The Singular Integral Operator Induced by Drury–Arveson Kernel
Tóm tắt
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.
Tài liệu tham khảo
Arcozzi, N., Rochberg, R., Sawyer, E.: Carleson measures for the Drury–Arveson Hardy space and other Besov–Sobolev spaces on complex balls. Adv. Math. 218, 1107–1180 (2008)
Arveson, W.: Subalgebras of \(C^*\)-algebras III: multivariable operator theory. Acta Math. 181, 159–228 (1998)
Cheng, G., Fang, X., Wang, Z., Yu, J.: Three measure theoretic properties for the Hardy kernel (2015) (under review)
Cheng, G., Fang, X., Wang, Z., Yu, J.: The hyper-singular cousin of the Bergman projection (2015) (under review)
Fang, Q., Xia, J.: Multipliers and essential norm on the Drury–Arveson space. Proc. Am. Math. Soc. 139, 2497–2504 (2011)
Rudin, W.: Function Theory in the Unit Ball of \(\mathbb{C}^n,\) Grundlehren der Mathematischen Wissenschaften, vol. 241. Springer, New York, Berlin (1980)
Zhu, K.: Spaces of Holomorphic Functions in the Unit Ball, Graduate Texts in Mathematics, vol. 226. Springer, New York (2005)