CHARACTERISATIONS OF HARDY GROWTH SPACES WITH DOUBLING WEIGHTS

Abstract

AbstractLet $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}H(\mathbb{D})$ denote the space of holomorphic functions on the unit disc $\mathbb{D}$. Given $p>0$ and a weight $\omega $, the Hardy growth space $H(p, \omega )$ consists of those $f\in H(\mathbb{D})$ for which the integral means $M_p(f,r)$ are estimated by $C\omega (r)$, $0<r<1$. Assuming that $p>1$ and $\omega $ satisfies a doubling condition, we characterise $H(p, \omega )$ in terms of associated Fourier blocks. As an application, extending a result by Bennett et al. [‘Coefficients of Bloch and Lipschitz functions’, Illinois J. Math. 25 (1981), 520–531], we compute the solid hull of $H(p, \omega )$ for $p\ge 2$.