Weighted norm inequalities for fractional integrals with an application to mean convergence of Laguerre series
The paper deals with the weighted norm inequalities in \(L_p\)-spaces for the Riemann-Liouville fractional integral \[ f_a(x)= {1\over\Gamma(a)}\int^x_0 {f(t)dt\over(x- t)^{1-a}} (a>0) \] [see, for example, Section 5 in the book by \textit{S. G. Samko}, \textit{A. A. Kilbas} and \textit{O. I. Marichev}: ``Fractional integrals and derivatives. Theory and Applications'' (Russian, 1987; Zbl 0617.26004, English translation, 1993; Zbl 0818.26003)]. Sufficient conditions are given for the validity of the inequality with power exponential weights \[ \int^\infty_0|x^{-(m+ l)}e^{-bx} f_a(x)|^q dx\leq K\Biggl(\int^\infty_0 |x^{-l} e^{-bx}f(x)|^p dx\Biggr)^{q/p} \] with \(p>1\), \(b>0\), \(m\geq 0\), \(l>1/p- 1\) and \(q\geq p\). Examples are given to show the accuracy of the conditions on parameters. The obtained results are applied on certain functions \(f(x)\) to show that their expansions with respect to Laguerre polynomials \(L^\lambda_k(x)\) are convergent in \(L_p\)-norm. It is proved the existence of constants \(c_k\) \((k= 0,1,2,\dots)\) so that \[ \Biggl|e^{-x/2} x^{\lambda/2}\Biggl( \sum^n_{k= 0} c_kL^\lambda_k- f\Biggr)\Biggr|_q\to 0\quad\text{as }n\to\infty. \] Such a relation for the standard range of \(q\in(4/3,4)\) was obtained by \textit{R. Askey} and \textit{S. Wainger} [Am. J. Math. 87, 695-708 (1965; Zbl 0125.31301)]. The author extends this result to some \(q\) outside of \((4/3,4)\). The example \(f(x)= x^\nu\) illustrates the above mean convergence of the Laguerre expansion.
scientific article; zbMATH DE number 1076154
scientific article