Orlicz boundedness for certain classical operators

Abstract

Let ɸ and ψ be functions defined on [0,∞) 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∞Ω, associated to an open bounded set Ω, to be bounded from the Orlicz space Lψ(Ω) into Lɸ(Ω), 0 ≤ α < n. For functions ɸ of finite upper type these results can be extended to the Hilbert transform f on the one-dimensional torus and to the fractional integral operator IαΩ, 0 < α < n. Since these operators are linear and self-adjoint we get, by duality, boundedness results near infinity, deriving in this way some generalized Trudinger type inequalities.