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 $$\begin{aligned} Kf(z) =\int _{\mathbb {B}_n} k(z, w) f(w) dv(w), \end{aligned}$$ where $$k(z, w)=\frac{1}{1-\langle z,w\rangle }, z,w\in \mathbb {B}_n,$$ 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 $$k(z,w)=\frac{1}{(1-z\bar{w})^{\alpha }}, z,w\in \mathbb {D}$$ and $$\alpha >0.$$ By using more higher dimensional techniques, we establish various and satisfactory boundedness results about Lebesgue spaces and Drury–Arveson space.