Compactness Characterizations of Commutators on Ball Banach Function Spaces

Abstract

Let $X$ be a ball Banach function space on ${\mathbb{R}}^n$. Let $\Omega$ be a Lipschitz function on the unit sphere of ${\mathbb{R}}^n$,which is homogeneous of degree zero and has mean value zero, and let $T_{\Omega }$ be the convolutional singular integral operator with kernel $\Omega (\cdot )/|\cdot {|}^n$. In this article, under the assumption that the Hardy--Littlewood maximal operator $\mathcal{M}$ is bounded on both $X$ and its associated space, the authors prove that the commutator $[b,T_{\Omega }]$ is compact on $X$ if and only if $b\in \mathrm{CMO}({\mathbb{R}}^n)$. To achieve this, the authors mainly employ three key tools: some elaborate estimates, given in this article, on the norm in $X$ of the commutators and the characteristic functions of some measurable subset,which are implied by the assumed boundedness of $\mathcal{M}$ on $X$ and its associated space as well as the geometry of ${\mathbb{R}}^n$; the complete John--Nirenberg inequality in $X$ obtained by Y. Sawano et al.; the generalized Fréchet--Kolmogorov theorem on $X$ also established in this article. All these results have a wide range of applications. Particularly, even when $X:=L^{p(\cdot )}({\mathbb{R}}^n)$ (the variable Lebesgue space), $X:=L^{\vec{p}}({\mathbb{R}}^n)$ (the mixed-norm Lebesgue space), $X:=L^{\Phi }({\mathbb{R}}^n)$ (the Orlicz space), and $X:=(E_{\Phi }^q{)}_t({\mathbb{R}}^n)$ (the Orlicz-slice space or the generalized amalgam space), all these results are new.

36 pages, Submitted. arXiv admin note: text overlap with arXiv:1911.04953, arXiv:1906.03653

References in corpus (1)

• The Fatou property of block spaces