TWO-WEIGHTED COMPOSITION OPERATORS ON SOBOLEV SPACES AND QUASICONFORMAL ANALYSIS

We establish some equivalent conditions for a homeomorphism φ : D → D ′ of Euclidean domains in R n , n ≥ 2 , to induce a bounded composition operator φ ∗ : L p 1 ( D ′ ; ω ) ∩ Lip l ( D ′ ) → L q 1 ( D ; θ ) , where 1 < q ≤ p < ∞ , by the composition rule: φ ∗ ( f ) = f ∘ φ . Here ω : D ′ → ( 0 , ∞ ) is an arbitrary weight function on the domain D ′ , and θ : D → ( 0 , ∞ ) is some weight function in Muckenhoupt’s A q -class on the domain D . Moreover, we prove that the class of homeomorphisms under consideration is completely determined by the controlled variation of the weighted capacity of cubical condensers whose shells are concentric cubes.