Integral Equations and Operator Theory

, Volume 64, Issue 2, pp 177–192

Multipliers of Fractional Cauchy Transforms


DOI: 10.1007/s00020-009-1681-2

Cite this article as:
Doubtsov, E. Integr. equ. oper. theory (2009) 64: 177. doi:10.1007/s00020-009-1681-2


Let Bn denote the unit ball of \(\mathbb{C}^n\), n ≥ 2. Given an α > 0, let \(\mathcal{K}_\alpha(n)\) denote the class of functions defined for \(z \in B_n\) by integrating the kernel \((1 - \langle z, \zeta \rangle)^{-\alpha}\) against a complex-valued measure on the sphere \(\{\zeta \in \mathbb{C}^n : |\zeta| = 1\}\). Let \(\mathcal{H}ol(B_n)\) denote the space of holomorphic functions in the ball. A function \(g \in \mathcal{H}ol(B_n)\) is called a multiplier of \(\mathcal{K}_\alpha (n)\) provided that \(fg \in \mathcal{K}_\alpha (n)\) for every \(f \in \mathcal{K}_\alpha (n)\). In the present paper, we obtain explicit analytic conditions on \(g \in \mathcal{H}ol(B_n)\) which imply that g is a multiplier of \(\mathcal{K}_\alpha(n)\). Also, we discuss the sharpness of the results obtained.

Mathematics Subject Classification (2000).

Primary 32A26 Secondary 32A37, 42B35, 46E15, 46J15 


Fractional Cauchy transform Hardy–Sobolev space holomorphic Lipschitz space pointwise multiplier 

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2009

Authors and Affiliations

  1. 1.St. Petersburg Department of V.A. Steklov Mathematical InstituteSt. PetersburgRussia

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