Proof of the estimate $\;\| D^{2} u \|_{L^{p}}\sim \| \Delta u \|_{L^{p}}$ when $p>1$

Question

Let $u\in C^{\infty}_{c}(\mathbb{R}^n)$. One can integrate by parts twice to discover that $$\;\| D^{2} u \|_{L^{2}}= \| \Delta u \|_{L^{2}}$$

Is the estimate

$$\;\| D^{2} u \|_{L^{p}}\sim \| \Delta u \|_{L^{p}}$$

true for $p>1$ ?

A reference for this or a hint of the proof would be appreciated.

Answer 1

The inequality holds true. See Proposition 3 page 72 in Stein's "singular integrals and differentiability properties of functions".