Generalized weighted Sobolev spaces and applications to Sobolev orthogonal polynomials. I (Q702088)

The authors give a definition of a Sobolev space with respect to a wide class of measures. Since the section ``Definitions'' contains more than 6 pages, it is not possible to give it here, but as an example the authors indicate the following: \(W^{2,2}([0,6],\mu)\), where \[ \begin{aligned} \| f\|^2_{ W^{2,2}([0,6],\mu)}&= \int_4^6| f| ^2+| f(6)| ^2+\int_0^1| f'(1)| ^2\sqrt{x}+\\ &+\int_3^5| f'| ^2\sqrt{x-3}+| f'(1)| ^2+\int_1^3| f''| ^2(3-x). \end{aligned} \] There are proved generalizations of classical results (for instance of Hardy and Muckenhoupt inequalities), the question ``when the evaluation functional of \(f\) (or \(f^{(j)})\) in a point is a bounded operator in \(W^{k,p}(\Omega,\mu)?\)'' is answered. The main result is a general condition under which the weighted Sobolev spaces are complete.