Commutators of Pseudodifferential Operators on Weighted Hardy Spaces

Yu-Long Deng,Antonio Masiello

doi:10.1155/2022/8851959

Abstract

In this paper, we establish an endpoint estimate for the commutator, b , T , of a class of pseudodifferential operators T with symbols in Hörmander class S ρ , δ m R n . In particular, there exists a nontrivial subspace of B M O R n such that, when b belongs to this subspace, the commutators b , T is bounded from H ω 1 R n into L ω 1 R n , which we extend the well-known result of Calderón-Zygmund operators.

Highlights

Remark that if the symbol a(x, ξ) satisfies some particular assumptions, pseudodifferential operator T in Lmρ,δ is a Calderon-Zygmund operator
BMO(Rn) such that, the commutators of pseudodifferential operators T is bounded on weighted Hardy space H1ω(Rn), where the operators T associated with the symbols in the Holmander class Smρ,δ(Rn) and ω ∈ Ap(Rn)
It is worthy to pointing out that in [21], Liang et al found a proper subspaces of BMO(Rn), such that, the commutators of Calderon-Zygmund operator is bounded on weighted

Summary

Introduction

Remark that if the symbol a(x, ξ) satisfies some particular assumptions, pseudodifferential operator T in Lmρ,δ is a Calderon-Zygmund operator (see, [7]). BMO(Rn) such that, the commutators of pseudodifferential operators T is bounded on weighted Hardy space H1ω(Rn), where the operators T associated with the symbols in the Holmander class Smρ,δ(Rn) and ω ∈ Ap(Rn). Given an infinitely differentiable function f ∈ Rn with compact supports and symbol a(x, ξ) ∈ Smρ,δ(Rn), the pseudodifferential operator T is defined by Let b ∈ BMO(Rn) and T be a Calderon-Zygmund operator.

Results

Conclusion