For the Sobolev-norm there holds $\Vert f\Vert_{W^{s,p}(\mathbb{R}^n)}\sim_{n,p,s}\left\Vert\left(\sum_{k\in\mathbb{N}_0}\left\vert (1+2^k)^sP_kf(x)\right\vert^2\right)^\frac{1}{2}\right\Vert_{L^p}$ with the Littlewood-Paley projections $P_kf$, which are defined as follows: Let $\phi$ and $\psi$ be a smooth and compactly supported functions such that the support of $\phi$ is included in a ball around the origin and the support of $\psi$ is in an annulus such that $1=\phi+\sum_{k\geq 1}\psi(2^{-k}\cdot)$. Then define $\widehat{P_0f}=\phi\hat f$ and $\widehat{P_kf}=\psi(2^{-k}\cdot)\hat f$.
The nonhomogeneous Besov norm ist defined by $\Vert f\Vert_{B^s_{p,r}}=\Vert (2^{ks}\Vert P_kf\Vert_{L^p})_{k\geq 0}\Vert_{\ell^r(\mathbb{N}_0)}$. The nonhomogeneous Besov space consists of all functions $f$ such that $\Vert f\Vert_{B^s_{p,r}}<\infty$. One can easily see that $\Vert f\Vert_{W^{s,2}}\sim\Vert f\Vert_{B^s_{2,2}}$. But how are $W^{s,p}$ and $B^s_{p,2}$ related if $p\neq 2$. Is there a way to show that $\Vert f\Vert_{W^{s,p}}\sim\Vert f\Vert_{B^s_{p,2}}$? It kinda looks like it but one has to interchange norms. Is that possible in this case?