Weighted inequalities for commutators of fractional integrals on spaces of homogeneous type

ANA BERNARDIS; SILVIA HARTZSTEIN; GLADIS PRADOLINI

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

Academic press

Año: 2006 vol. 322 p. 825 - 846

0022-247X

Let $0<gamma<1$, $b$ a BMO function and $I_{gamma,b}^m$ thecommutator of order $m$ for the fractional integral. We prove twotype of weighted $L^p$ inequalities for $I_{gamma,b}^m$ in thecontext of the spaces of homogeneous type. The first oneestablishes that, for $A_infty$ weights, the operator$I_{gamma,b}^m$ is bounded in weighted $L^p$ norm by the maximaloperator $M_gamma(M^m)$, where $M_gamma$ is the fractionalmaximal operator and $M^m$ is the Hardy-Littlewood maximaloperator iterated $m$ times. The other inequality is a consequenceof the first one and shows that the operator $I_{gamma,b}^m$ isbounded from $L^pleft[M_{gammap}(M^{[(m+1)p]}w)(x)dmu(x) ight]$ to $L^p[w(x)dmu(x)]$, where$[(m+1)p]$ is the integer part of $(m+1)p$, and no condition onthe weight $w$ is required. From the first inequality we alsoobtain weighted $L^p$ - $L^q$ estimates for $I_{gamma,b}^m$generalizing the classical results of Muckenhoupt and Wheeden forthe fractional integral operator.