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Meredith Sargent: Escaping non-tangentiality: a different approach to Julia-Caratheodory theory

Time: Wed 2019-11-27 13.15 - 14.15

Location: F11, KTH

Participating: Meredith Sargent, University of Arkansas

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A classical Julia-Carathéodory theorem states that if there is a sequence tending to \(\tau\) in the boundary of a domain \(D\) along which the Julia quotient is bounded, then the function \(\phi\) can be extended to \(\tau\) such that \(\phi\) is nontangentially continuous and differentiable at \(\tau\) and \(\phi(\tau)\) is in the boundary of \(\Omega\).
We develop a theory in the case of Pick functions where we consider sequences that approach the boundary in a controlled tangential way, yielding necessary and sufficient conditions for higher order regularity. In this talk, I will discuss the proof, including some of the technical details involved: amortization of the Julia Quotient, \(\gamma\)-regularity, and \(\gamma\)-auguries. I will also speak about some applications, including moment theory and the fractional Laplacian.