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

Sobolev spaces with respect to measures in curves and zeros of Sobolev orthogonal polynomials