
> 
> 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 $W^{2,2}(\Bbb R^d)$.



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


