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.

Full Text
Published version (Free)

Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call