Orlicz boundedness for certain classical operators

Abstract

Letand be functions defined on (0,1) taking the value zero at zero and with non-negative continuous derivative. Under very mild extra assumptions we find necessary and sufficient conditions for the fractional maximal operator M � , associated to an open bounded set , to be bounded from the Orlicz space L () into L � (), 0 � � < n. For functionsof finite upper type these results can be extended to the Hilbert transform e f on the one-dimensional torus and to the fractional integral operator I � ,