Are $\| \Delta u \|_{L^p(\Bbb R^d)} + \| u \|_{L^p(\Bbb R^d)}$ and $\| u \|_{W^{2,p}(\Bbb R^d)}$ equivalent norms?

Question

Is it true that $$\| \Delta u \|_{L^p(\Bbb R^d)} + \| u \|_{L^p(\Bbb R^d)}\quad\text{and}\quad\| u \|_{W^{2,p}(\Bbb R^d)}$$ are equivalent norms?

This results is pretty easy and straightforward for $p=2$ using techniques via Fourier transform and Plancherel. But what could we use in place of Fourier transform when $p\neq 2?$

Please prove or disprove or provide me with some good reference where I can fine.

I fact I need to show that the domain of the generator of the Gauss-Weierstrass semigroup in $L^p(\Bbb R^d)$ is $W^{2,p}(\Bbb R^d)$. This result is counterpart for the case $p=2$ where the domain of the generator is $W^{2,2}(\Bbb R^d)$.

Answer 1

First, the classical Calderón–Zygmund estimate gives $$\left\|D^2 u\right\|_{L^p(\mathbb{R}^d)}\leq C(p,d) \left\|\Delta u\right\|_{L^p(\mathbb{R}^d)}.$$ By interpolation, $\left\|u\right\|_{W^{2,p}(\mathbb{R})}$ is equivalent to $$\left\|\Delta^2 u\right\|_{L^p(\mathbb{R}^d)}+\left\|u\right\|_{L^p(\mathbb{R}^d)}.$$ For detail proof you can look up Gilbarg-Trudinger Chapter 9 for $L^p$ estimate, and Adams for Sobolev spaces.