smooth compactly supported data. For $u\in C_c^{\mathrm{\infty }}$ the function $f=-\mathrm{\Delta }u$ lies in $C_c^{\mathrm{\infty }}\subset L^p\cap L^2$, so both operators in step 1.1 are defined on $L^p$ and the identity holds a.e.; using twice the $L^p$ bound of [F1], $\mathrm{\parallel }{\mathrm{\partial }}_i{\mathrm{\partial }}_{j}u{\mathrm{\parallel }}_{L^p}=\mathrm{\parallel }{R}_i{R}_{j}f{\mathrm{\parallel }}_{L^p}\le C_{n,p}^2\mathrm{\parallel }f{\mathrm{\parallel }}_{L^p}=C_{n,p}^2\mathrm{\parallel }\mathrm{\Delta }u{\mathrm{\parallel }}_{L^p}$. Taking the maximum over $i,j$ gives $\mathrm{\parallel }{D}^{2}u{\mathrm{\parallel }}_{L^p}\le C_{n,p}^2\mathrm{\parallel }\mathrm{\Delta }u{\mathrm{\parallel }}_{L^p}$ for every $u\in C_c^{\mathrm{\infty }}({\mathbb{R}}^n)$.

Density. Let $u\in W^{2,p}({\mathbb{R}}^n)$ and let $u_k\in C_c^{\mathrm{\infty }}({\mathbb{R}}^n)$ satisfy $u_k\to u$ in $W^{2,p}({\mathbb{R}}^n)$, which exists by [F3]. Then $\mathrm{\Delta }{u}_k\to \mathrm{\Delta }u$ and ${\mathrm{\partial }}_i{\mathrm{\partial }}_j{u}_k\to {\mathrm{\partial }}_i{\mathrm{\partial }}_{j}u$ in $L^p$ by the definition of the Sobolev norm [F3], and applying step 2.1 to $u_k$ and passing to the limit gives $\mathrm{\parallel }{D}^{2}u{\mathrm{\parallel }}_{L^p}\le C_{n,p}^2\mathrm{\parallel }\mathrm{\Delta }u{\mathrm{\parallel }}_{L^p({\mathbb{R}}^n)}$ and the same bound for each ${\mathrm{\partial }}_i{\mathrm{\partial }}_{j}u$. Since $\mathrm{\Delta }u\in L^p$ holds automatically for $W^{2,p}$ classes by [F3], the inequality applies to every such class.

Conclusion and constants. The two displayed inequalities hold with $C=C_{n,p}^2$, which is finite for every $1<p<\mathrm{\infty }$ by [F1]. Squaring an upper bound supplies an upper bound only; no sharp growth rate or endpoint estimate is inferred. The proof uses the Riesz-transform $L^p$ theory, whose choice assumption is the Countable Choice of [A1] together with those of the Fourier interfaces; no compactness, no extension operator and no maximal-function argument is used.