Sharp two weight inequalities for commutators of Riemann-Liouville and Weyl fractional integral operators

ANA L. BERNARDIS; MARÍA LORENTE

INTEGRAL EQUATIONS AND OPERATOR THEORY

Birkhäuser

Año: 2008 p. 449 - 475

0378-620X

Let $b$ be a BMO function, $0<alpha<1$ and $I_{alpha ,b}^{+,k}$the commutator of order $k$ for the Weyl fractional integral. Inthis paper we prove weighted strong type $(p,p)$ inequalities($p>1$) and weighted endpoint estimates ($p=1$) for the operator$I_{alpha ,b}^{+,k}$ and for the pairs of weights of the type$(w,mathcal{M}w)$, where $w$ is any weight and $mathcal{M}$ is asuitable one-sided maximal operator. We also prove that, for$A_infty^+$ weights, the operator $I_{alpha ,b}^{+,k}$ iscontrolled in the $L^p(w)$ norm by a composition of the one-sidedfractional maximal operator and the one-sided Hardy-Littlewoodmaximal operator iterated $k$ times. These results improve thoseobtained by an immediate  application of the correspondingtwo-sided results and provide a different way to obtain knownresults about the operators $I_{alpha ,b}^{+,k}$. The sameresults can be obtained for the commutator of order $k$ for theRiemann-Liouville fractional integral $I_{alpha ,b}^{-,k}$.