Complex Analysis and Operator Theory

, Volume 12, Issue 4, pp 917–929 | Cite as

The Singular Integral Operator Induced by Drury–Arveson Kernel

Article

Abstract

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.

Keywords

Singular integral operator Unit ball Reproducing kernel  Drury–Arveson space 

Mathematics Subject Classification

47B34 47G10 

Notes

Acknowledgments

G. Cheng is supported by NSFC (11471249) and Zhejiang Provincial NSFC (LY14A010021).

References

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Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  1. 1.School of MathematicsWenzhou UniversityWenzhouPeople’s Republic of China
  2. 2.City College of Wenzhou UniversityWenzhouPeople’s Republic of China

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