On an integral representation of functions from the Hardy class \(H_p\), \(p\leq 1\). (English) Zbl 1178.30060
Summary: In this paper, we show, for example, that if the function \(f(z)\) is analytic in the half-plane \(\text{Im}\,z>0\) and belongs to the Hardy space \(H_ p,\;0<p\leq1\), then it can be represented in the form
\[
f(z)={s!\over\pi i}\int^ \infty_ {-\infty}{h(t)\,dt\over(t-z)^ {s+1}},\;\text{Im}\,z>0,
\]
where \(s=[1/p]\) and \(h(t)\) is real-valued and belongs to the class \(L_ 1+L_ \infty\). Moreover, \(h(t)\) is completely characterized by means of the maximal function that measures smoothness. This paper is closely linked to the results of A. J. Miyachi, A. B. Aleksandrov, V. G. Krotov, and the author.
MSC:
30H10 | Hardy spaces |
30E20 | Integration, integrals of Cauchy type, integral representations of analytic functions in the complex plane |