On geometric characterizations of mappings generating composition operators on Sobolev spaces (Q6598492)

The author investigates the family of linear operators \(\varphi^*: f \mapsto f \circ \varphi\), \(f \in L^1_p (\tilde{\Omega})\). Here \( {\Omega}, \tilde{\Omega}\) are domains in \(\mathbb{R}^n\), \(\varphi: \Omega \to \tilde{\Omega}\) is a homeomorphism and \(L^1_p (\Omega)\) denotes the first order Sobolev space built on \(L_p (\Omega)\). To prove \(\varphi^* \in \mathcal{L}(L^1_p (\tilde{\Omega}),L^1_q (\Omega)) \), a major role is played by the geometric \((p,q)\)-dilatation, \(q <p\),\N\[\NH_{p,q}^\lambda (x,r;\Phi_{p,q}):= \frac{L^p_\varphi (x,r)\, r^{n-p}}{|\varphi (B(x,\lambda r))|}\Big( \frac{|B(x,r)|}{\Phi_{p,q}(B(x,\lambda r))}\Big)^\frac{p-q}{q} \, , \quad \lambda \ge 1\, , \ x \in \Omega\, , \ r>0\, ,\N\]\Nwhere \(L_\varphi (x,r):= \max_{|x-y|=r}\, |\varphi (x)- \varphi (y)|\) and \(\Phi_{p,q}\) is a bounded monotone countable-additive absolutely continuous set function defined on open subsets of \(\Omega\). Then the author proves the following. Let \(1 <q < p < \infty\). If \(\varphi: \Omega \to \tilde{\Omega}\) is a homeomorphism that satisfies\N\[\N\limsup_{r\to 0} \, H_{p,q}^\lambda (x,r;\Phi_{p,q}) \le H_{p,q}^\lambda (x,r;\Phi_{p,q}) < \infty \quad\mbox{for all }x \in \Omega,\quad 1<q<p<\infty,\N\]\Nfor some constant \( H_{p,q}^\lambda (x,r;\Phi_{p,q}) \), then \(\varphi\) generates a bounded composition operator \(\varphi^* \in \mathcal{L}(L^1_p (\tilde{\Omega}),L^1_q (\Omega))\). A converse statement is also true. In addition, the limiting case \(1 < p=q< \infty\) is treated.