Journé Commutators and Polynomial Approximations with Characterizations of Hardy Space and BMO

Abstract

A characterization of product and multi-paraproducts of bounded mean oscillations with upper bound for multi-parameter and higher order Journé commutators are studied. We describe an algorithm of nonlinear piecewise polynomial approximations by refinable functions, of functions of the classes 𝑊𝑝𝛼 and multivariate bound variation spaces of Wiener-Young type. We show the classes and maximal function characterization of Hardy spaces associated with operators, nonnegative self-adjoint operators satisfying Gaussian estimates with characterizations of Hardy and bounded mean oscillations by weak factorizations and commutators.