The Singular Integral Operator Induced by Drury–Arveson Kernel

In this paper, we study the singular integral operator induced by the reproducing kernel of the Drury–Arveson space Kf(z)=∫Bnk(z,w)f(w)dv(w),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} Kf(z) =\int _{\mathbb {B}_n} k(z, w) f(w) dv(w), \end{aligned}$$\end{document}where k(z,w)=11-⟨z,w⟩,z,w∈Bn,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k(z, w)=\frac{1}{1-\langle z,w\rangle }, z,w\in \mathbb {B}_n,$$\end{document} which can be viewed as a higher dimensional continuation of Cheng et al. (Three measure theoretic properties for the Hardy kernel, preprint, 2015, The hyper-singular cousin of the Bergman projection, preprint, 2015), in which the authors consider the singular integral operators with the kernels as k(z,w)=1(1-zw¯)α,z,w∈D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k(z,w)=\frac{1}{(1-z\bar{w})^{\alpha }}, z,w\in \mathbb {D}$$\end{document} and α>0.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha >0.$$\end{document} By using more higher dimensional techniques, we establish various and satisfactory boundedness results about Lebesgue spaces and Drury–Arveson space.