Calderón–Zygmund estimates

Remarks

On the Fourier side, $\widehat{\partial_i \partial_j u}(\xi) = \frac{\xi_i \xi_j}{\abs{\xi}^2} \widehat{\Delta u}(\xi)$. So $\partial_i \partial_j u = R_i R_j \Delta u$, up to sign, where $R_i$ is the Riesz transform with multiplier $-i \xi_i / \abs{\xi}$. Each $R_i R_j$ is a singular integral operator, and Calderón–Zygmund theory shows that such operators are bounded on $L^p$ for $1 < p < \infty$. The proof of that bound uses the Calderón–Zygmund lemma to get a weak type $(1,1)$ bound, then interpolation with the $L^2$ bound from Plancherel, then duality. The estimate fails for $p = 1$ and for $p = \infty$: a function with bounded Laplacian need not have bounded second derivatives. Combined with cut-off functions, the estimate gives the local $W^{2,p}$ estimates for solutions of $\Delta u = f$, and by freezing coefficients for elliptic equations with continuous leading coefficients.

Going deeper

For a student

These estimates are the $L^p$ counterpart of the Schauder estimates in Hölder spaces. They are the main tool to show that a weak solution of an elliptic equation with an $L^p$ right-hand side has second derivatives in $L^p$. They are also a model case of the general theory of singular integrals, which treats operators like the Riesz and Hilbert transforms.