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We study non-regularity of growth of the fractional Cauchy transform $$\begin{aligned} f(z)=\int _{-\pi }^{\pi } \frac{d\psi (t)}{(1-ze^{-it})^\alpha }, \quad \alpha >0, \psi \in BV[-\pi ,\pi ], \end{aligned}$$f(z)=∫-ππdψ(t)(1-ze-it)α,α>0,ψ∈BV[-π,π],in terms of the… Click to show full abstract
We study non-regularity of growth of the fractional Cauchy transform $$\begin{aligned} f(z)=\int _{-\pi }^{\pi } \frac{d\psi (t)}{(1-ze^{-it})^\alpha }, \quad \alpha >0, \psi \in BV[-\pi ,\pi ], \end{aligned}$$f(z)=∫-ππdψ(t)(1-ze-it)α,α>0,ψ∈BV[-π,π],in terms of the modulus of continuity of the function $$\psi $$ψ. Sharp estimates of the lower logarithmic order of f are found. In the case $$\alpha \in (0,1)$$α∈(0,1) the estimates are of different form than that for the logarithmic order.
Journal Title: Analysis and Mathematical Physics
Year Published: 2019
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