Weighted Sobolev spaces on curves
The aim of this paper is to define weighted Sobolev spaces \(W^{k,p}(\gamma,\mu)\) on a simple curve that has a locally absolutely continuous parametrization, \(\gamma\colon I\to\mathbb{C}\). The authors need to introduce the classes \(\text{AC}^{(k)}(\gamma), {\text{AC}}^{(k)}_{\text{loc}}(\gamma)\) of \(k\) times differentiable functions on \(\gamma\) in order to define the derivatives along \(\gamma\). If \(1\leq p\leq\infty\), for every arc \([z_1,z_2]\subset\gamma\), by means of a Muckenhoupt-type weighted norm inequality they also define the class \(B_p([z_1,z_2])\) of weights \(w\), which contains the classical \(A_p\) class when \(\gamma\) assumes values in \({\mathbb R}\). Next, if \(\mu=(\mu_0,\mu_1,\dots,\mu_k)\) is a given vector measure on \(\gamma\), they denote by \(\Omega_j\) the set of those points in \(\gamma\) with a neighborhood \(V\subset\gamma\) such that \(w_j\in B_p(V)\). Then they introduce the completion of \(\mu\) and the set \(\Omega^{(j)}\) of \(j\)-regular points \(z_0\in\gamma\), \(1\leq j\leq k\). These sets are used in the definition of a \(p\)-admissible vector measure \(\mu\), and the corresponding weighted Sobolev space \(W^{k,p}(\gamma,\mu)\) contains all the equivalence classes of functions \(f\colon\gamma\to\mathbb{C}\) such that \(f^{(j)}\in{\text{AC}}^{(1)}_{\text{loc}}(\Omega^{(j)})\) for \(0\leq j<k\) and \(\|f^{(j)}\|_{L^p(\gamma,\mu_j)}<\infty\) for \(0\leq j\leq k\), with respect to the seminorm \(\|f\|_{W^{k,p}(\gamma,\mu)}:=\sum^k_{j=0}\|f^{(j)}\|_{L^p(\gamma,\mu_j)}\). With these definitions, the authors find conditions under which these new spaces are complete, they study the density of the polynomials and the problem of the boundedness of the multiplication operator \(f\mapsto zf\).
scientific article; zbMATH DE number 1891506
scientific article