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In this paper we present a definition of weighted Sobolev spaces on curves and find general conditions under which the spaces are complete for non-closed compact curves. We also prove the density of the polynomials in these spaces and, finally, we find conditions under which the multiplication operator is bounded in the space of polynomials. # 2002 Elsevier Science (USA)

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Abstract.Let Tf(x) = supiî>0 \SRf(x)\ where SR is the spherical partial sum operator. We show that T is bounded from the Lorentz space L ,(R") into L ooiR"), » = 0,1 when acting on radial functions and where p0 = _2iL " -_?«_

We generalize the classical Muckenhoupt inequality with two measures to three under appropriate conditions. As a consequence, we prove a 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.