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 , associated to an open bounded set Ω, to be bounded from the Orlicz space into , 0 ≤ α < n. For functions ϕ of finite upper type these results can be extended to the Hilbert transform f̃ on the one-dimensional torus and to the fractional integral operator , 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm91-2-6,
author = {E. Harboure and O. Salinas and B. Viviani},
title = {Orlicz boundedness for certain classical operators},
journal = {Colloquium Mathematicae},
volume = {91},
year = {2002},
pages = {263-282},
zbl = {1044.42016},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm91-2-6}
}
E. Harboure; O. Salinas; B. Viviani. Orlicz boundedness for certain classical operators. Colloquium Mathematicae, Tome 91 (2002) pp. 263-282. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm91-2-6/