Integral Equations and Operator Theory

, Volume 64, Issue 2, pp 177–192

Multipliers of Fractional Cauchy Transforms

Article

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

Abstract.

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 

Keywords.

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

Personalised recommendations