Norm inequalities for fractional integrals of Laguerre and Hermite expansions
The paper is devoted to the study of the fractional integration operators \(I^{\sigma}\) and \(I^{\sigma}_{H}\) defined by \[ I^{\sigma}\left(\sum^{\infty}_{k=0}a_{k}L^{\alpha}_{k}(x)\right) =\sum^{\infty}_{k=0}(k+1)^{-\sigma}a_{k}L^{\alpha}_{k}(x) \] and \[ I^{\sigma}_{H}\left(\sum^{\infty}_{k=0}a_{k}H_{k}(x)\right) =\sum^{\infty}_{k=0}(k+1)^{-\sigma}a_{k}H_{k}(x), \] respectively, where \(L^{\alpha}_{k}(x)\) is the classical Laguerre polynomial while \(H_{k}(x)\) is the Hermite polynomial. Sufficient conditions are given for the operator \(I^{\sigma}\) to be bounded from the one weighted space \(L_{p}(w,(0,\infty))\) with the power-exponential weight \(w(x)=x^{\gamma}e^{-px/2}\) to another one. The proof of this statement is based on the representation of the Laguerre polynomial \(L^{\alpha}_{k}(x)\) via the Erdélyi-Kober fractional integral - see, for example,the relation (18.5) in the book by \textit{S. G. Samko}, the reviewer and \textit{O. I. Marichev} [``Fractional integrals and derivatives. Theory and applications'' (1993; Zbl 0818.26003; Russian original 1987; Zbl 0617.26004)] and the use of the known inequality for this operator. The above result is applied to prove a similar statement for the operator inverse to \(I^{\sigma}_{H}\) in the weighted space \(L_{p}(w,(0,\infty))\) with the exponential weight \(w(x)=e^{-px^{2}/2}\). The connection of the Laguerre fractional integration \(I^{\sigma}\) with the modified Hankel transform is given and the corresponding weighted estimate for the operator \(I^{\sigma}\) in the weighted \(L_p\)-space with the power weight \(x^{2\gamma +1}\) is derived. Note that another approach to the fractional integration \(I^{\sigma}\) of the Laguerre series was developed by \textit{G. Gasper, K. Stempak} and \textit{W. Trebels} [Methods Appl. Anal. 2, No. 1, 67-75 (1995; Zbl 0837.42014)] and by \textit{Y. Kanjin} and \textit{E. Sato} [Proc. Am. Math. Soc. 123, No. 7, 2165-2171 (1995; Zbl 0856.42018)].
scientific article; zbMATH DE number 1533112
scientific article