The boundedness of multilinear Calderón-Zygmund operators on weighted and variable Hardy spaces

Abstract

We prove norm estimates for multilinear fractional integrals acting on weighted and variable Hardy spaces. In the weighted case we develop ideas we used for multilinear singular integrals [7]. For the variable exponent case, a key element of our proof is a new multilinear, off-diagonal version of the Rubio de Francia extrapolation theorem. γ−mn−N for some sufficiently large integer N. We define the multilinear fractional Calderón-Zygmund operator T γ by T γ (f 1 , . . . , f m )(x) = R mn K γ (x, y 1 , . . . , y m )f 1 (y 1 ) · · · f m (y m ) d y.

Multilinear fractional type operators and their commutators on Hardy spaces with variable exponents

“…The first goal of this article is to show that multilinear fractional type operators are bounded from product of Hardy spaces with variable exponents into Lebesgue or Hardy spaces with variable exponents via atomic decompositions theory. We also remark that some boundedness of many types of multilinear operators on some variable Hardy spaces have established in [9,36,37,39]. After we were completing this paper we learned that some of similar results had been also established independently by Cruz-uribe et al [10] independently though our approaches are very different.…”

On the atomic and molecular decomposition of weighted Hardy spaces

“…The verification of the convergence in L s for the infinite atomic decomposition was sometimes an overlooked detail. As far as the author knows, the above result has been proved for w-(p, ∞, d) atoms in R by J. García-Cuerva in [5], and for w-(p, ∞, d) atoms in R n by D. Cruz-Uribe et al in [3].…”

A new approach to norm inequalities on weighted and variable Hardy spaces

“…The advantage of such a decomposition is that it allows the interchange of the operator and the sum without having to worry about the convergence of the sum, and reduces the problem to estimating the operator on individual atoms. In a previous paper [9] we extended the finite atomic decomposition to weighted Hardy spaces; here we prove it for variable Hardy spaces, generalizing a result proved in [10].…”

“…Theorem 1.11 is new. However, a slightly weaker result was implicitly proved as a special case of a result for multilinear Calderón-Zygmund operators recently proved in [9].…”