TWO-WEIGHTED COMPOSITION OPERATORS ON SOBOLEV SPACES AND QUASICONFORMAL ANALYSIS

Abstract

We establish some equivalent conditions for a homeomorphism \(\varphi : D\rightarrow D'\) of Euclidean domains in \(\mathbb R^n\), \(n\ge 2\), to induce a bounded composition operator \(\varphi ^*: {L}^1_p(D';\omega ) \cap \text {Lip}_l(D')\rightarrow L^1_q(D;\theta )\), where \(1< q \le p<\infty\), by the composition rule: \(\varphi ^*(f)=f\circ \varphi\). Here \(\omega :D'\rightarrow (0,\infty )\) is an arbitrary weight function on the domain \(D'\), and \(\theta :D\rightarrow (0,\infty )\) 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.