Orlicz boundedness for certain classical operators

E. Harboure, O. Salinas, and B. Viviani. "Orlicz boundedness for certain classical operators." Colloquium Mathematicae 91.2 (2002): 263-282. <http://eudml.org/doc/283677>.

@article{E2002,
abstract = {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 $M_\{Ω\}^\{α\}$, associated to an open bounded set Ω, to be bounded from the Orlicz space $L^\{ψ\}(Ω)$ into $L^\{ϕ\}(Ω)$, 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 $I_\{Ω\}^\{α\}$, 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.},
author = {E. Harboure, O. Salinas, B. Viviani},
journal = {Colloquium Mathematicae},
keywords = {Orlicz spaces; boundedness; maximal function; fractional integral; Hilbert transform; Trudinger-type inequalities},
language = {eng},
number = {2},
pages = {263-282},
title = {Orlicz boundedness for certain classical operators},
url = {http://eudml.org/doc/283677},
volume = {91},
year = {2002},
}

TY - JOUR
AU - E. Harboure
AU - O. Salinas
AU - B. Viviani
TI - Orlicz boundedness for certain classical operators
JO - Colloquium Mathematicae
PY - 2002
VL - 91
IS - 2
SP - 263
EP - 282
AB - 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 $M_{Ω}^{α}$, associated to an open bounded set Ω, to be bounded from the Orlicz space $L^{ψ}(Ω)$ into $L^{ϕ}(Ω)$, 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 $I_{Ω}^{α}$, 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.
LA - eng
KW - Orlicz spaces; boundedness; maximal function; fractional integral; Hilbert transform; Trudinger-type inequalities
UR - http://eudml.org/doc/283677
ER -