Asymptotic behavior inside the disk for Lebesgue Sobolev orthogonal polynomials

In the present paper we study the behavior, inside and on the unit disk, of the monic orthogonal polynomials with respect to the following Sobolev inner product where is a finite positive Borel measure on with infinite support and is the normalized Lebesgue measure. Since strong asymptotics for the monic Sobolev orthogonal polynomials are well-known outside the unit disk, our aim, in this paper, is to study when the asymptotic formula can be extended up to the boundary and inside the disk.