$L^p$ estimates for the Laplacian via blow-up

In this note we provide a new proof of the $W^{2,p}$ Calder\'on-Zygmund regularity estimates for the Laplacian, i.e., $\Delta u=f$ and its parabolic counterpart $\partial_t u-\Delta u=f$. Our proof is an adaptation of a contradiction and compactness argument that so far had been only used to prove estimates in H\"older spaces. This new approach is simpler than previous ones, and avoids the use of any interpolation theorem.