Besov spaces induced by doubling weights

Abstract

Let $1\le p<\infty$, $0<q<\infty$ and $\nu$ be a two-sided doubling weight satisfying $$\sup_{0\le r<1}\frac{(1-r)^q}{\int_r^1\nu(t)\,dt}\int_0^r\frac{\nu(s)}{(1-s)^q}\,ds<\infty.$$ The weighted Besov space $\mathcal{B}_{\nu}^{p,q}$ consists of those $f\in H^p$ such that $$\int_0^1 \left(\int_{0}^{2\pi} |f'(re^{i\theta})|^p\,d\theta\right)^{q/p}\nu(r)\,dr<\infty.$$ Our main result gives a characterization for $f\in \mathcal{B}_{\nu}^{p,q}$ depending only on $|f|$, $p$, $q$ and $\nu$. As a consequence of the main result and inner-outer factorization, we obtain several interesting by-products. In particular, we show the following modification of a classical factorization by F. and R. Nevanlinna: If $f\in \mathcal{B}_{\nu}^{p,q}$, then there exist $f_1,f_2\in \mathcal{B}_{\nu}^{p,q} \cap H^\infty$ such that $f=f_1/f_2$. In addition, we give a sufficient and necessary condition guaranteeing that the product of $f\in H^p$ and an inner function belongs to $\mathcal{B}_{\nu}^{p,q}$. Applying this result, we make some observations on zero sets of $\mathcal{B}_{\nu}^{p,p}$.