In the book Wavelets and Operators authored by Yves Meyer and translated by D.Salinger page 47 the author presents the following inequalities without proof: $$ \left \| \Lambda^{s}f \right \|_{p} \leq C(s,n) \times \left \| \left( I- \Delta \right)^{ \frac{s}{2} } f\right \|_{p} $$
$$ \left \| \left( I- \Delta \right)^{ \frac{s}{2} } f\right \|_{p} \leq C'(s,n) \times \left \| \left( I + \Lambda^{s} \right) f\right \|_{p} $$
Where $ \Delta $ is the Laplacian operator defined on $ \mathbb{R}^{n}$, $ \Lambda= \left(- \Delta \right)^{ \frac{1}{2} } $, $ s \geq 0$, $ p \geq 1 $ and $ \left \| . \right \|_{p} $ is the $ L^{p} $ norm. For convenience one might take $f$ to be in a small dense subset like $ C^{\infty}_{c} $.
P.S: For the first one the author suggests to compare Fourier transform of $ \Lambda^{s}f $ and $ \left( I- \Delta \right)^{ \frac{s}{2} } f$