Norm inequalities for fractional integrals of Laguerre and Hermite expansions

Abstract

Suppose the fractional integration operator I^σ is generated by the sequence {(k+1)^-σ } in the setting of Laguerre and Hermite expansions. Then, via projection formulas, the problem of the norm boundedness of I^σ 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.