A simple characterization of weighted Sobolev spaces with bounded multiplication operator

In this paper we give a simple characterization of weighted Sobolev spaces (with piecewise monotone weights) such that the multiplication operator is bounded: it is bounded if and only if the support of @m"0 is large enough. We also prove some basic properties of the appropriate weighted Sobolev spaces. 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.

[1]  José M. Rodríguez,et al.  The Multiplication Operator in Sobolev Spaces with Respect to Measures , 2001, J. Approx. Theory.

[2]  Ana Portilla,et al.  Weierstrass's Theorem in Weighted Sobolev Spaces With $k$ Derivatives , 2007 .

[3]  T. Kilpeläinen,et al.  Weighted Sobolev spaces and capacity. , 1994 .

[4]  H. P. Cabrera,et al.  Zero Location and nth Root Asymptotics of Sobolev Orthogonal Polynomials , 1999 .

[5]  Lance L. Littlejohn,et al.  Orthogonal polynomials and approximation in Sobolev spaces , 1993 .

[6]  José M. Rodríguez,et al.  Weighted Sobolev Spaces on Curves , 2002, J. Approx. Theory.

[7]  G. Lo.  Zero Location and n th Root Asymptotics of Sobolev Orthogonal Polynomials , 1999 .

[8]  A Kolmogorov-Szegö-Krein type condition for weighted Sobolev spaces , 2005 .

[9]  Arieh Iserles,et al.  On polynomials orthogonal with respect to certain Sobolev inner products , 1991 .

[10]  J. García-cuerva,et al.  Weighted norm inequalities and related topics , 1985 .

[11]  Weierstrass' Theorem in Weighted Sobolev Spaces , 2001 .

[12]  H. Triebel.  Interpolation Theory, Function Spaces, Differential Operators , 1978 .

[13]  José M. Rodríguez,et al.  Weierstrass' theorem with weights , 2004, J. Approx. Theory.

[14]  Bohumír Opic,et al.  How to define reasonably weighted Sobolev spaces , 1984 .

[15]  Sobolev orthogonal polynomials in the complex plane , .

[16]  Arieh Iserles,et al.  Orthogonality and approximation in a Sobolev space , 1990 .

[17]  Sobolev orthogonal polynomial in the complex plane , 2001 .

[18]  José M. Rodríguez,et al.  Generalized Weighted Sobolev Spaces and Applications to Sobolev Orthogonal Polynomials I , 2004 .

[19]  B. Muckenhoupt,et al.  Weighted norm inequalities for the Hardy maximal function , 1972 .

[20]  The Multiplication Operator in Sobolev Spaces with Respect to Measures , 2001 .

[21]  VenancioAlvarez.  GENERALIZED WEIGHTED SOBOLEV SPACES AND APPLICATIONS TO SOBOLEV ORTHOGONAL POLYNOMIALS II , 2002 .

[22]  A. Kufner,et al.  Some applications of weighted Sobolev spaces , 1987 .

[23]  J. Mason,et al.  Algorithms for approximation , 1987 .

[24]  Weighted Weierstrass' theorem with first derivatives , 2007 .

[25]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[26]  Weighted Sobolev Spaces and Capacity , 1994 .

[27]  José M. Rodríguez,et al.  Approximation by polynomials and smooth functions in Sobolev spaces with respect to measures , 2003, J. Approx. Theory.

[28]  J. Heinonen,et al.  Nonlinear Potential Theory of Degenerate Elliptic Equations , 1993 .

[29]  A. Kufner.  Weighted Sobolev Spaces , 1985 .