Weighted inequalities through factorization.

Hernández, Eugenio. "Weighted inequalities through factorization.." Publicacions Matemàtiques 35.1 (1991): 141-153. <http://eudml.org/doc/41680>.

@article{Hernández1991,
abstract = {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 &lt; p &lt; ∞ 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) &lt; ∞, 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 &lt; p &lt; ∞, 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 &lt; p &lt; q &lt; ∞ 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 ∞.},
author = {Hernández, Eugenio},
journal = {Publicacions Matemàtiques},
keywords = {weighted inequalities; factorization theorem for weights; integral transform; Hardy operator; fractional integrals; Laplace transform; Riemann-Liouville operator},
language = {eng},
number = {1},
pages = {141-153},
title = {Weighted inequalities through factorization.},
url = {http://eudml.org/doc/41680},
volume = {35},
year = {1991},
}

TY - JOUR
AU - Hernández, Eugenio
TI - Weighted inequalities through factorization.
JO - Publicacions Matemàtiques
PY - 1991
VL - 35
IS - 1
SP - 141
EP - 153
AB - 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 &lt; p &lt; ∞ 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) &lt; ∞, 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 &lt; p &lt; ∞, 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 &lt; p &lt; q &lt; ∞ 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 ∞.
LA - eng
KW - weighted inequalities; factorization theorem for weights; integral transform; Hardy operator; fractional integrals; Laplace transform; Riemann-Liouville operator
UR - http://eudml.org/doc/41680
ER -