Orlicz boundedness for certain classical operators

E. Harboure¹ ; O. Salinas¹ ; B. Viviani¹

¹ Instituto de Matemática Aplicada del Litoral Universidad Nacional del Litoral Güemes 3450 3000 Santa Fe, Argentina

Colloquium Mathematicum, Tome 91 (2002) no. 2, pp. 263-282.

Résumé

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.

Détail

DOI : 10.4064/cm91-2-6

Keywords: phi psi functions defined infty taking value zero zero non negative continuous derivative under mild extra assumptions necessary sufficient conditions fractional maximal operator mit omega alpha associated bounded set mit omega bounded orlicz space psi mit omega phi mit omega leq alpha functions phi finite upper type these results extended hilbert transform widetilde one dimensional torus fractional integral operator mit omega alpha alpha since these operators linear self adjoint get duality boundedness results near infinity deriving generalized trudinger type inequalities