  In this paper we obtain some practical criteria to bound the multiplication
operator in Sobolev spaces with respect to measures in curves. As a consequence
of these results, we characterize the weighted Sobolev spaces with bounded
multiplication operator, for a large class of weights. To have bounded
multiplication operator has important consequences in Approximation Theory: it
implies the uniform bound of the zeros of the corresponding Sobolev orthogonal
polynomials, and this fact allows to obtain the asymptotic behavior of Sobolev
orthogonal polynomials. We also obtain some non-trivial results about these
Sobolev spaces with respect to measures; in particular, we prove a main result
in the theory: they are Banach spaces.
