Sharp maximal estimates for multilinear commutators of multilinear strongly singular Calderón–Zygmund operators and applications

Abstract

In this paper, we aim to establish the sharp maximal pointwise estimates for the multilinear commutators generated by multilinear strongly singular Calderón–Zygmund operators and BMO functions or Lipschitz functions, respectively. As applications, the boundedness of these multilinear commutators on product of weighted Lebesgue spaces are obtained. It is interesting to note that there is no size condition assumption for the kernel of the multilinear strongly singular Calderón–Zygmund operator. Due to the stronger singularity for the kernel of the multilinear strongly singular Calderón–Zygmund operator, we need to be more careful in estimating the mean oscillation over the small balls to get the sharp maximal function estimates.

Weighted norm inequalities for multilinear strongly singular Calderón–Zygmund operators on RD‐spaces

“…The above works inspired us to extend the multilinear strongly singular Calderón-Zygmund operators in [30,33] to the context of RD-spaces; more precisely, we get the following results in this paper.…”

“…In particular, their boundedness properties in the setting of unweighted Lebesgue spaces as well as endpoint mappings involving the space of functions of bounded mean oscillations $$BMO(\mathbb {R}^n)$$ and the Hardy space $$H^1(\mathbb {R}^n)$$ were established. Another kind of multilinear strongly singular Calderón–Zygmund operators can be found in [30, 33]. These works provided a comprehensive approach to the sharp maximal estimate for this class of multilinear singular integrals and constructed the strong type and endpoint estimates for these multilinear operators and their iterated commutators with $$BMO$$ functions.…”

“…These works provided a comprehensive approach to the sharp maximal estimate for this class of multilinear singular integrals and constructed the strong type and endpoint estimates for these multilinear operators and their iterated commutators with $$BMO$$ functions. For other results on strongly singular Calderón–Zygmund operators and related topics, one can refer to [29–32, 34–36] and so on.…”

Multilinear strongly singular integral operators on non-homogeneous metric measure spaces

“…The strongly singular non-convolution operators were introduced and researched by Alvarez and Milman [1,2], whose properties are similar to those of Calderón-Zygmund operators, but the kernel is more singular near the diagonal than those of the standard case. Furthermore, Lin and Lu [25][26][27][28] obtained the boundedness for the strongly singular integral and its commutators on Lebesgue spaces, Morrey spaces, and Hardy spaces.…”

The Weighted Morrey Boundedness of Multilinear Singular Integral Operators on RD-Spaces

“…Since then, a lot of papers focus on this field of research. The following are some recent development: Grafakos et al [8] provided a self-contained proof of the multilinear extension of the Marcinkiewicz real method interpolation theorem; Mo and Lu [18] established the boundedness of commutators generated by the multilinear Calderón-Zygmund type singular integrals and Lipschitz functions on the Triebel-Lizorkin space and Lipschitz spaces; Lin et al [17] obtained some sharp maximal estimates for multilinear commutators of multilinear strongly singular Calderón-Zygmund operators.…”