Inequalities for weighted \(L^ 2\)-functions on the half-line (Q580535)

The author applies the general theory of reproducing kernel spaces to give an extension of a theorem of Paley and Wiener on holomorphic functions in half-planes. This theory also yields sharp inequalities for weighted \(L^ 2\)-functions on the half-line. In particular, the following sharp convolution inequality is proved: For any \(q_ j>0\) and any \(F_ j\) such that \(t^{(1-q_ j)/2}F_ j\) is in \(L^ 2(0,\infty)\), \(j=1,...,n\), we have \[ \int^{\infty}_{0}| \prod^{n}_{j=1}*F_ j(t)|^ 2t^{1-(q_ 1+...+q_ n)}dt\leq \frac{\Gamma (q_ 1)...\Gamma (q_ n)}{\Gamma (q_ 1+...+q_ n)}\prod^{n}_{j=1}\int^{\infty}_{0}| F_ j(t)|^ 2t^{1-q_ j}dt \] where \(\prod^{n}_{j=1}*F_ j\) denotes the iterated convolution of \(F_ 1,...,F_ n\) on (0,\(\infty)\). Equality holds if and only if each \(F_ j\) is of the form \[ F_ j(t)=c_ jt^{q_ j-1} \exp (-t{\bar \zeta})\quad (0<t<\infty) \] for some constants \(c_ j\) and some \(\zeta\in {\mathbb{C}}\) with Re \(\zeta\) \(>0\), which is independent of j. This extends a result of \textit{S. Saitoh} [Proc. Am. Math. Soc. 91, 285-286 (1984; Zbl 0537.42004)] when \(n=2m\), \(m=1,..\). and \(q_ 1=q_ 2=...=q_{2m}=1\). An additional interest in the above result may be due to its possible relation with the Genchev isomorphism [see \textit{M. M. Dzhrbashyan} and \textit{V. M. Martirosyan}, Anal. Math. 12, 191-212 (1986; Zbl 0609.30039)].