Weak and strong type estimates for multilinear Calder\xf3n\u2013Zygmund operators on differential forms

Xuexin Li,Yuming Xing,Jinling Niu

doi:10.1186/s13660-021-02628-5

Abstract

In this paper, we define the multilinear Calderón–Zygmund operators on differential forms and prove the end-point weak type boundedness of the operators. Based on nonhomogeneous A-harmonic tensor, the Poincaré-type inequalities for multilinear Calderón–Zygmund operators on differential forms are obtained.

Highlights

1 Introduction The multilinear Calderón–Zygmund theory was originally introduced by Coifman and Meyer [1,2,3] in their study of certain singular integral operators, such as Calderón commutators, paraproducts, and pseudodifferential operators
In [5] and [6], the maximal operator associated with multilinear Calderón–Zygmund singular integrals was introduced and used to obtain the weighted norm estimates for multilinear singular integrals
Stockdale and Wick [7] provided an alternative proof of the weaktype (1, . . . , 1; 1/m) estimates for m-multilinear Calderón–Zygmund operators on Rn first proved by Grafakos and Torres

Summary

Introduction

The multilinear Calderón–Zygmund theory was originally introduced by Coifman and Meyer [1,2,3] in their study of certain singular integral operators, such as Calderón commutators, paraproducts, and pseudodifferential operators. Grafakos and Torres studied systematically on the multilinear Calderón–Zygmund operators in [4] They proved an end-point weak type estimate and obtained the strong type Lp1 × · · · × Lpm → Lp boundedness results for multilinear Calderón–Zygmund operators by the classical interpolation method. By combining the Calderón–Zygmund decomposition with some skillful techniques, we establish the end-point weak type boundedness of multilinear Calderón–Zygmund operators on differential forms which includes the result of multilinear Calderón–Zygmund. 3, we define the multilinear Calderón– Zygmund operators on differential forms and prove the end-point weak type boundedness of multilinear Calderón–Zygmund operators on differential forms in Theorem 1. 4. the result is extended to obtain the Poincaré-type inequality for the multilinear Calderón–Zygmund operator on a bounded convex domain in Theorem 3

Preliminaries

Methods