Weighted Estimates for Singular Integral Operators Satisfying Hörmander’s Conditions of Young Type

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.