Do we have that $$\| \Delta u \|_{L^p(\Bbb R^d)} + \| u \|_{L^p(\Bbb R^d)}~~~~and~~\| u \|_{W^{2,p}(\Bbb R^d)}$$ 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 this .
I fact I need to show that Domain of the generator of the Gauss-Weierstrass semi-group in $L^p(\Bbb R^d)$ is $W^{2,p}(\Bbb R^d)$. this result is counter part for the case $p=2$ where the domain of the generator is $W^{2,2}(\Bbb R^d)$ any help regarding this is welcome