Let $||u||_{B^{-\alpha}_{p, \infty}}$ be a Besov-type seminorm where $0<\alpha<1$, and $1\leq p \leq \infty$. It is defined as follows:
$||u||_{B^{-\alpha}_{p, \infty}} : = || 2^{-\alpha j } ||P_j u || _{L^p} ||_{l^\infty _j} = \sup _{j \geq 0} 2^{-\alpha j } ||P_j u || _{L^p}$ where $P_j$ denotes the Littlewood-Paley projection to $\xi \sim 2^j$.
Let $||u|| $ be the $L^p$- Holder seminorm where $||u|| := \sup _{0 < |h| \leq 1} \frac{||\tau_h f - f||_{L^p} }{ |h|^\alpha} $ where $\tau_h$ denotes the translation operator.
I want to prove that $||u|| \sim ||u||_{B^{-\alpha}_{p, \infty}}$ (up to some multiplicative constant). Where can I find the relevant theories? I want some good references.
Thank you!