Besov spaces induced by doubling weights

Let $1le p<infty$, $0<q<infty$ and $ u$ be a two-sided doubling weight satisfying $$sup_{0le r<1}frac{(1-r)^q}{int_r^1 u(t),dt}int_0^rfrac{ u(s)}{(1-s)^q},ds<infty.$$ The weighted Besov space $mathcal{B}_{ u}^{p,q}$ consists of those $fin H^p$ such that $$int_0^1 left(int_{0}^{2pi} |f(re^{itheta})|^p,dthetaright)^{q/p} u(r),dr<infty.$$ Our main result gives a characterization for $fin mathcal{B}_{ u}^{p,q}$ depending only on $|f|$, $p$, $q$ and $ u$. 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 $fin mathcal{B}_{ u}^{p,q}$, then there exist $f_1,f_2in mathcal{B}_{ u}^{p,q} cap H^infty$ such that $f=f_1/f_2$. In addition, we give a sufficient and necessary condition guaranteeing that the product of $fin H^p$ and an inner function belongs to $mathcal{B}_{ u}^{p,q}$. Applying this result, we make some observations on zero sets of $mathcal{B}_{ u}^{p,p}$.