The authors study the properties of the mapping which to a function \(f\) analytic in the unit disk associates the function given by \[ T_g f(z)= \int_0^z f(\zeta) g'(\zeta) d\zeta , \] where \(g\) is a given analytic function (the symbol of the operator). In the case where \(g\) is the identity, this operator just amounts to taking the primitive of \(f\), and Hardy and Littlewood had already shown that this sends the usual Hardy space \(H^p\) to \(H^{p/(1-p)}\), for \(0<p<1\). Other special cases of the operator \(T_g\) have been studied since, see A. G. Siskakis, Proc. Am. Math. Soc. 110, 461–462 (1990; Zbl 0719.47020), Miao Jie, Proc. Am. Math. Soc. 116, 1077–1079 (1992; Zbl 0787.47029), and also for case where \(p=2\) and \(g \in BMOA\), C. Pommerenke, Comment. Math. Helv. 52, 591–602 (1977; Zbl 0369.30012).
The authors prove the following results, which extend the results in [A. Aleman and A. Siskakis, An integral operator on \(H^p\), Complex Variables, Theory Appl. 28, 149–158 (1995; Zbl 0837.30024)] and yield an improvement of the Hardy-Littlewood theorem:
(i) For \(1/q > 1/p\), \(T_g\) maps \(H^p\) into \(H^q\) if and only if \(g \in H^s\), where \(s=1/q-1/p\);
(ii) For \(p=q\), \(T_g\) maps \(H^p\) into itself if and only if \(g \in BMOA\);
(iii) For \(1/p > 1/q \geq 1/p -1\), \(T_g\) maps \(H^p\) into \(H^q\) if and only if \(g\) is in the analytic Lipschitz class \(\Lambda_{1/p-1/q}\);
(iv) For \(1/q < 1/p -1\), \(T_g\) maps \(H^p\) into \(H^q\) if and only if \(g\) is constant (and thus \(T_g=0\)).
Furthermore, they characterize the cases where \(T_g\) is a compact operator between the spaces in question (in case (i), \(T_g\) is compact whenever it is bounded, in cases (ii) and (iii), one needs the ”little oh” versions of the ”big oh” conditions given on \(g\)). They also derive from their main results mapping properties of \(T_g\) from \(H^p\) to the analytic Lipschitz spaces, namely, if one assumes that \(g \in \Lambda_\alpha\) for some \(\alpha \in (0,1)\), then
(v) for \(p=1/\alpha\), \(T_g\) maps \(H^p\) into \(BMOA\) (and this cannot be improved to \(H^\infty\) in general) ;
(vi) for \(p>1/\alpha\), \(T_g\) maps \(H^p\) into \(\Lambda_{\alpha - 1/p}\).
The proof of the sufficiency part of results (ii) and (iii) (the hardest part of the paper) uses as an intermediate step the following result :
For \(0<p\leq q <\infty\), \(1/q \geq 1/p -1\), let \(f\in H^p\) and \(g \in \Lambda_{1/p-1/q}\). Suppose that \(F\) is analytic and zero-free in the unit disk with \(\log F \in BMOA\) and that \(Ff \in H^{p'}\) for some \(p' \geq p\). Then there exist \(\beta >0\) and \(q' \geq q\) such that \(F^\beta T_g f \in H^{q'}\) (moreover, the possible range of \(q'\) can be given a precise, if complicated, description).
In the application, \(F\) is usually a negative power of an outer function defined using the boundary values of \(| f| \). Result (i) is deduced from (ii) and (iii) thanks to a lemma that shows that the derivative of a function in some \(H^s\) can be decomposed as the sum of 4 terms which are products of functions in \(H^s\) itself by derivatives of functions in \(BMOA\).