In [4] P. Jones solved the question posed by B. Muckenhoupt in [7] concerning the factorization of Ap weights. We recall that a non-negative measurable function w on Rn is in the class Ap, 1 < p < ∞ if and only if the Hardy-Littlewood maximal operator is bounded on Lp(Rn, w). In what follows, Lp(X, w) denotes the class of all measurable functions f defined on X for which ||fw1/p||Lp(X) < ∞, where X is a measure space and w is a non-negative measurable function on X.
It has recently been proved that the factorization of Ap weights is a particular case of a general factorization theorem concerning positive sublinear operators. The case in which the operator is bounded from Lp(X, v) to Lp(Y, u), 1 < p < ∞, for u and v non-negative measurable functions on X and Y respectively, is treated in [8]. The case in which the operator is bounded from Lp(X, v) to Lq(X, u), 1 < p < q < ∞ is treated in [3].
Our first result is a factorization theorem for weights u and v associated to operators bounded from Lp(X, v) to Lq(Y, u) where X and Y are two, possibly different, measure spaces and p and q are any index between 1 and ∞.
@article{urn:eudml:doc:41680,
title = {Weighted inequalities through factorization.},
journal = {Publicacions Matem\`atiques},
volume = {35},
year = {1991},
pages = {141-153},
mrnumber = {MR1103612},
zbl = {0722.42012},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:41680}
}
Hernández, Eugenio. Weighted inequalities through factorization.. Publicacions Matemàtiques, Tome 35 (1991) pp. 141-153. http://gdmltest.u-ga.fr/item/urn:eudml:doc:41680/