## Tohoku Mathematical Journal 2000 June SECOND SERIES VOL. 52, NO. 2

 Tohoku Math. J. 52 (2000), 251-260

Title NORM INEQUALITIES FOR FRACTIONAL INTEGRALS OF LAGUERRE AND HERMITE EXPANSIONS

Author George Gasper and Walter Trebels

(Received October 5, 1998)
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.

1991 Mathematics Subject Classification. Primary 26A33; Secondary 33C45, 42A38, 42C10.

Key words and phrases. Fractional integration, Laguerre and Hermite expansions, Hankel transforms, multipliers.