Homogenization of iterated singular integrals with applications to random quasiconformal maps
Kari Astala
University of Helsinki, FinlandSteffen Rohde
University of Washington, Seattle, USAEero Saksman
University of Helsinki, FinlandTerence Tao
University of California Los Angeles, USA
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Abstract
We study homogenization of iterated randomized singular integrals and homeomorphic solutions to the Beltrami differential equation with a random Beltrami coefficient. More precisely, let be a sequence of normalized homeomorphic solutions to the planar Beltrami equation , where the random dilatation satisfies and has locally periodic statistics, for example of the type
where decays rapidly in , the random variables are i.i.d., and . We establish the almost sure and local uniform convergence as of the maps to a deterministic quasiconformal limit . This result is obtained as an application of our main theorem, which deals with homogenization of iterated randomized singular integrals. As a special case of our theorem, let be translation and dilation invariant singular integrals on , and consider a -dimensional version of , e.g., as defined above or within a more general setting, see Definition 3.4 below. We then prove that there is a deterministic function such that almost surely, as weakly in , for .
Cite this article
Kari Astala, Steffen Rohde, Eero Saksman, Terence Tao, Homogenization of iterated singular integrals with applications to random quasiconformal maps. Rev. Mat. Iberoam. 38 (2022), no. 7, pp. 2285–2336
DOI 10.4171/RMI/1392