Local $W^{2,p}$ regularity of weak solutions of the Poisson equation

Localization. Choose a ball $B^{\mathrm{\prime }\mathrm{\prime }}$ with $B^{\mathrm{\prime }}⋐B^{\mathrm{\prime }\mathrm{\prime }}⋐B$ and, by [F5], a cutoff $\eta \in C_c^{\mathrm{\infty }}(B)$ with $\eta =1$ on a neighbourhood of $\overline{B^{\mathrm{\prime }\mathrm{\prime }}}$; put $g:=\eta f$, extended by zero to ${\mathbb{R}}^n$, so that $g\in L_c^p({\mathbb{R}}^n)$ and $g=f$ on $B^{\mathrm{\prime }\mathrm{\prime }}$. Let $w:=N(g)$ be the Newtonian potential of $g$.

Hessian bound for smooth data without dividing by the frequency variable. Let $g\in C_c^{\mathrm{\infty }}$ and $w=Ng$. The classical-potential supplier Hölder data give a classical Newtonian solution gives $w\in C^2$ and $-\mathrm{\Delta }w=g$. Fix $\chi \in C_c^{\mathrm{\infty }}(B_2)$ equal to one on $B_1$, and put ${\chi }_T(x)=\chi (x\mathrm{/}T)$. For large $T$ containing the support of $g$, $\mathrm{\Delta }({\chi }_{T}w)=-g+2\mathrm{\nabla }{\chi }_T\cdot \mathrm{\nabla }w+w\mathrm{\Delta }{\chi }_T$. On $T\le \mathrm{\mid }x\mathrm{\mid }\le 2T$, the kernel formulas and differentiation away from the support give $\mathrm{\mid }w(x)\mathrm{\mid }\le C_g{T}^{2-n}$ for $n\ge 3$, $\mathrm{\mid }w(x)\mathrm{\mid }\le C_g(1+\log T)$ for $n=2$, and $\mathrm{\mid }\mathrm{\nabla }w(x)\mathrm{\mid }\le C_g{T}^{1-n}$ in both cases. Thus the commutator has $L^p$ norm at most $C_g{T}^{-n+n\mathrm{/}p}(1+{\mathbf{1}}_{n=2}\log T)$, which tends to zero for $p>1$. The whole-space estimate Global $W^{2,p}$ estimate for the Laplacian on Euclidean space applies to the compactly supported $C^2$ function ${\chi }_{T}w$ (its classical derivatives are weak derivatives by integration by parts). On any fixed ball $B_M$, ${\chi }_{T}w=w$ for $T>M$, so $\mathrm{\parallel }{D}^{2}w{\mathrm{\parallel }}_{L^p(B_M)}\le C_{n,p}(\mathrm{\parallel }g{\mathrm{\parallel }}_p+o(1))$. First let $T\to \mathrm{\infty }$, then $M\to \mathrm{\infty }$; Monotone convergence for the integral applied to the increasing ball indicators times the nonnegative Hessian integrands gives $\mathrm{\parallel }{D}^{2}w{\mathrm{\parallel }}_{L^p({\mathbb{R}}^n)}\le C_{n,p}\mathrm{\parallel }g{\mathrm{\parallel }}_p$. This includes $p=2$ and avoids any two-dimensional Fourier inversion at zero.

Second derivatives of $w$: the $L^p$ case. For general $g\in L_c^p$, choose $g_k\in C_c^{\mathrm{\infty }}$ with $g_k\to g$ in $L^p$ (possible by [F5] after multiplying by a cutoff). By [F1] the potentials $N{g}_k$ converge to $Ng$ in $L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1$, and by the $L^p$ bound of step 2.1 the fields $D_{ij}N{g}_k$ are Cauchy in $L^p({\mathbb{R}}^n)$ (apply step 2.1 to $g_k-g_{\mathrm{\ell }}$). Completeness of scalar $L^p$ follows from Riesz-Fischer completeness of $L^p$ for $1\le p\le \mathrm{\infty }$ for real components and Complex Lp completeness and almost-everywhere subsequences for complex data under Countable Choice. Since distributional differentiation is continuous [F3], the limit is $D_{ij}Ng\in L^p$, so $w=Ng\in W_{\mathrm{l}\mathrm{o}\mathrm{c}}^{2,p}({\mathbb{R}}^n)$ with, if $\mathrm{supp}g\subseteq B_{\rho }(0)$, for every ball $B_R(0)$, $\mathrm{\parallel }w{\mathrm{\parallel }}_{W^{2,p}(B_R)}\le C(n,p,R,\rho )\mathrm{\parallel }g{\mathrm{\parallel }}_{L^p}$ (the zero- and first-order terms are controlled by the Young bounds of [F1] and the second-order terms by step 2.1).

The remainder is harmonic. Since $\eta =1$ on $B^{\mathrm{\prime }\mathrm{\prime }}$ we have $g=f$ there, so $-\mathrm{\Delta }(u-w)=f-g=0$ in ${\mathcal{D}}^{\mathrm{\prime }}(B^{\mathrm{\prime }\mathrm{\prime }})$ by [F1]; hence $h:=u-w$ is a weakly harmonic function on $B^{\mathrm{\prime }\mathrm{\prime }}$ and therefore $C^{\mathrm{\infty }}$ there by [F4]. The interior derivative estimates give $\mathrm{\parallel }h{\mathrm{\parallel }}_{W^{2,p}(B^{\mathrm{\prime }})}\le C(n,B^{\mathrm{\prime }},B^{\mathrm{\prime }\mathrm{\prime }})\mathrm{\parallel }h{\mathrm{\parallel }}_{L^1(B^{\mathrm{\prime }\mathrm{\prime }})}$, and $\mathrm{\parallel }h{\mathrm{\parallel }}_{L^1(B^{\mathrm{\prime }\mathrm{\prime }})}\le \mathrm{\mid }B{\mathrm{\mid }}^{1\mathrm{/}2}\mathrm{\parallel }u{\mathrm{\parallel }}_{L^2(B)}+\mathrm{\parallel }w{\mathrm{\parallel }}_{L^1(B^{\mathrm{\prime }\mathrm{\prime }})}\le C(B,n,p)(\mathrm{\parallel }u{\mathrm{\parallel }}_{L^2(B)}+\mathrm{\parallel }f{\mathrm{\parallel }}_{L^p(B)})$ by the local Young bound [F1], Hölder on the bounded supports, and $\mathrm{\parallel }u{\mathrm{\parallel }}_{L^1(B^{\mathrm{\prime }\mathrm{\prime }})}\le \mathrm{\mid }B{\mathrm{\mid }}^{1\mathrm{/}2}\mathrm{\parallel }u{\mathrm{\parallel }}_{L^2(B)}$.

Conclusion. On $B^{\mathrm{\prime }}$ one has $u=w+h$ with $w\in W^{2,p}(B^{\mathrm{\prime }})$ by step 3.1 and $h\in W^{2,p}(B^{\mathrm{\prime }})$ by step 4.1, so $u\in W^{2,p}(B^{\mathrm{\prime }})$ and $\mathrm{\parallel }u{\mathrm{\parallel }}_{W^{2,p}(B^{\mathrm{\prime }})}\le \mathrm{\parallel }w{\mathrm{\parallel }}_{W^{2,p}(B^{\mathrm{\prime }})}+\mathrm{\parallel }h{\mathrm{\parallel }}_{W^{2,p}(B^{\mathrm{\prime }})}\le C(n,p,B^{\mathrm{\prime }},B)(\mathrm{\parallel }f{\mathrm{\parallel }}_{L^p(B)}+\mathrm{\parallel }u{\mathrm{\parallel }}_{L^2(B)})$, using $\mathrm{\parallel }g{\mathrm{\parallel }}_{L^p}\le \mathrm{\parallel }\eta {\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{\parallel }f{\mathrm{\parallel }}_{L^p(B)}$. No boundary condition on $u$ was used, and the constants depend only on $n,p$ and the balls.