Definition 1

Let $s,b\in \mathbb R$ and $p,q\in(0,\infty]$ . Assume that $\vec \varphi \in A(\mathbb R^n)$ . The $logarithmic \,Besov\,space\, B^{s,b}_{p,q}(\mathbb R^n)$ is defined as the collection of all the tempered distributions $f$ on $\mathbb R^n$ with finite quasi-norm

\[\|f\|_{B^{s,b}_{p,q}\mathbb (R^n)}:=\left[\sum\limits^\infty_{j=0}2^{jsq}(1+j)^{bq}\|\varphi_j*f\|^q_{L^p(\mathbb R^n)}\right]^{1/q},\]

where the usual modification is made when $p=\infty$ or $q=\infty$ .

Theorem 1

Let $s,b\in \mathbb R$ and $p,q\in(0,\infty]$ . For any $f\in B^{s,b}_{p,q}(\mathbb R^n)$ , we have

\[\|f\|_{B^{s,b}_{p,q}(\mathbb R^n)}\sim\|f\|_{L^p(\mathbb R^n}+[f]_{B^{s,b}_{p,q}(\mathbb R^n)}\]

with

\[[f]_{B^{s,b}_{p,q}(\mathbb R^n)}:=\left(\int_{\mathbb R^n}\|\Delta^M_hf\|^q_{L^p(\mathbb R^n)}\left[log\left(e+\frac 1{|h|}\right)\right]^{bq}\frac{\mathrm d h}{|h|^{n+sq}}\right)^{1/q},\]

where $M$ is a natural number larger than $s$ , $\Delta ^M_h f$ denotes the M-order difference of $f$ , and the positive equivalence constants are independent of $f$ .