  In this note we provide a new proof of the $W^{2,p}$
\textit{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.
