Multipliers of \(H^ p\) and BMOA
For functions \(f(z)=\sum^{\infty}_{0}a_ nz^ n\), \(g(z)=\sum^{\infty}_{0}b_ nz^ n\) analytic in the unit disk \({\mathbb{D}}\) write \(f*g(z)=\sum^{\infty}_{0}a_ nb_ nz^ n\) and \(M_ p(r,f)=(1/2\pi)\int^{2\pi}_{0}| f(re^{i\theta})^ p d\theta)^{1/p}\). The space \(H^ p\) is the set of analytic f with \(M_ p(r,f)=O(1)\) as \(r\to 1\). For function spaces X and Y denote by (X,Y) the set of analytic g in \({\mathbb{D}}\) such that f*g\(\in Y\forall f\in X.\) Theorems of Hardy-Littlewood and Stein-Zygmund characterize \((H^ 1,H^ q)\), for \(2\leq q<\infty\), as the set of g with \(M_ q(r,g')=O((1-r)^{- 1})\). The set of g with \(M_{\infty}(r,g')=O((1-r)^{-1})\) is the Bloch space \({\mathcal B}\). One might expect that \((H^ 1,H^{\infty})={\mathcal B}\), but it follows from Fefferman's duality theorem \((H^ 1)^*=BMO\) that \((H^ 1,H^{\infty})\) is in fact the smaller space BMOA of analytic functions of bounded mean oscillation. In this paper the authors complete this circle of results by proving that \((H^ 1,BMOA)={\mathcal B}\). Tools consist of old and new relations between membership of functions in spaces and growth of means of their derivatives. For example, they use both the Hardy-Littlewood result that \[ f\in H^ 1\quad \Rightarrow \int^{1}_{0}(1-r)M^ 2_ 1(r,f')dr<\infty, \] and a dual result that \[ \int^{1}_{0}(1-r)M^ 2_{\infty}(r,h')dr<\infty \quad \Rightarrow \quad h\in BMOA. \] The authors also give a characterization of functions in \((H^ p,H^ q)\) when \(0<p<1\) and \(p<q\leq \infty\). For \(q\geq 1\) this is due to Duren and Shields. The case \(q=\infty\) is equivalent to the duality result \((H^ p)^*=A^{m,\alpha}\), where \(m=[1/p]-1\), \(\alpha =(1/p)-m\); with \(A^{m,0}\) understood to be a Zygmund rather than a Lipschitz space when \(\alpha =0\).
scientific article; zbMATH DE number 4208465