Orlicz boundedness for certain classical operators
Volume 91 / 2002
Abstract
Let $\phi $ and $\psi $ be functions defined on $[ 0,\infty ) $ 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 $M_{{\mit \Omega }}^{\alpha }$, associated to an open bounded set ${\mit \Omega } $, to be bounded from the Orlicz space $L^{\psi }({\mit \Omega } )$ into $L^{\phi }({\mit \Omega })$, $0\leq \alpha < n$. For functions $\phi $ of finite upper type these results can be extended to the Hilbert transform $\widetilde {f}$ on the one-dimensional torus and to the fractional integral operator $I_{{\mit \Omega } }^{\alpha }$, $0<\alpha < n$. 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.
