Abstract. This is the fourth article of our series. Here, we apply the results of [AM1] to study weighted norm inequalities for the Riesz transform of the Laplace-Beltrami operator on Riemannian manifolds and of subelliptic sum of squares on Lie groups, under the doubling volume property and Poincaré inequalities. 1. Introduction and

Abstract. We show that if an operator T is bounded on weighted Lebesgue space L2 (w) and obeys a linear bound with respect to the A2 constant of the weight, then its commutator [b, T] with a function b in BMO will obey a quadratic bound with respect to the A2 constant of the weight. We also prove that the kth-order commutator T k k−1 b = [b, Tb] will obey a bound that is a power (k + 1) of the A2 constant of the weight. Sharp extrapolation provides corresponding Lp (w) estimates. The results are sharp in terms of the growth of the operator norm with respect to the Ap constant of the weight for all 1 < p < ∞, all k, and all dimensions, as examples involving the Riesz transforms, power functions and power weights show. date: Feb-8-2010 1.

Abstract. A classical theorem of Coifman, Rochberg, and Weiss on commutators of singular integrals is extended to the case of generalized L p spaces with variable exponent. 1.

Abstract. We prove weighted norm inequalities for fractional powers of elliptic operators together with their commutators with BMO functions, encompassing what is known for the classical Riesz potentials and elliptic operators with Gaussian domination by the classical heat operator. The method relies upon a good-λ method that does not use any size or smoothness estimates for the kernels. 1.

Abstract. In this paper, we establish a sharp inequality for some multilinear operators related to certain integral operators. The operators include Calderón-Zygmund singular integral operator, Littlewood-Paley operator, Marcinkiewicz operator and Bochner-Riesz operator. As application, we obtain the weighted norm inequalities and Llog L type estimate for the multilinear operators.

inequalities and dyadic harmonic analysis

In this paper we extend a Calderón-Zygmund commutator-type estimate. This estimate enables us to prove an embedding result concerning weighted function spaces.