A class of multilinear bounded oscillation operators on measure spaces and applications

In recent years, dyadic analysis has attracted a lot of attention due to the A2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_2$$\end{document} conjecture. It has been well understood that in the Euclidean setting, Calderón–Zygmund operators can be pointwise controlled by a finite number of dyadic operators with a very simple structure, which leads to some significant weak and strong type inequalities. Similar results hold for Hardy–Littlewood maximal operators and Littlewood–Paley square operators. These owe to good dyadic structure of Euclidean spaces. Therefore, it is natural to wonder whether we could work in general measure spaces and find a universal framework to include these operators. In this paper, we develop a comprehensive weighted theory for a class of Banach-valued multilinear bounded oscillation operators on measure spaces, which merges multilinear Calderón–Zygmund operators with a quantity of operators beyond the multilinear Calderón–Zygmund theory. We prove that such multilinear operators and corresponding commutators are locally pointwise dominated by two sparse dyadic operators, respectively. We also establish three kinds of typical estimates: local exponential decay estimates, mixed weak type estimates, and sharp weighted norm inequalities. Beyond that, based on Rubio de Francia extrapolation for abstract multilinear compact operators, we obtain weighted compactness for commutators of specific multilinear operators on spaces of homogeneous type. A compact extrapolation allows us to get weighted estimates in the full range of exponents, while weighted interpolation for multilinear compact operators is crucial to the compact extrapolation. These are due to a weighted Fréchet–Kolmogorov theorem in the quasi-Banach range, which gives a characterization of relative compactness of subsets in weighted Lebesgue spaces. As applications, we illustrate multilinear bounded oscillation operators with examples including multilinear Hardy–Littlewood maximal operators on measure spaces, multilinear ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega $$\end{document}–Calderón–Zygmund operators on spaces of homogeneous type, multilinear Littlewood–Paley square operators, multilinear Fourier integral operators, higher order Calderón commutators, maximally modulated multilinear singular integrals, and q-variation of ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega $$\end{document}-Calderón–Zygmund operators.