Besov Spaces Induced by Doubling Weights

Abstract

AbstractLet $$1\leqslant p<\infty $$ 1 ⩽ p < ∞ , $$0<q<\infty $$ 0 < q < ∞ , and $$\nu $$ ν be a two-sided doubling weight satisfying $$\begin{aligned} \sup _{0\leqslant r<1}\frac{(1-r)^q}{\int _r^1\nu (t)\,dt}\int _0^r\frac{\nu (s)}{(1-s)^q}\,ds<\infty . \end{aligned}$$ sup 0 ⩽ r < 1 ( 1 - r ) q ∫ r 1 ν ( t ) d t ∫ 0 r ν ( s ) ( 1 - s ) q d s < ∞ . The weighted Besov space $$\mathcal {B}_{\nu }^{p,q}$$ B ν p , q consists of those $$f\in H^p$$ f ∈ H p such that $$\begin{aligned} \int _0^1 \left( \int _{0}^{2\pi } |f'(re^{i\theta })|^p\,d\theta \right) ^{q/p}\nu (r)\,dr<\infty . \end{aligned}$$ ∫ 0 1 ∫ 0 2 π | f ′ ( r e i θ ) | p d θ q / p ν ( r ) d r < ∞ . Our main result gives a characterization for $$f\in \mathcal {B}_{\nu }^{p,q}$$ f ∈ B ν p , q depending only on |f|, p, q, and $$\nu $$ ν . As a consequence of the main result and inner-outer factorization, we obtain several interesting by-products. For instance, we show the following modification of a classical factorization by F. and R. Nevanlinna: If $$f\in \mathcal {B}_{\nu }^{p,q}$$ f ∈ B ν p , q , then there exist $$f_1,f_2\in \mathcal {B}_{\nu }^{p,q} \cap H^\infty $$ f 1 , f 2 ∈ B ν p , q ∩ H ∞ such that $$f=f_1/f_2$$ f = f 1 / f 2 . Moreover, we give a sufficient and necessary condition guaranteeing that the product of $$f\in H^p$$ f ∈ H p and an inner function belongs to $$\mathcal {B}_{\nu }^{p,q}$$ B ν p , q . Applying this result, we make some observations on zero sets of $$\mathcal {B}_{\nu }^{p,p}$$ B ν p , p .