Convolution Operators with Fractional Measures Associated to Holomorphic Functions

Abstract

Let Ω be an open set in the complex plane and let ρ be a holomorphic function on Ω. Let K be a compact subset of Ω with nonempty interior such that 0 ∉ ∂K. Let μ be the Borel measure of R ⁴ ≃ C ² given by

μ(E = ∫ _K χE(z, ρ(z))|z|^γ−2 dσ(z)

where 0 < γ ≦ 2 and dσ(x ₁ + ix ₂) = dx ₁ dx ₂ denotes the Lebesgue measure on C. Let T _μ be the convolution operator T _μ f = μ * f. In this paper we characterize the type set E _μ associated to T _μ.