Authors:

Rodriguez, Jose M.

Alvarez, Venancio

Romera, Elena

Pestana, Domingo

Alvarez, Venancio

Romera, Elena

Pestana, Domingo

Pub Date:

08/01/2006

Publication:

Name: Electronic Transactions on Numerical Analysis Publisher: Institute of Computational Mathematics Audience: Academic Format: Magazine/Journal Subject: Computers; Mathematics Copyright: COPYRIGHT 2006 Institute of Computational Mathematics ISSN: 1068-9613

Issue:

Date: August, 2006 Source Volume: 24

Accession Number:

187843927

Full Text:

Abstract. In this paper we present a definition of Sobolev spaces
with respect to general measures, prove some useful technical results,
some of them generalizations of classical results with Lebesgue measure
and find general conditions under which these spaces are complete. These
results have important consequences in Approximation Theory. We also
find conditions under which the evaluation operator is bounded.

Key words. Sobolev spaces, weights, orthogonal polynomials

AMS subject classifications. 41A10, 46E35, 46G10

1. Introduction. Weighted Sobolev spaces are an interesting topic in many fields of Mathematics. In the classical books [7], [8], we can find the point of view of Partial Differential Equations. (See also [20] and [6]). We are interested in the relationship between this topic and Approximation Theory in general, and Sobolev Orthogonal Polynomials in particular.

The specific problems we want to solve are the following:

1) Given a Sobolev scalar product with general measures in R, find hypotheses on the measures, as general as possible, so that we can define a Sobolev space whose elements are functions.

2) If a Sobolev scalar product with general measures in R is well defined for polynomials, what is the completion, P, of the space of polynomials with respect to the norm associated to that scalar product? This problem has been studied in some particular cases (see e.g. [4], [3], [5]), but at this moment no general theory has been built.

Our study has as application an answer to the question of finding the most general conditions under which the multiplication operator, M f (x) = x f (x), is bounded in the space [P.sup.k,2]. We know by a theorem in [10] that, the zeroes of the Sobolev orthogonal polynomials are contained in the disk {z : |z| < ||M||}. The location of these zeroes allows to prove results on the asymptotic behaviour of Sobolev orthogonal polynomials (see [9]). In the second part of this paper, [15], and in [17] and [1], we answer the question stated also in [9] about general conditions for M to be bounded.

The completeness that we study now is one of the central questions in the theory of weighted Sobolev spaces, together with the density of [C.sup.[infinity]] functions. In particular, when all the measures are finite, have compact support and are such that [C.sup.[infinity]sub.c]] (R). is dense in a Sobolev space that is complete, then the closure of the polynomials is the whole Sobolev space. This is deduced from Bernstein's proof of Weierstrass' theorem, where the polynomials he builds approximate uniformly up to the k-th derivative any function in [C.sup.k]([a, b]) (see e.g. [2], p.113).

In the paper we also prove some inequalities which generalize classical results about Sobolev spaces with respect to Lebesgue measure (see Theorem 3.2).

What we present here is an abridged version of the paper [ 14], where the complete proofs of the results may be found, together with the corresponding lemmas and related results.

In the first part of the article we obtain a good definition of Sobolev space with respect to very general measures. We allow the measures to be almost independent of each other. The main result that we present in the paper is Theorem 3.1. It states very general conditions on the measures under which this Sobolev space is complete.

2. Definitions and previous results. The main concepts that we need to understand the statement of our results are contained in the following definitions. The first one is a class of weights that will be the absolutely continuous part of our measures.

DEFINITION 2.1. We say that a weight w belongs to [B.sub.P](J), if and only if,

[w.sup.-1] [member of] [L.sup.1/(p-1).sub.loc] (J), for 1 [less than or equal to] p < [infinity]

[w.sup.-1] [member of] [L.sup.1.sub.loc] (J), for p < [infinity]

This class contains the classical [A.sub.p] weights appearing in Harmonic Analysis, but is larger. We consider vectorial measures [micro] = ([[micro].sub.0], ..., [[micro].sub.k]) in the definition of our Sobolev space and make for each one the decomposition d[micro]p; = [d([[micro].sub.j]).sub.s] + [w.sub.j]dx, where [([[micro].sub.j]).sub.s] is singular with respect to the Lebesgue measure and [w.sub.j] is a Lebesgue measurable function.

DEFINITION 2.2. Let us consider 1 [less than or equal to] p [less than or equal to] [infinity] and a vectorial measure [micro] = ([[micro].sub.0], ..., [[micro].sub.k]).

For 0 [less than or equal to] j [less than or equal to] k, we define the open set

[[OMEGA].sub.j] = {x [member of] R : [there exists] an open neighbourhood V of x with [w.sub.j] [member of] [B.sub.p](V)} .

Observe that we always have [w.sub.j] [member of] [B.sub.p] ([[OMEGA].sub.j]), for any 0 [less than or equal to] j [less than or equal to] k. In fact, [[OMEGA].sub.j] is the largest open set U with [w.sub.j] [member of] [B.sub.p] (U). Obviously, [[OMEGA].sub.j] depends on p and [mu], although p and [mu] do not appear explicitly in the symbol [[OMEGA].sub.j]. It is easy to check that if [f.sup.(j)] [member of] [L.sup.P]([[OMEGA].sub.j], [w.sub.j]) with [0 [less than or equal to] j [less than or equal to] k, then, [f.sup.(j)) [member of] [L.sup.1.sub.loc. ([[OMEGA].sub.j]), and therefore [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. The notation [AC.sub.loc] refers to the class of locally absolutely continuous functions.

We denote by [[OMEGA].sub.j] the set of "good" points at the level j for the vectorial weight ([w.sub.0], ..., [w.sub.k]). These are in essence the points x for which there exists a weight [w.sub.i] with j < i [less than or equal to] k that is, in a neighbourhood of x, in the class [B.sub.p].

Let us present now the class of measures that we use and the definition of Sobolev space.

DEFINITION 2.3. We say that the vectorial measure [micro] = ([[micro].sub.0], ..., [[micro].sub.k]) is p-admissible if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Remarks.

1. The hypothesis of p-admissibility is natural. It would not be reasonable to consider Dirac deltas in [[micro].sub.j] in the points where [f.sup.(j)] is not continuous.

2. Observe that there is not any restriction on supp[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

3. Every absolutely continuous measure is p-admissible.

DEFINITION 2.4. Let us consider 1 [less than or equal to] p [less than or equal to] [infinity], an open set [OMEGA] [subset] R and a p-admissible vectorial measure [micro] = ([[micro].sub.0], ..., [[micro].sub.k]) in [OMEGA]. We define the Sobolev space [W.sup.k,P]([OMEGA], [micro]) as the space of equivalence classes of

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] with respect to the seminorms

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Here

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where ess sup refers to Lebesgue measure, and we assume the usual convention sup 0 = -[infinity].

Before we state our theorems, let us recall a classical result that will be generalized in our Theorem 3.2.

Muckenhoupt inequality. ([12], [11]) Let us consider, 1 [less than or equal to] p < [infinity] and [[micro].sub.0], [micro] measures in (a, b] with [w.sub.1] := a[[micro].sub.1]/ax. Then there exists a positive constant c such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

for any measurable function g in (a, b], if and only if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

3. Completeness of the Sobolev spaces. And now, here is our main theorem in the paper. In it and in Theorem 3.2 we consider special classes that we call C and [C.sub.0]. The conditions ([OMEGA], [micro]) [member of] [C.sub.0] and ([OMEGA], [micro]) [member of] C are not very restrictive. The first one consists, roughly speaking, in considering measures [micro] such that [parallel] * [parallel] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] is a norm for some sequence of compact sets {[M.sub.n] growing to [OMEGA]. As to the class C, it is a slight modification of [C.sub.0], in which we consider measures [micro] = ([[micro].sub.0], ..., [[micro].sub.k]) such that by adding a minimal amount of deltas to [[micro].sub.0] we obtain a measure in the class [C.sub.0].

THEOREM 3.1. Let us consider 1 [less than or equal to] p < [infinity], an open set [OMEGA] [subset] R and a p-admissible vectorial measure [micro] = ([[micro].sub.0], ..., [[micro].sub.k]) in [OMEGA] with ([OMEGA], [micro]) [member of] C. Then the Sobolev space [W.sup.k,p] ([OMEGA], [micro]) is complete.

The main ingredient of the proof of this result is Theorem 3.2. It allows us to control the [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] norm (in appropriate sets) of a function and its derivatives in terms of its Sobolev norm. It is also useful by its applications in the papers [15], [16], [17], [18], [1], [19] and [13]. Furthermore, it is important by itself, since it answers to the following main question: when the evaluation functional of f (or [f.sup.(j)]) in a point is a bounded operator in [W.sup.k,p] ([OMEGA], [micro])?

THEOREM 3.2. Let us consider 1 [less than or equal to] p [less than or equal to] [infinity], an open set [OMEGA] [subset] R and a p-admissible vectorial measure [micro] in [OMEGA]. If [K.sub.j] is a finite union of compact intervals contained in [[OMEGA].sup.(j)], for 0 [less than or equal to] j < k, then:

(a) If ([OMEGA], [micro]) [subset] [C.sub.0] there exists a positive constant [c.sub.1] = [c.sub.1] ([K.sub.0], ..., [K.sub.k-1]) such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

for all g [member of] [V.sup.k,p]([OMEGA], [mu]).

(b) If ([OMEGA], [mu]) [member of] C there exists a positive constant [c.sub.2] = [c.sub.2] ([K.sub.0], ..., [K.sub.k-1]) such that for every g [member of] [V.sup.k,p] ([OMEGA], [mu]), there exists [g.sub.0] [V.sup.k,p]([OMEGA], [mu]), independent of [K.sub.0], ..., [K.sub.k-1] and [c.sub.2], with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Furthermore, if [g.sub.0], fo are these representatives of g, f respectively, we have for the same constant [c.sub.2]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

This theorem has the following corollary, that we use in the proof of Theorem 3.1:

COROLLARY 3.3. Let us consider, 1 [less than or equal to] p [less than or equal to] [infinity], an open set OMEGA] [subset] R and a p-admissible vectorial measure [mu] in [OMEGA]. If [K.sub.j] is a finite union of compact intervals contained in [[OMEGA].sup.j], for 0 [less than or equal to] j [less than or equal to] k, then:

(a) If ([OMEGA], [mu]) [member of] C there exists a positive constant [c.sub.z] = [c.sub.z] ([K.sub.0], ..., [K.sub.k-1]) such that,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(b) If ([OMEGA], [mu]) [member of] C there exists a positive constant [c.sub.z] = [c.sub.z] ([K.sub.0], ..., [K.sub.k-1]) such that for every g [member of] [V.sup.k,p] ([OMEGA], [mu]), there exists [g.sub.0] [member of] [V.sup.k,p]([OMEGA], [mu]) (the same function as in Theorem 3.2), with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Furthermore, if [g.sub.0], [f.sub.0] are the representatives of g, f respectively, we have for the same constant [c.sub.2]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

As a consequence of theorems 3.2 and 3.1, we can prove the density of the space of polynomials in these Sobolev spaces (see [15], [16], [18], [1] and [19]) and the boundedness of the multiplication operator (see [15], [17] and [1]).

Proof of Theorem 3.1: Let {[f.sub.n]} be a Cauchy sequence in [W.sup.k,p]([OMEGA],[mu]). Then, for each [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] is a Cauchy sequence in [L.sup.p] ([OMEGA],[mu]) and it converges to a function [y.sub.j] [member of] [L.sup.p]([OMEGA],[mu]).

First of all, let us show that [g.sub.j] can be extended to a function in [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] if 0 [less than or equal to] j [less than or equal to] k) and in [L.sup.1.sub.loc]([[OMEGA].SUP.(j-1]) (if 0 < j [less than or equal to] k).

If 0 [less than or equal to] j < k, let us consider any compact interval K [subset] [[OMEGA].sup.(j). By part (b) of Theorem 3.2 we know there exists a representative (independent of K) of the class of [f.sub.n] [member of] [W.sup.k,p] ([OMEGA], [micro]) (which we also denote by [f.sub.n]) and a positive constant c such that for every n, m [member of] N

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

As {[f.sup.(j).sub.n]} [subset] C(K), there exists a function [h.sub.j] [member of] C(K) such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Since we can take as K any compact interval contained in [[OMEGA].sup(j)], we obtain that the function [h.sub.j]; can be extended to [[OMEGA].sup.(j)] and we have in fact [h.sub.j] [member of] C([[OMEGA].sup.(j)]). It is obvious that [g.sub.j] = [h.sub.j]; in [[OMEGA].sup.(j)]) (except for at most a set of zero [[micro].sub.j]-measure), since [f.sup.(j).sub.n] converges to [g.sub.j] in the norm of [L.sup.p] ([OMEGA], [[micro].sub.j]) and to [h.sub.j] uniformly on each compact interval K C [subset] [[OMEGA].sup.(J)] . Therefore we can assume that [g.sub.j] [member of] C([[OMEGA].sup.(j)]).

If 0 < j [less than or equal to] k, let us consider any compact interval J [subset] [[OMEGA].sup.(j-1)]. Now, part (b) of Corollary 3.3 gives

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

As {[f.sup.(j).sub.n]} [member of] [L.sup.1] (J), there exists a function [u.sub.j] [member of] [L.sup.1](J) such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Since we can take as J any compact interval contained in [[OMEGA].sup.(j-1)], we obtain that the function [u.sub.j] can be extended to [[OMEGA].sup.(j-1)] and we have in fact [u.sub.j] [member of] [L.sup.1.sub.loc] ([[OMEGA].sup.(j-1)]). It is obvious that [g.sub.j] = [u.sub.j] in ([(OMEGA].sup.(j)]) (except for at most a set of zero Lebesgue measure), since [f.sup.j.sub.n] converges to [u.sub.j] in [L.sup.1.sub.loc],([[OMEGA].sup.(j)]) and to [g.sub.j] locally uniformly in [[OMEGA].sup.(j)]. Let us consider a set A which concentrates the mass of ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]), with |A| = 0; we can take [u.sub.j] = [g.sub.j] in A. We only need to show [u.sub.j] = [g.sub.j] in [w.sub.j] [member of] ([B.sub.p]) [[OMEGA].sub.j]) (recall that by hypothesis [w.sub.j] = 0 in R\ [[OMEGA].sub.j]), but this is immediate since [w.sub.j] [member of] [B.sub.p]([[OMEGA].sub.j]) and the convergence in [L.sup.P]([[OMEGA].sub.j], [w.sub.j]) implies the convergence in [L.sup.1.sub.loc] ([[OMEGA].sub.j]). Therefore we can assume that [g.sub.j] [member of] [L.sup.1.sub.loc] ([[OMEGA].sup.(j-1)]).

In fact, we have seen that {[f.sup.(j).sub.n]} converges to [g.sub.j] in [L.sup.[infinity].sub.loc] ([[OMEGA].sup.(j)]) (if 0 [less than or equal to] j < k) and in [L.sup.1.sub.loc] ([[OMEGA].sup.(j-1)]) (if 0 < j [less than or eqaul to] k).

Let us see now that [g'.sub.j] = [g.sub.j+] in the interior of [[OMEGA].sup.(j)] for 0 [less than or equal to] j < k. Let us consider a connected component I of int ([[OMEGA].sup.(j)]). Given [phi] [C.sup.[infinity].sub.C] (I), let us consider the convex hull K of supp [phi]. We have that K is a compact interval contained in I [subset] [[OMEGA].sup.(j)]. The uniform convergence of {[J.sup.(j).sub.n])} in K and the [L.sup.-] convergence of {[Jn.sup.(j + 1)]} in K gives that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Then [g.sub.j+] = [g.sub.0.sup.(j + 1)] in int ([[OMEGA].sup.(j)]) and [g.sub.0.sup.(j)]) [member of] [AC.sub.loc] (int ([[OMEGA].sup(j)])) for 0 [less than or eqaul to] j < k. In order to see that [g.sub.0.sup.(j)] [member of] [AC.sub.loc] ([[OMEGA].sup.j]), it is enough to recall that [g.sub.0.sup.(j)]' = [g.sub.j+1] [member of] [L.sup.1.sub.loc]([[OMEGA].sup.j]).

* Received November 30, 2004. Accepted for publication February 28, 2005. Recommended by J. Arvesu.

REFERENCES

[1] V. ALVAREZ, D. PESTANA, J. M. RODRIGUEZ, AND E. ROMERA, Weighted Sobolev spaces on curves, J. Approx. Theory, 119 (2002) pp. 41-85.

[2] P. J. DAVIS, Interpolation and Approximation, Dover, New York, 1975.

[3] W. N. EVERITT AND L. L. LITTLEJOHN,The density of polynomials in a weighted Sobolev space, Rendiconti di Matematica, Serie VII, 10 (1990), pp. 835-852.

[4] W. N. EVERITT, L. L. LITTLEJOHN, AND S. C. WILLIAMS, Orthogonal polynomials in weighted Sobolev spaces, in Lecture Notes in Pure and Applied Mathematics, 117, Marcel Dekker, 1989, pp. 53-72.

[5] W. N. EVERITT, L. L. LITTLEJOHN, AND S. C. WILLIAMS, Orthogonal polynomials and approximation in Sobolev spaces, J. Comput. Appl. Math., 48 (1993), pp. 69-90.

[6] J. HEINONEN, T. KILPELi1INEN, AND O. MARTIO, Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford Science Pubt, Clarendon Press, 1993.

[7] A. KUFNER, Weighted Sobolev Spaces, Teubner Verlagsgesellschaft, Teubner-Texte zur Mathematik (Band 31), Leipzig, 1980; also published by John Wiley & Sons, New York, 1985.

[8] A. KUFNER AND A. M. SANDIG, Some Applications of Weighted Sobolev Spaces, Teubner Verlagsgesellschaft, Teubner-Texte zur Mathematik (Band 100), Leipzig, 1987.

[9] G. LOPEZ LAGOMASINO AND H. PIJEIRA, Zero location and n-th root asymptotics of Sobolev orthogonal polynomials, J. Approx. Theory, 99 (1999), pp. 30-43.

[10] G. LOPEZ LAGOMASINO, H. PIJEIRA, AND I. PEREZ, Sobolev orthogonal polynomials in the complex plane, J. Comp. Appl. Math., 127 (2001), pp. 219-230.

[11] V. G. MAZ'JA, Sobolev Spaces, Springer-Verlag, New York, 1985.

[12] B. MUCKENHOUPT, Hardy's inequality with weights, Studia Math., 44 (1972), pp. 31-38.

[13] A. PORTILLA, Y. QUINTANA, J. M. RODRIGUEZ, AND E. TOURIS, Weierstrass' theorem with weights, J. Approx. Theory, 127 (2004), pp. 83-107.

[14] J. M. RODRIGUEZ, V. ALVAREZ, E. ROMERA, AND D. PESTANA, Generalized Weighted Sobolev spaces and applications to Sobolev orthogonal polynomials I, Acta Appl. Math., 80 (2004), pp. 273-308.

[15] J. M. RODRIGUEZ, V. ALVAREZ, E. ROMERA, AND D. PESTANA, Generalized Weighted Sobolev spaces and applications to Sobolev orthogonal polynomials II, Approx. Theory Appl., 18 (2002), pp. 1-32.

[16] J. M. RODRIGUEZ, Weierstrass' theorem in weighted Sobolev spaces, J. Approx. Theory, 108 (2001), pp. 119-160.

[17] J. M. RODRIGUEZ, The multiplication operator in Sobolev spaces with respect to measures, J. Approx. Theory, 109 (2001), pp. 157-197.

[18] J. M. RODRIGUEZ, Approximation by polynomials and smooth functions in Sobolev spaces with respect to measures, J. Approx. Theory, 120 (2003), pp. 185-216.

[19] J. M. RODRIGUEZ AND V. A. YAKUBOVICH, A Kolmogorov-Szeg6-Krein type condition for weighted Sobolev spaces, Indiana U. Math. J., 54 (2005), pp. 575-598.

[20] H. TRIEBEL, Interpolation Theory, Function Spaces and Differential Operators, North-Holland Mathematical Library, 1978.

JOSE M. RODRIGUEZ ([dagger]), VENANCIO ALVAREZ ([dagger]), ELENA ROMERA ([dagger]), AND DOMINGO PESTANA ([paragraph])

([dagger]) Dep. de Matemeticas, Univ. Carlos III de Madrid, Avda. de la Universidad, 30, 28911 Leganes (Madrid), SPAIN (jomaro@math.uc3m.es). Research partially supported by grants from DGI (BFM 2003-06335-C03-02 and BFM 2003-04870), Spain.

([double dagger]) Dep. de Analisis Matematico, Facultad de Ciencias, Campus de Teatinos, 29071 Malaga, SPAIN (nancho@anamat.cie.uma.es). Research partially supported by grants from MCYT (MTM 2004-00078) and Junta de Andalucia (FQM-210), Spain.

([section]) Dep. de Matematicas, Univ. Carlos III de Madrid, Avda. de la Universidad, 30, 28911 Leganes (Madrid), SPAIN (eromera@math.uc3m.es). Research partially supported by a grant from DGI (BFM 2003-06335-C03-02), Spain.

([paragraph]) Dep. de Matematicas, Univ. Carlos III de Madrid, Avda. de la Universidad, 30, 28911 Leganes (Madrid), SPAIN (dompes@math.uc3m.es). Research partially supported by grants from DGI (BFM 2003-06335-C03-02 and BFM 2003-04870), Spain.

Key words. Sobolev spaces, weights, orthogonal polynomials

AMS subject classifications. 41A10, 46E35, 46G10

1. Introduction. Weighted Sobolev spaces are an interesting topic in many fields of Mathematics. In the classical books [7], [8], we can find the point of view of Partial Differential Equations. (See also [20] and [6]). We are interested in the relationship between this topic and Approximation Theory in general, and Sobolev Orthogonal Polynomials in particular.

The specific problems we want to solve are the following:

1) Given a Sobolev scalar product with general measures in R, find hypotheses on the measures, as general as possible, so that we can define a Sobolev space whose elements are functions.

2) If a Sobolev scalar product with general measures in R is well defined for polynomials, what is the completion, P, of the space of polynomials with respect to the norm associated to that scalar product? This problem has been studied in some particular cases (see e.g. [4], [3], [5]), but at this moment no general theory has been built.

Our study has as application an answer to the question of finding the most general conditions under which the multiplication operator, M f (x) = x f (x), is bounded in the space [P.sup.k,2]. We know by a theorem in [10] that, the zeroes of the Sobolev orthogonal polynomials are contained in the disk {z : |z| < ||M||}. The location of these zeroes allows to prove results on the asymptotic behaviour of Sobolev orthogonal polynomials (see [9]). In the second part of this paper, [15], and in [17] and [1], we answer the question stated also in [9] about general conditions for M to be bounded.

The completeness that we study now is one of the central questions in the theory of weighted Sobolev spaces, together with the density of [C.sup.[infinity]] functions. In particular, when all the measures are finite, have compact support and are such that [C.sup.[infinity]sub.c]] (R). is dense in a Sobolev space that is complete, then the closure of the polynomials is the whole Sobolev space. This is deduced from Bernstein's proof of Weierstrass' theorem, where the polynomials he builds approximate uniformly up to the k-th derivative any function in [C.sup.k]([a, b]) (see e.g. [2], p.113).

In the paper we also prove some inequalities which generalize classical results about Sobolev spaces with respect to Lebesgue measure (see Theorem 3.2).

What we present here is an abridged version of the paper [ 14], where the complete proofs of the results may be found, together with the corresponding lemmas and related results.

In the first part of the article we obtain a good definition of Sobolev space with respect to very general measures. We allow the measures to be almost independent of each other. The main result that we present in the paper is Theorem 3.1. It states very general conditions on the measures under which this Sobolev space is complete.

2. Definitions and previous results. The main concepts that we need to understand the statement of our results are contained in the following definitions. The first one is a class of weights that will be the absolutely continuous part of our measures.

DEFINITION 2.1. We say that a weight w belongs to [B.sub.P](J), if and only if,

[w.sup.-1] [member of] [L.sup.1/(p-1).sub.loc] (J), for 1 [less than or equal to] p < [infinity]

[w.sup.-1] [member of] [L.sup.1.sub.loc] (J), for p < [infinity]

This class contains the classical [A.sub.p] weights appearing in Harmonic Analysis, but is larger. We consider vectorial measures [micro] = ([[micro].sub.0], ..., [[micro].sub.k]) in the definition of our Sobolev space and make for each one the decomposition d[micro]p; = [d([[micro].sub.j]).sub.s] + [w.sub.j]dx, where [([[micro].sub.j]).sub.s] is singular with respect to the Lebesgue measure and [w.sub.j] is a Lebesgue measurable function.

DEFINITION 2.2. Let us consider 1 [less than or equal to] p [less than or equal to] [infinity] and a vectorial measure [micro] = ([[micro].sub.0], ..., [[micro].sub.k]).

For 0 [less than or equal to] j [less than or equal to] k, we define the open set

[[OMEGA].sub.j] = {x [member of] R : [there exists] an open neighbourhood V of x with [w.sub.j] [member of] [B.sub.p](V)} .

Observe that we always have [w.sub.j] [member of] [B.sub.p] ([[OMEGA].sub.j]), for any 0 [less than or equal to] j [less than or equal to] k. In fact, [[OMEGA].sub.j] is the largest open set U with [w.sub.j] [member of] [B.sub.p] (U). Obviously, [[OMEGA].sub.j] depends on p and [mu], although p and [mu] do not appear explicitly in the symbol [[OMEGA].sub.j]. It is easy to check that if [f.sup.(j)] [member of] [L.sup.P]([[OMEGA].sub.j], [w.sub.j]) with [0 [less than or equal to] j [less than or equal to] k, then, [f.sup.(j)) [member of] [L.sup.1.sub.loc. ([[OMEGA].sub.j]), and therefore [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. The notation [AC.sub.loc] refers to the class of locally absolutely continuous functions.

We denote by [[OMEGA].sub.j] the set of "good" points at the level j for the vectorial weight ([w.sub.0], ..., [w.sub.k]). These are in essence the points x for which there exists a weight [w.sub.i] with j < i [less than or equal to] k that is, in a neighbourhood of x, in the class [B.sub.p].

Let us present now the class of measures that we use and the definition of Sobolev space.

DEFINITION 2.3. We say that the vectorial measure [micro] = ([[micro].sub.0], ..., [[micro].sub.k]) is p-admissible if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Remarks.

1. The hypothesis of p-admissibility is natural. It would not be reasonable to consider Dirac deltas in [[micro].sub.j] in the points where [f.sup.(j)] is not continuous.

2. Observe that there is not any restriction on supp[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

3. Every absolutely continuous measure is p-admissible.

DEFINITION 2.4. Let us consider 1 [less than or equal to] p [less than or equal to] [infinity], an open set [OMEGA] [subset] R and a p-admissible vectorial measure [micro] = ([[micro].sub.0], ..., [[micro].sub.k]) in [OMEGA]. We define the Sobolev space [W.sup.k,P]([OMEGA], [micro]) as the space of equivalence classes of

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] with respect to the seminorms

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Here

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where ess sup refers to Lebesgue measure, and we assume the usual convention sup 0 = -[infinity].

Before we state our theorems, let us recall a classical result that will be generalized in our Theorem 3.2.

Muckenhoupt inequality. ([12], [11]) Let us consider, 1 [less than or equal to] p < [infinity] and [[micro].sub.0], [micro] measures in (a, b] with [w.sub.1] := a[[micro].sub.1]/ax. Then there exists a positive constant c such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

for any measurable function g in (a, b], if and only if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

3. Completeness of the Sobolev spaces. And now, here is our main theorem in the paper. In it and in Theorem 3.2 we consider special classes that we call C and [C.sub.0]. The conditions ([OMEGA], [micro]) [member of] [C.sub.0] and ([OMEGA], [micro]) [member of] C are not very restrictive. The first one consists, roughly speaking, in considering measures [micro] such that [parallel] * [parallel] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] is a norm for some sequence of compact sets {[M.sub.n] growing to [OMEGA]. As to the class C, it is a slight modification of [C.sub.0], in which we consider measures [micro] = ([[micro].sub.0], ..., [[micro].sub.k]) such that by adding a minimal amount of deltas to [[micro].sub.0] we obtain a measure in the class [C.sub.0].

THEOREM 3.1. Let us consider 1 [less than or equal to] p < [infinity], an open set [OMEGA] [subset] R and a p-admissible vectorial measure [micro] = ([[micro].sub.0], ..., [[micro].sub.k]) in [OMEGA] with ([OMEGA], [micro]) [member of] C. Then the Sobolev space [W.sup.k,p] ([OMEGA], [micro]) is complete.

The main ingredient of the proof of this result is Theorem 3.2. It allows us to control the [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] norm (in appropriate sets) of a function and its derivatives in terms of its Sobolev norm. It is also useful by its applications in the papers [15], [16], [17], [18], [1], [19] and [13]. Furthermore, it is important by itself, since it answers to the following main question: when the evaluation functional of f (or [f.sup.(j)]) in a point is a bounded operator in [W.sup.k,p] ([OMEGA], [micro])?

THEOREM 3.2. Let us consider 1 [less than or equal to] p [less than or equal to] [infinity], an open set [OMEGA] [subset] R and a p-admissible vectorial measure [micro] in [OMEGA]. If [K.sub.j] is a finite union of compact intervals contained in [[OMEGA].sup.(j)], for 0 [less than or equal to] j < k, then:

(a) If ([OMEGA], [micro]) [subset] [C.sub.0] there exists a positive constant [c.sub.1] = [c.sub.1] ([K.sub.0], ..., [K.sub.k-1]) such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

for all g [member of] [V.sup.k,p]([OMEGA], [mu]).

(b) If ([OMEGA], [mu]) [member of] C there exists a positive constant [c.sub.2] = [c.sub.2] ([K.sub.0], ..., [K.sub.k-1]) such that for every g [member of] [V.sup.k,p] ([OMEGA], [mu]), there exists [g.sub.0] [V.sup.k,p]([OMEGA], [mu]), independent of [K.sub.0], ..., [K.sub.k-1] and [c.sub.2], with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Furthermore, if [g.sub.0], fo are these representatives of g, f respectively, we have for the same constant [c.sub.2]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

This theorem has the following corollary, that we use in the proof of Theorem 3.1:

COROLLARY 3.3. Let us consider, 1 [less than or equal to] p [less than or equal to] [infinity], an open set OMEGA] [subset] R and a p-admissible vectorial measure [mu] in [OMEGA]. If [K.sub.j] is a finite union of compact intervals contained in [[OMEGA].sup.j], for 0 [less than or equal to] j [less than or equal to] k, then:

(a) If ([OMEGA], [mu]) [member of] C there exists a positive constant [c.sub.z] = [c.sub.z] ([K.sub.0], ..., [K.sub.k-1]) such that,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(b) If ([OMEGA], [mu]) [member of] C there exists a positive constant [c.sub.z] = [c.sub.z] ([K.sub.0], ..., [K.sub.k-1]) such that for every g [member of] [V.sup.k,p] ([OMEGA], [mu]), there exists [g.sub.0] [member of] [V.sup.k,p]([OMEGA], [mu]) (the same function as in Theorem 3.2), with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Furthermore, if [g.sub.0], [f.sub.0] are the representatives of g, f respectively, we have for the same constant [c.sub.2]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

As a consequence of theorems 3.2 and 3.1, we can prove the density of the space of polynomials in these Sobolev spaces (see [15], [16], [18], [1] and [19]) and the boundedness of the multiplication operator (see [15], [17] and [1]).

Proof of Theorem 3.1: Let {[f.sub.n]} be a Cauchy sequence in [W.sup.k,p]([OMEGA],[mu]). Then, for each [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] is a Cauchy sequence in [L.sup.p] ([OMEGA],[mu]) and it converges to a function [y.sub.j] [member of] [L.sup.p]([OMEGA],[mu]).

First of all, let us show that [g.sub.j] can be extended to a function in [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] if 0 [less than or equal to] j [less than or equal to] k) and in [L.sup.1.sub.loc]([[OMEGA].SUP.(j-1]) (if 0 < j [less than or equal to] k).

If 0 [less than or equal to] j < k, let us consider any compact interval K [subset] [[OMEGA].sup.(j). By part (b) of Theorem 3.2 we know there exists a representative (independent of K) of the class of [f.sub.n] [member of] [W.sup.k,p] ([OMEGA], [micro]) (which we also denote by [f.sub.n]) and a positive constant c such that for every n, m [member of] N

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

As {[f.sup.(j).sub.n]} [subset] C(K), there exists a function [h.sub.j] [member of] C(K) such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Since we can take as K any compact interval contained in [[OMEGA].sup(j)], we obtain that the function [h.sub.j]; can be extended to [[OMEGA].sup.(j)] and we have in fact [h.sub.j] [member of] C([[OMEGA].sup.(j)]). It is obvious that [g.sub.j] = [h.sub.j]; in [[OMEGA].sup.(j)]) (except for at most a set of zero [[micro].sub.j]-measure), since [f.sup.(j).sub.n] converges to [g.sub.j] in the norm of [L.sup.p] ([OMEGA], [[micro].sub.j]) and to [h.sub.j] uniformly on each compact interval K C [subset] [[OMEGA].sup.(J)] . Therefore we can assume that [g.sub.j] [member of] C([[OMEGA].sup.(j)]).

If 0 < j [less than or equal to] k, let us consider any compact interval J [subset] [[OMEGA].sup.(j-1)]. Now, part (b) of Corollary 3.3 gives

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

As {[f.sup.(j).sub.n]} [member of] [L.sup.1] (J), there exists a function [u.sub.j] [member of] [L.sup.1](J) such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Since we can take as J any compact interval contained in [[OMEGA].sup.(j-1)], we obtain that the function [u.sub.j] can be extended to [[OMEGA].sup.(j-1)] and we have in fact [u.sub.j] [member of] [L.sup.1.sub.loc] ([[OMEGA].sup.(j-1)]). It is obvious that [g.sub.j] = [u.sub.j] in ([(OMEGA].sup.(j)]) (except for at most a set of zero Lebesgue measure), since [f.sup.j.sub.n] converges to [u.sub.j] in [L.sup.1.sub.loc],([[OMEGA].sup.(j)]) and to [g.sub.j] locally uniformly in [[OMEGA].sup.(j)]. Let us consider a set A which concentrates the mass of ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]), with |A| = 0; we can take [u.sub.j] = [g.sub.j] in A. We only need to show [u.sub.j] = [g.sub.j] in [w.sub.j] [member of] ([B.sub.p]) [[OMEGA].sub.j]) (recall that by hypothesis [w.sub.j] = 0 in R\ [[OMEGA].sub.j]), but this is immediate since [w.sub.j] [member of] [B.sub.p]([[OMEGA].sub.j]) and the convergence in [L.sup.P]([[OMEGA].sub.j], [w.sub.j]) implies the convergence in [L.sup.1.sub.loc] ([[OMEGA].sub.j]). Therefore we can assume that [g.sub.j] [member of] [L.sup.1.sub.loc] ([[OMEGA].sup.(j-1)]).

In fact, we have seen that {[f.sup.(j).sub.n]} converges to [g.sub.j] in [L.sup.[infinity].sub.loc] ([[OMEGA].sup.(j)]) (if 0 [less than or equal to] j < k) and in [L.sup.1.sub.loc] ([[OMEGA].sup.(j-1)]) (if 0 < j [less than or eqaul to] k).

Let us see now that [g'.sub.j] = [g.sub.j+] in the interior of [[OMEGA].sup.(j)] for 0 [less than or equal to] j < k. Let us consider a connected component I of int ([[OMEGA].sup.(j)]). Given [phi] [C.sup.[infinity].sub.C] (I), let us consider the convex hull K of supp [phi]. We have that K is a compact interval contained in I [subset] [[OMEGA].sup.(j)]. The uniform convergence of {[J.sup.(j).sub.n])} in K and the [L.sup.-] convergence of {[Jn.sup.(j + 1)]} in K gives that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Then [g.sub.j+] = [g.sub.0.sup.(j + 1)] in int ([[OMEGA].sup.(j)]) and [g.sub.0.sup.(j)]) [member of] [AC.sub.loc] (int ([[OMEGA].sup(j)])) for 0 [less than or eqaul to] j < k. In order to see that [g.sub.0.sup.(j)] [member of] [AC.sub.loc] ([[OMEGA].sup.j]), it is enough to recall that [g.sub.0.sup.(j)]' = [g.sub.j+1] [member of] [L.sup.1.sub.loc]([[OMEGA].sup.j]).

* Received November 30, 2004. Accepted for publication February 28, 2005. Recommended by J. Arvesu.

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JOSE M. RODRIGUEZ ([dagger]), VENANCIO ALVAREZ ([dagger]), ELENA ROMERA ([dagger]), AND DOMINGO PESTANA ([paragraph])

([dagger]) Dep. de Matemeticas, Univ. Carlos III de Madrid, Avda. de la Universidad, 30, 28911 Leganes (Madrid), SPAIN (jomaro@math.uc3m.es). Research partially supported by grants from DGI (BFM 2003-06335-C03-02 and BFM 2003-04870), Spain.

([double dagger]) Dep. de Analisis Matematico, Facultad de Ciencias, Campus de Teatinos, 29071 Malaga, SPAIN (nancho@anamat.cie.uma.es). Research partially supported by grants from MCYT (MTM 2004-00078) and Junta de Andalucia (FQM-210), Spain.

([section]) Dep. de Matematicas, Univ. Carlos III de Madrid, Avda. de la Universidad, 30, 28911 Leganes (Madrid), SPAIN (eromera@math.uc3m.es). Research partially supported by a grant from DGI (BFM 2003-06335-C03-02), Spain.

([paragraph]) Dep. de Matematicas, Univ. Carlos III de Madrid, Avda. de la Universidad, 30, 28911 Leganes (Madrid), SPAIN (dompes@math.uc3m.es). Research partially supported by grants from DGI (BFM 2003-06335-C03-02 and BFM 2003-04870), Spain.

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