Muckenhoupt inequality with three measures and applications to Sobolev orthogonal polynomials

We generalize the Muckenhoupt inequality with two measures to three under certain conditions. As a consequence, we prove a very simple characterization of the boundedness of the multiplication operator and thus of the boundedness of the zeros and the asymptotic behavior of the Sobolev orthogonal polynomials, for a large class of measures which includes the most usual examples in the literature.