Norm inequalities for fractional integrals of Laguerre and Hermite expansions

Abstract

Suppose the fractional integration operator $I^{\sigma}$ is generated by the sequence $\{(k+1)^{-\sigma}\}$ in the setting of Laguerre and Hermite expansions. Then, via projection formulas, the problem of the norm boundedness of $I^{\sigma}$ is reduced to the well-known fractional integration on the half-line. A corresponding result with respect to the modified Hankel transform is derived and its connection with the Laguerre fractional integration is indicated.