On some second order integral operators and Hardy classes. (English) Zbl 0728.30025

Fekete, Otto

Semin. Oper. Liniari Anal. Armonică, Univ. Timişoara 1989, No. 1, 9 p. (1989).

Let \({\mathcal H}\) denote the set of analytic functions f in the unit disc \(\Delta\), and let \({\mathcal H}_ 0\) denote the subset of \({\mathcal H}\) consisting of functions f such that \(f(0)=0\) and \(f'(0)=1\). With suitable conditions on the constants \(\beta\) and \(\gamma\) and on the analytic functions \(\phi\) (z) and \(\psi\) (z), the author determines the Hardy classes of the integral operators, \[ [I_{\phi}(f)]_{\beta},\quad [I_{\phi}(f)]_{\beta}*[I_{\psi}(f)]_{\gamma},\quad [J_{\phi}(f(]_{\beta},\quad J_{\phi}(f)]_{\beta}*[J_{\psi}(f)]_{\gamma}, \] defined on \({\mathcal H}\), and it’s derivatives in some cases, where \[ [I_{\phi}(f)]_{\beta}(z)=\frac{1}{z^{\beta}\phi (z)}\int^{z}_{0}f(t)t^{\beta -1}\phi (t)dt \] and \[ [J_{\phi}(f)](z)=\frac{1}{\phi (z)}\int^{z}_{0}f(t)\phi '(t)dt, \] when the Hardy classes of f are known. For instance the author proves: “Let \(f\in H^ p\) (p\(\geq 1)\), \(\phi\in {\mathcal H}\), \(\phi\) (z)\(\neq 0\) in \(\Delta\), \(\phi '/\phi \in H^ q\) (q\(\geq 1)\) and Re \(\beta\) \(>0\). If \(p<q/(q-1)\), then \([I_{\phi}(f)]_{\beta}\in H^{pq/(p+q-pq)}\) and \([I_{\phi}(f)]'_{\beta}\in H^{pq/(p+q)}\); if \(p\geq q/(q-1)\) then \([I_{\phi}(f)]_{\beta}\in H^{\infty}\) and \([I_{\phi}(f)]_{\beta}'\in H^{pq/(p+q)''}\). Similar conclusions continue to hold for the operator \(J_{\phi}(f)\) with the assumptions that \(p>1\), \(q>1\), \(\phi\in {\mathcal H}_ 0\) with \(\phi (z)\phi '(z)\neq 0\) in \(\Delta\), \(f\in H^ p\) and \(\phi '/\phi \in H^ q\).

Reviewer: S.Ponnusamy (Madras)

MSC:

30D55

\(H^p\)-classes (MSC2000)

Keywords:

integral operators