BIRS Workshop Lecture Videos

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BIRS Workshop Lecture Videos

Extensions of BMO-functions and fixed points of contractive mappings in L2, I Shvartsman, Pavel


Let E be a closed subset of Rn of positive Lebesgue measure. We discuss a constructive algorithm which to every function f defined on E assigns its almostoptimalextensiontoafunctionF(f)∈BMO(Rn). Weobtaintheextension F(f) as a fixed point of a certain contractive mapping Tf : L2(Rn) → L2(Rn). The extension operator f → F(f) is non-linear, and in general it is not known whether there exists a continuous linear extension operator BMO(Rn)|E → BMO(Rn) for an arbitrary set E. In these talk we present a rather wide family of sets for which such extension op- erators exist. In particular, this family contains closures of domains with arbitrary internal and external cusps. The proof of this result is based on a solution to a similar problem for spaces of Lipschitz functions defined on subsets of a hyperbolic space.

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