Brennan's conjecture for composition operators on Sobolev spaces (Q2839013)

For conformal homeomorphisms \(\varphi:\Omega\to\mathbb D\) of a simply connected domain \(\Omega\subset\mathbb R^2\) with nonempty boundary onto the unit disk \(\mathbb D\), Brennan's conjecture reads NEWLINE\[NEWLINE\int_{\Omega}|\varphi'(z)|^sd\mu<\infty,\;\;z=x+iy,\;\;{4\over3}<s<4,NEWLINE\]NEWLINE where \(d\mu\) is the Lebesgue measure.NEWLINENEWLINEFor an Euclidean domain \(\Omega\subset\mathbb R^n\), \(n\geq2\), the homogeneous Sobolev space \(L_p^1(\Omega)\), \(1\leq p\leq\infty\), is the space of all locally summable, weakly differentiable functions \(f:\Omega\to\mathbb R\) with the finite seminorm NEWLINE\[NEWLINE\|f|L_p^1(\Omega)\|=\left(\int_{\Omega}|\nabla f(z)|^pd\mu\right)^{{1\over p}},\;\;1\leq p<\infty,\;\;\|f|L^1_{\infty}(\Omega)\|=\text{esssup}_{z\in\Omega}|\nabla f(z)|.NEWLINE\]NEWLINE Let \(\varphi:\Omega\to\Omega'\) be a homeomorphism of Euclidean domains \(\Omega,\Omega'\subset\mathbb R^n\). It is said that \(\varphi\) generates a bounded composition operator \(\varphi^*:L_p^1(\Omega')\to L_q^1(\Omega)\), \(1\leq q\leq p\leq\infty\), by the composition rule \(\varphi^*(f)=f\circ\varphi\), if, for any \(f\in L_p^1(\Omega')\), \(\varphi^*(f)\in L_q^1(\Omega)\) and there is \(K<\infty\) such that \(\|\varphi^*(f)|L_q^1(\Omega)\|\leq K\|f|L_p^1(\Omega')\|\).NEWLINENEWLINEIt is proved that Brennan's conjecture is equivalent to the boundedness of composition operators generated by conformal homeomorphisms \(\varphi:\Omega\to\mathbb D\). Namely, Brennan's conjecture holds for \(s\in(4/3,4)\) if and only if any conformal homeomorphism \(\varphi:\Omega\to\mathbb D\) generates a bounded composition operator \(\varphi^*:L_p^1(\mathbb D)\to L^1_{q(p,s)}(\Omega)\) for any \(p\in(2,\infty)\) and \(q(p,s)=ps/(p+s-2)\).NEWLINENEWLINEThe authors give a geometric interpretation of Brennan's conjecture in terms of integrability of \(p\)-distortion and prove the Inverse Composition Theorem: A diffeomorphism \(\varphi:\Omega\to\Omega'\) of plane domains \(\Omega,\Omega'\) generates a bounded composition operator \(\varphi^*:L_p^1(\Omega')\to L_q^1(\Omega)\), \(1<q<p<\infty\), if and only if \(\varphi^{-1}:\Omega'\to\Omega\) generates a bounded composition operator \((\varphi^{-1})^*:L_{q'}^1(\Omega)\to L_{p'}^1(\Omega')\), \(1/p+1/p'=1\), \(1/q+1/q'=1\).