Newtonian Hessian kernels fit the Calderón–Zygmund framework

Differentiating $\mathrm{\Gamma }(x)=\mathrm{\mid }x{\mathrm{\mid }}^{2-n}\mathrm{/}((n-2){\omega }_{n-1})$ twice off the origin gives $k_{ij}(x)={\mathrm{\partial }}_{ij}\mathrm{\Gamma }(x)={\omega }_{n-1}^{-1}(n{x}_i{x}_j\mathrm{\mid }x{\mathrm{\mid }}^{-n-2}-{\delta }_{ij}\mathrm{\mid }x{\mathrm{\mid }}^{-n})$ for $x\mathrm{\ne }0$: the first derivative is $(2-n)(n-2{)}^{-1}{\omega }_{n-1}^{-1}{x}_i\mathrm{\mid }x{\mathrm{\mid }}^{-n}=-{\omega }_{n-1}^{-1}{x}_i\mathrm{\mid }x{\mathrm{\mid }}^{-n}$, and differentiating once more with ${\mathrm{\partial }}_j(x_i\mathrm{\mid }x{\mathrm{\mid }}^{-n})={\delta }_{ij}\mathrm{\mid }x{\mathrm{\mid }}^{-n}-n{x}_i{x}_j\mathrm{\mid }x{\mathrm{\mid }}^{-n-2}$ gives the displayed expression. This function is smooth on ${\mathbb{R}}^n\setminus \{0\}$ and homogeneous of degree $-n$.

Size and gradient bounds: $\mathrm{\mid }{k}_{ij}(x)\mathrm{\mid }\le {\omega }_{n-1}^{-1}(n\mathrm{\mid }{x}_i{x}_j\mathrm{\mid }\mathrm{\mid }x{\mathrm{\mid }}^{-n-2}+\mathrm{\mid }x{\mathrm{\mid }}^{-n})\le (1+n){\omega }_{n-1}^{-1}\mathrm{\mid }x{\mathrm{\mid }}^{-n}$ by $\mathrm{\mid }{x}_i{x}_j\mathrm{\mid }\le \mathrm{\mid }x{\mathrm{\mid }}^2$; and since $k_{ij}$ is $C^{\mathrm{\infty }}$ off the origin and homogeneous of degree $-n$, each partial derivative is homogeneous of degree $-n-1$, continuous on the compact unit sphere, and therefore satisfies $\mathrm{\mid }\mathrm{\nabla }{k}_{ij}(x)\mathrm{\mid }\le C_n\mathrm{\mid }x{\mathrm{\mid }}^{-n-1}$ for all $x\mathrm{\ne }0$ with $C_n:=n{\sup }_{\mathrm{\mid }z\mathrm{\mid }=1}\mathrm{\mid }\mathrm{\nabla }{k}_{ij}(z)\mathrm{\mid }<\mathrm{\infty }$.

The spherical mean vanishes: for $r>0$, substituting $x=r\omega$ and using the homogeneity, ${\int }_{S^{n-1}}{k}_{ij}(r\omega )d{\sigma }_{n-1}(\omega )=r^{-n}{\omega }_{n-1}^{-1}{\int }_{S^{n-1}}(n{\omega }_i{\omega }_j-{\delta }_{ij})d{\sigma }_{n-1}(\omega )$. The surface measure is invariant under coordinate permutations and under the sign change ${\omega }_i\mapsto -{\omega }_i$, so ${\int }_{S^{n-1}}{\omega }_i{\omega }_{j}d\sigma =0$ for $i\mathrm{\ne }j$ and all $n$ integrals $\int {\omega }_i^{2}d\sigma$ are equal to ${\omega }_{n-1}\mathrm{/}n$ since their sum is $\int \mathrm{\mid }\omega {\mathrm{\mid }}^{2}d\sigma ={\omega }_{n-1}$; hence $\int (n{\omega }_i{\omega }_j-{\delta }_{ij})d\sigma =n{\delta }_{ij}{\omega }_{n-1}\mathrm{/}n-{\delta }_{ij}{\omega }_{n-1}=0$, and the spherical mean of $k_{ij}$ vanishes.

Principal value and the local delta term. By step 2.2, ${\int }_{\epsilon <\mathrm{\mid }x\mathrm{\mid }<1}{k}_{ij}(x)dx=0$. Thus for every Schwartz test $\phi$, the limit $⟨V_{ij},\phi ⟩={\lim }_{\epsilon \downarrow 0}{\int }_{\mathrm{\mid }x\mathrm{\mid }>\epsilon }{k}_{ij}(x)\phi (x)dx$ exists: near zero subtract $\phi (0)$, giving a majorant $C\mathrm{\mid }x{\mathrm{\mid }}^{1-n}{\sup }_{\mathrm{\mid }z\mathrm{\mid }\le 1}\mathrm{\mid }\mathrm{\nabla }\phi (z)\mathrm{\mid }$, and the tail is integrable by Schwartz decay. These bounds also prove $V_{ij}\in {\mathcal{S}}^{\mathrm{\prime }}$. Integration by parts on $\{\epsilon <\mathrm{\mid }x\mathrm{\mid }<R\}$ using [F2] first shows ${\mathrm{\partial }}_i\mathrm{\Gamma }$ is the regular distribution of $-{\omega }_{n-1}^{-1}{x}_i\mathrm{\mid }x{\mathrm{\mid }}^{-n}$: the inner boundary term from $\mathrm{\Gamma }$ is $O(\epsilon )$, and outer terms vanish as $R\to \mathrm{\infty }$. Applying integration by parts once more gives $⟨{\mathrm{\partial }}_{ij}\mathrm{\Gamma },\phi ⟩={\lim }_{\epsilon \downarrow 0}[{\int }_{\mathrm{\mid }x\mathrm{\mid }>\epsilon }{k}_{ij}\phi -{\omega }_{n-1}^{-1}{\int }_{S^{n-1}}{\omega }_i{\omega }_j\phi (\epsilon \omega )d\sigma ]$. The inward normal of the exterior region at radius $\epsilon$ is $-\omega$, which fixes the minus sign. Step 2.2 evaluates the boundary limit as ${\delta }_{ij}\phi (0)\mathrm{/}n$. Hence ${\mathrm{\partial }}_{ij}\mathrm{\Gamma }=V_{ij}-({\delta }_{ij}\mathrm{/}n){\delta }_0$; summing the diagonal identities, whose off-origin kernels have zero trace, proves $-\mathrm{\Delta }\mathrm{\Gamma }={\delta }_0$.

For $\mathrm{\mid }x\mathrm{\mid }\ge 2\mathrm{\mid }y\mathrm{\mid }>0$, the segment from $x-y$ to $x$ stays in $\{\mathrm{\mid }z\mathrm{\mid }\ge \mathrm{\mid }x\mathrm{\mid }\mathrm{/}2\}$. The mean value theorem and step 2.1 give $\mathrm{\mid }{k}_{ij}(x-y)-k_{ij}(x)\mathrm{\mid }\le 2^{n+1}{C}_n\mathrm{\mid }y\mathrm{\mid }\mathrm{\mid }x{\mathrm{\mid }}^{-n-1}$. For every $y\mathrm{\ne }0$, [F4] therefore gives ${\int }_{\mathrm{\mid }x\mathrm{\mid }\ge 2\mathrm{\mid }y\mathrm{\mid }}\mathrm{\mid }{k}_{ij}(x-y)-k_{ij}(x)\mathrm{\mid }dx\le 2^{n+1}{C}_n\mathrm{\mid }y\mathrm{\mid }{\omega }_{n-1}\mathrm{/}(2\mathrm{\mid }y\mathrm{\mid })=2^n{C}_n{\omega }_{n-1}$. The size bound gives annular constant $(1+n)\log 2$, and step 1.1 supplies smoothness off zero. Thus $k_{ij}$ is a base Calderón–Zygmund kernel; its first-difference bound then makes it standard $1$-Hölder with constant $A_2^{\mathrm{\prime }}=2^{n+1}{C}_n$.

There is no frequency-zero ambiguity. The locally integrable kernel $\mathrm{\Gamma }$, bounded at infinity, defines a tempered distribution. From step 3.1, $4{\pi }^2\mathrm{\mid }\xi {\mathrm{\mid }}^2\mathcal{F}\mathrm{\Gamma }=1$, so $\mathcal{F}\mathrm{\Gamma }$ agrees off zero with $a(\xi )=(4{\pi }^2\mathrm{\mid }\xi {\mathrm{\mid }}^2{)}^{-1}$. Since $n\ge 3$, $a$ is locally integrable even at zero and tempered. Set $U=\mathcal{F}\mathrm{\Gamma }-u_a$; multiplication by $\mathrm{\mid }\xi {\mathrm{\mid }}^2$ annihilates $U$, so it is supported at zero. Homogeneity of $\mathrm{\Gamma }$ and change of variables in its pairing give $⟨\mathcal{F}\mathrm{\Gamma },\phi (\cdot \mathrm{/}\lambda )⟩={\lambda }^{n-2}⟨\mathcal{F}\mathrm{\Gamma },\phi ⟩$ for $\lambda >0$: the Fourier transform of $\phi (\cdot \mathrm{/}\lambda )$ is ${\lambda }^n\widehat{\phi }(\lambda \cdot )$. The same scaling holds for $u_a$ and hence $U$. By [F5], $U=\sum c_{\alpha }{\mathrm{\partial }}^{\alpha }{\delta }_0$, while each summand pairs with $\phi (\cdot \mathrm{/}\lambda )$ as ${\lambda }^{-\mathrm{\mid }\alpha \mathrm{\mid }}{\mathrm{\partial }}^{\alpha }{\delta }_0(\phi )$. Uniqueness of the coefficients gives $c_{\alpha }({\lambda }^{-\mathrm{\mid }\alpha \mathrm{\mid }}-{\lambda }^{n-2})=0$ for every $\lambda >0$. Taking $\lambda =2$ and $n\ge 3$ forces all coefficients to vanish, so $\mathcal{F}\mathrm{\Gamma }=u_a$.

By [F2] and step 4.1, $\mathcal{F}({\mathrm{\partial }}_{ij}\mathrm{\Gamma })=u_{-{\xi }_i{\xi }_j\mathrm{/}\mathrm{\mid }\xi {\mathrm{\mid }}^2}$. Step 3.1 therefore gives $\mathcal{F}{V}_{ij}=u_{-{\xi }_i{\xi }_j\mathrm{/}\mathrm{\mid }\xi {\mathrm{\mid }}^2+{\delta }_{ij}\mathrm{/}n}$. This bounded real symbol is that of $R_i{R}_j+({\delta }_{ij}\mathrm{/}n)I$, so [F3,F5] identify $V_{ij}\ast f$ on Schwartz functions with an $L^2$ multiplier of norm at most $1+1\mathrm{/}n$.

The multiplier $T=R_i{R}_j+({\delta }_{ij}\mathrm{/}n)I$ is a Calderón–Zygmund operator with this kernel. Its symbol is real, so Plancherel's pairing gives $T^{\ast }=T$. For compactly supported $f\in L^2$ and a smooth compactly supported test $\phi$ supported away from $\mathrm{supp}f$, [F5] and step 5.1 give $⟨Tf,\phi ⟩=⟨f,T\phi ⟩$. On the support of $f$, $T\phi =V_{ij}\ast \phi$ is the ordinary off-support kernel integral. The kernel is real and even, so Fubini yields $⟨Tf,\phi ⟩=\int [\int k_{ij}(x-y)f(y)dy]\overline{\phi (x)}dx$. Positive separation and the size bound make this double integral absolutely convergent. The kernel integral is locally integrable off the support, so injectivity of the distribution pairing proves the required almost-everywhere off-support representation.

The preceding steps prove the kernel estimates, principal-value distribution, local delta correction, Fourier symbol, boundedness and off-support operator representation. This proves the example.