Speaker: Professor Alexander Menovschikov, HSE University

Title: On Mappings Generating Embedding Operators in Sobolev Classes on Metric
Measure Spaces

Abstract:

Let X =(X,ρ,µ) be a doubling metric measure space which supports the weak p-Poincar´e inequality. We consider bi-measurable homeomorphisms φ : Ω → Ω, of bounded domains Ω, Ω ⊂ X, which generate bounded composition operators on Newtonian–Sobolev spaces N1,p(Ω) → N1,q(Ω),1 < q ≤ p < ∞. We prove the Luzin N−1-property of such mappings with respect to capacities and obtain necessary and sufficient conditions on bi-measurable homeomorphisms that generate bounded composition operators on Newtonian–Sobolev spaces. We prove this by using special test functions generated by distance functions. On the base of the composition operators we consider Sobolev type embedding theorems in weak (p,q)-quasiconformal α-regular domains

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