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K. Astala

Publications and source records attributed to K. Astala.

4 recordsLinked to original sources
arXiv2015

Chord-arc curves and the Beurling transform

We study the relation between the geometric properties of a quasicircle~$Γ$ and the complex dilatation~$μ$ of a quasiconformal mapping that maps the real line onto~$Γ$. Denoting by~$S$ the Beurling transform, we characterize Bishop-Jones quasicircles in terms of the boundedness of the operator~$(I-μS)$ on a particular weighted $L^2$~space, and chord-arc curves in terms of its invertibility. As an application we recover the~$L^2$ boundedness of the Cauchy integral on chord-arc curves.

math.CA↗
arXiv2009

Random Curves by Conformal Welding

We construct a conformally invariant random family of closed curves in the plane by welding of random homeomorphisms of the unit circle given in terms of the exponential of Gaussian Free Field. We conjecture that our curves are locally related to SLE$(κ)$ for $κ<4$.

math.CV↗
arXiv2009

Random Conformal Weldings

We construct a conformally invariant random family of closed curves in the plane by welding of random homeomorphisms of the unit circle. The homeomorphism is constructed using the exponential of $βX$ where $X$ is the restriction of the two dimensional free field on the circle and the parameter $β$ is in the "high temperature" regime $β<\sqrt 2$. The welding problem is solved by studying a non-uniformly elliptic Beltrami equation with a random complex dilatation. For the existence a method of Lehto is used. This requires sharp probabilistic estimates to control conformal moduli of annuli and they are proven by decomposing the free field as a sum of independent fixed scale fields and controlling the correlations of the complex dilation restricted to dyadic cells of various scales. For uniqueness we invoke a result by Jones and Smirnov on conformal removability of Hölder curves. We conjecture that our curves are locally related to SLE$(κ)$ for $κ<4$.

math.CV↗