We show that the class of weights w for which the Calderón operator is bounded on L p (w) can be used to develop a theory of real interpolation which is more general and exhibits new features when compared to the usual variants of the Lions-Peetre methods. In particular we obtain extrapolation theorems (in the sense of Rubio de Francia's theory) and reiteration theorems for these methods. We also consider interpolation methods associated with the classes of weights for which the Calderón operator is bounded on weighted Lorentz spaces and obtain similar results. We extend the commutator theorems associated with the real method of interpolation in several directions. We obtain weighted norm inequalities for higher order commutators as well as commutators of fractional order. One application of our results gives new weighted norm inequalities for higher order commutators of singular integrals with multiplications by BMO functions. We also introduce analogs of the space BMO in order to consider the relationship between commutators for Calderón type operators and their corresponding classes of weights.
Abstract. The class of functions for which the commutator with the HardyLittlewood maximal function or the maximal sharp function are bounded on L q are characterized and proved to be the same. is bounded in L q , for some (and for all) q ∈ (1, ∞). The cancellation implied by the commutator operation and the properties of singular integrals are crucial for the validity of the result. Later in [3], using real interpolation techniques, Milman and Schonbeck proved a commutator result that applies to the Hardy-Littlewood maximal operator M as well as the sharp maximal operator. In fact the commutator result is valid for a large class of nonlinear operators which we now describe.Let us say that T is a positive quasilinear operator if it is defined on a suitable class of locally integrable functions D(T ) and satisfiesWe have (cf.[3]) Proposition 1. Let b be a nonnegative BM O function and suppose that T is a positive quasilinear operator which is bounded on L q (w), for some 1 ≤ q < ∞ and for all w weights belonging to the Muckenhoupt class A r for some r ∈ [1, +∞).In particular the result applies to the maximal operator and the sharp maximal function. Note that since the Hardy-Littlewood maximal operator M is a positive
Abstract. We show that any random vector uniformly distributed on any hyperplane projection of B n 1 or B n ∞ verifies the variance conjecture Var |X| 2 ≤ C supFurthermore, a random vector uniformly distributed on a hyperplane projection of B n ∞ verifies a negative square correlation property and consequently any of its linear images verifies the variance conjecture.
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