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

Abstract

The following open question was implicit in the literature: Are there singular integrals whose kernels satisfy the L r -Hormander condition for any r> 1 but not the L ∞ - Hormander 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 -Hormander condition for any r> 1 but not the L ∞ -Hormander condition. For a Young function A we introduce the notion of L A -Hormander. 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 Lp(w) norm of an iterate of the one-sided Hardy-Littlewood Maximal Operator, for any w in the A + class.