Composition operators in weighted Sobolev spaces on Carnot groups
Let \(\Omega\) be an open set in the Carnot group \({\mathbb G}\). Let \(w\) be a Muckenhoupt weight. Let \(L^1_q (\Omega, w)\) be the weighted homogeneous Sobolev space and let \(W^1_q (\Omega, w)\) be the usual weighted Sobolev space for any \(1<q<\infty\). Let \(\varphi^*: f \mapsto f \circ \varphi\) be a composition operator. The first main result of the paper characterizes which \(\varphi\) generate bounded composition operators \[ \varphi^*: \quad L^1_p (\Omega', v) \hookrightarrow L^1_q (\Omega, w), \qquad 1<q\leq p < \infty. \] The second main result studies under which circumstances the embedding \[ \text{id}: \quad W^1_p (\Omega', v) \hookrightarrow L_s (\Omega', v) \] is bounded or, in addition, compact.
scientific article; zbMATH DE number 5990916
scientific article