Orlicz boundedness for certain classical operators

ELEONOR HARBOURE; OSCAR SALINAS; BEATRIZ VIVIANI

COLLOQUIUM MATHEMATICUM

Institute of Mathematics - Polish Academy of Sciencies

Año: 2002 vol. 91 p. 263 - 282

0010-1354

Let ϕ 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 MαΩ, associated to an open bounded set Ω, to be bounded from the Orlicz space Lψ(Ω) into Lϕ(Ω), 0≤α<n. For functions ϕ 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.