We present a general framework to deal with commutators of singular integral operators with BMO functions. Hörmander type conditions associated with Young functions are assumed on the kernels. Coifman type estimates, weighted norm inequalities and two-weight estimates are considered. We give applications to homogeneous singular integrals, Fourier multipliers and one-sided operators.

The following open question was implicit in the literature: Are there singular integrals whose kernels satisfy the L r -Hörmander condition for any r > 1 but not the L ∞ -Hörmander condition? We prove that the one-sided discrete square function, studied in ergodic theory, is an example of a vector-valued singular integral whose kernel satisfies the L r -Hörmander condition for any r > 1 but not the L ∞ -Hörmander condition. For a Young function A we introduce the notion of L A -Hörmander. We prove that if an operator satisfies this condition, then one can dominate the L p (w) norm of the operator by the L p (w) norm of a maximal function associated to the complementary function of A, for any weight w in the A ∞ class and 0 < p < ∞. We use this result to prove that, for the one-sided discrete square function, one can dominate the L p (w) norm of the operator by the L p (w) norm of an iterate of the one-sided Hardy-Littlewood Maximal Operator, for any w in the A + ∞ class.

We extend the results by Jones and Rosenblatt about the series of the differences of differentiation operators along lacunary sequences to BMO and to the setting of weighted L pspaces. We use a different approach which allows to establish that the one-sided Sawyer A p weights are the natural ones to study the boundedness and convergence of that series in weighted spaces.

The purpose of this paper is to prove strong type inequalities with pairs of related weights for commutators of one-sided singular integrals (given by a Calder6n-Zygmund kernel with support in (-oo, 0)) and the one-sided discrete square function. The estimate given by C. Segovia and J. L. Torrea is improved for these one-sided operators giving a wider class of weights for which the inequality holds.2000 Mathematics subject classification: primary 42B20.

No abstract