ETNA
Electronic Transactions on Numerical Analysis.
Volume 24, pp. 88-93, 2006.
Copyright  2006, Kent State University.
ISSN 1068-9613.
Kent State University
etna@mcs.kent.edu
GENERALIZED WEIGHTED SOBOLEV SPACES AND APPLICATIONS TO
SOBOLEV ORTHOGONAL POLYNOMIALS: A SURVEY
JOSÉ M. RODRı́GUEZ , VENANCIO ÁLVAREZ , ELENA ROMERA , AND DOMINGO PESTANA
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 , 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 is well defined for polynomials,
what is the completion,
, 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,
, is bounded in the space
. We know by a theorem in [10] that, the zeroes of the Sobolev orthogonal polynomials
are contained in the disk
. 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
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
functions. In particular, when all
the measures are finite, have compact support and are such that
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
"!$#%&#('
)+*
) , * -.
/ Received November 30, 2004. Accepted for publication February 28, 2005. Recommended by J. Arvesú.
Dep. de Matemáticas, Univ. Carlos III de Madrid, Avda. de la Universidad, 30, 28911 Leganés (Madrid),
SPAIN (jomaro@math.uc3m.es). Research partially supported by grants from DGI (BFM 2003-06335-C03-02
and BFM 2003-04870), Spain.
Dep. de Análisis Matemático, Facultad de Ciencias, Campus de Teatinos, 29071 Málaga, SPAIN
(nancho@anamat.cie.uma.es). Research partially supported by grants from MCYT (MTM 2004-00078)
and Junta de Andalucı́a (FQM-210), Spain.
Dep. de Matemáticas, Univ. Carlos III de Madrid, Avda. de la Universidad, 30, 28911 Leganés (Madrid),
SPAIN (eromera@math.uc3m.es). Research partially supported by a grant from DGI (BFM 2003-06335-C0302), Spain.
Dep. de Matemáticas, Univ. Carlos III de Madrid, Avda. de la Universidad, 30, 28911 Leganés (Madrid),
SPAIN (dompes@math.uc3m.es). Research partially supported by grants from DGI (BFM 2003-06335-C03-02
and BFM 2003-04870), Spain.
88
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89
GENERALIZED WEIGHTED SOBOLEV SPACES
0
)132 4576789
(see e.g. [2],
approximate uniformly up to the -th derivative any function in
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.
D EFINITION 2.1. We say that a weight belongs to
, if and only if,
:
;<=?>@
:ACBDFE JLB3K G%, H < ACB3I ?>@ 5 for MN!POFQSRT5
:ACBDFEUJLB K , ?>@ 5
for OVRXW
This class contains the classical YZ< weights appearing in Harmonic Analysis, but is
larger. We consider vectorial measures [\]9[@^_5%W%W`W%5a[ in the definition of our Sobolev
space and make for each one the decomposition b_[Ccdeb"[c3fhg\:ic`b_ , where 9[ c3f is
singular with respect to the Lebesgue measure and :Uc is a Lebesgue measurable function.
D EFINITION 2.2. Let us consider M!POF!SR and a vectorial measure [jk[@^_5%W`W%W%5a[ .
For lm!onp!d0 , we define the open set
q cNrs
FDtuv an open neighbourhood w of with :xcDF;y<z-w{'|W
: c
}
: c Do; < ~}
l!sn€!$0
H c ID‚EUJLB K , q c
Y ) LJ K , q c
HI
: q c
D\; < q c q
ld!&nS!X0
O c [ < q O
[
c
q c
H
I
P
D
E
3
5
:
c
c
ƒH c CA B3IDdY) JLK , q c
M„!Vn…!$0
n
:Z^†5%W%W`W%5
:‡
nˆQ‚‰|!S0
;<
[jŠ9[@^_5%W%W`W%5a[ O
[ 3fySl
Observe that we always have
, for any
. In fact,
is the
largest open set with
. Obviously,
depends on and , although and
do not appear explicitly in the symbol . It is easy to check that if
with
, then,
, and therefore
if
. The
notation
refers to the class of locally absolutely continuous functions.
We denote by
the set of “good” points at the level for the vectorial weight
. These are in essence the points for which there exists a weight
with
that
is, in a neighbourhood of , in the class .
Let us present now the class of measures that we use and the definition of Sobolev space.
D EFINITION 2.3. We say that the vectorial measure
is -admissible if
, for
, and
.
Remarks.
1. The hypothesis of -admissibility is natural. It would not be reasonable to consider
Dirac deltas in
in the points where
is not continuous.
2. Observe that there is not any restriction on supp
.
3. Every absolutely continuous measure is -admissible.
D EFINITION 2.4. Let us consider
, an open set
and a -admissible
vectorial measure
in . We define the Sobolev space
as the
space of equivalence classes of
9[c
af(Œ‹ q H c I |l
[c
M!€npQŽ0
O
ƒH c I
O
Mˆq !‚O…!kR
9[ ^ f
[‘[C^_5%W`W%W%5a[ w < q 5a[@x“” j qd•—–d˜ H c I DFY) JLK , q H c I # H c I #%ž Ÿ H ¡¢ £¤3I QR
q$ O
’Š
< q a5 [@
n l=5%Mš5`W%W`W›570œ‚M
™
for n™l=5%Mš5`W%W`W›570@¥+5
for
and
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90
J.M. RODRÍGUEZ, V. ALVAREZ, E. ROMERA, AND D. PESTANA
with respect to the seminorms
#%.#
¦ˆ§{¨ Ÿ H ¡C £(I rª©
«
BaG < 5 for MN!€OFQSRX5
#% H c I # <ž Ÿ
H ¡C £ ¤ I?­
ca¢^
#%.#`¦ §{¨ ® H ¡C £(I r°^`ˆ
¯ z³ c±(³ ² #% H c I #%ž ® H ¡¢ £ ¤ I W
Here
#{´¢# ž® Hµ¡C £ ¤ I rS¯ˆ±(²¶.·
¸a»š¸"¼ ¸3¹zº ´ ½:|c 9`r5 »¾¼ ¸a¹zº ´C9 %µÀZ5
supp Hµ£ ¤ I-¿
¡
¸a¹zºZÁkœR
where ess sup refers to Lebesgue measure, and we assume the usual convention
.
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,
and
measures in
with
. Then there exists a positive constant such that
45{678
: B r b_[ B ˜ bš
Mm!ÂOoQsR
Ã
ÄÄ
[@^š53[ B
Ä
!‚ÃË#{´¢# ž1Ÿ H9HÌÈ
£šÍ½I
āł» Æ ´C9Ç31b_Ç ÄÄ ž Ÿ
H HµÈ
Æ É £(Ê3I
ÆÉ
for any measurable function ´ in 4576{8 , if and only if
Î < 9[ ^ 53[ y“ϸa¹zº [ ^ a453Ò
89%#›: ACB # ž ŸÕ
B
B Í~Ó½Ô ÍÖ H9× Ñ3 Æ I9I QŽRXW
È
ÐÑ%Ð Æ
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 and . The
conditions
and
are not very restrictive. The first one consists, roughly
is a norm for some sequence of
speaking, in considering measures such that
compact sets
growing to . As to the class , it is a slight modification of , in which
we consider measures
such that by adding a minimal amount of deltas to
we obtain a measure in the class .
T HEOREM 3.1. Let us consider
, an open set
and a -admissible vectorial measure
in with
. Then the Sobolev space
is complete.
The main ingredient of the proof of this result is Theorem 3.2. It allows us to control
the
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 (or
) in a point is a bounded operator in
?
T HEOREM 3.2. Let us consider
, an open set
and a -admissible
vectorial measure in . If
is a finite union of compact intervals contained in
, for
, then:
(a) If
there exists a positive constant
such that
q 5a[@xDÙØ
µÛÚaµ ¦ §›¨ Ÿ HÌÜÞÝ £(I
q [
Ø
[Œª[ ^ 5`W%W%W`53[ Ø^
qà M!POF!Sq R
q
[P9[ ^ 5`W%W`W%53[ 5a[@ZDtØ
Ø
q 35 [@xDÙØ1^
(…ß"'
[^
Ø=^
Ø=^
O
’Š
< q a5 [@
Ei*
[
lm!€npQŽ0 q
a5 [@xDØ ^
q
âmc
´DFw+ < q 35 [@ .
qᐠÃB 
à B -â ^ 5`W%W%W`5aâ ACB ÃB
for all
H c I
Mt!sOS!&R
« ACB c ž ®
#{´ H I # µH ã ¤ I !#{´¢#`¦ {§ ¨ Ÿ H ¡C £(I 5
ca¬C^
’k < q 35 [@
O q c
HI
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91
GENERALIZED WEIGHTED SOBOLEV SPACES
ä q 5a[@åˆq D€Ø there exists a positive constant
à ŠÃ -â ^ 5`W%W`W%5aâ ACB such that for
´…D‚w
< 5a[@ , there exists ´ ^ D‚w
< q 53[@ , independent of â ^ 5%W`W%W57â A¢B and à ,
(b) If
every
with
#{´ ^ œP´C# ¦ {§ ¨ Ÿ H ¡¢ £(I àly5
« A¢B H c I ž ®
#›´ ^ # HLã.¤½I !\#{´ ^ #
¦ˆ§{¨ Ÿ H ¡@ £(I $#{´¢#`¦p§{¨ Ÿ H ¡C £(I W
ca¬C^
Furthermore, if ´ ^ 57 ^ are these representatives of ´ 57 respectively, we have for the same
constant Ã
« ACB
à #›´ ^ H c I œ… ^ H c I # ž"® HLã.¤3I !s#›´+œ….# ¦ §›¨ Ÿ H ¡@ £(I W
ca¬C^
Ã
q\ This theorem has the following corollary, that we use in the proof of Theorem 3.1:
, an open set
and a -admissible
C OROLLARY 3.3. Let us consider,
vectorial measure in . If
is a finite union of compact intervals contained in
, for
, then:
(a) If
there exists a positive constant
such that,
[
q
M™!…OP!VR
âˆc
O
q Hc I
l™!onpQŽ0 q
35 [@xD„Ø1^
à B Ã B âp^š5%W`W%W`5aâ A¢B « A¢B
à B #›´ H caæ B½I # ž Í HLã.¤aI !#{´¢# ¦ §›¨ Ÿ H ¡C £(I 5 ç"´pDFw < q 5a[@›W
ca¬C^
q
(b) If ä 5a[@åˆD€Ø there exists a positive constant à ŠÃ -â ^ 5`W%W`W%5aâ
ACB such that for
< q
< q
every ´PDŒw
5a[@ , there exists ´ ^ D…w
5a[@Z the same function as in Theorem 3.2 ,
with
« ACB H ac æ 3B I ž
#{´ ^ # Í LH ã.¤3I !#{´¾^_# ¦ §{¨ Ÿ H ¡@ £(I $#{´¢# ¦ §{¨ Ÿ H ¡C £(I W
ca¬C^
Furthermore, if ´š^_57^ are the representatives of ´ 57 respectively, we have for the same constant Ã
« ACB
à #›´ ^ H caæ B½I œ… ^ H caæ B3I # ž Í HLã.¤3I !s#›´+œ….#`¦ §›¨ Ÿ H ¡@ £(I W
ca¬C^
#{´¾^UœP´C# ¦ {§ ¨ Ÿ H ¡¢ £(I ly5
Ã
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
be a Cauchy sequence in
. Then, for each
,
is a Cauchy sequence in
and it converges to a function
.
First of all, let us show that can be extended to a function in
(if
)
and in
(if
).
l< !\q no!$0 ß H c I '
E 5a[c
ExJµB K , q H c A¢B½Ia
(ß"'
(´ c
l™Q…nˆ!d0
E < q 53[c
’k
< q 35 [@
)m q H c 3I ´(ctD
l!ŽntQV0
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92
J.M. RODRÍGUEZ, V. ALVAREZ, E. ROMERA, AND D. PESTANA
lm!onpQd0
 q Hc I
â d
â
ß DS’
< q a5 [@
è.53é‘Dtê
, let us consider any compact interval
. By part (b) of Theorem 3.2
If
we know there exists a representative (independent of ) of the class of
(which we also denote by ) and a positive constant such that for every
¾ß
Ã
«
ÃË#` ß H c I œ… ë H c I # ž ® HLãyI !
#% ß H ‡ I œo ë H ‡ I #%ž Ÿ H ¡C £(ì9I W
µ‡ ¬C^
c  )mâP , there exists a function í c Dj)m-âP such that
As ( ß H I '
«
ÃË#% ß H c I œ…í c_# ž ® HLãyI !
#% ß H ‡ I œ´š‡7#ž Ÿ H ¡¢ £ ì I W
‡µ¬C^
q
Since we can take as â q any compact interval contained qin H c I , we obtain that the funcc
c
tion í1c can be extended to H I and we have in fact í1cDF)m H I3 . It is obvious that ´cÞàí1c
q
c
in H c I q (except for at most a set of zero [Cc -measure), since ß H I converges
to ´c in the norm
îq H c I . Therefore
<
of E 53[ c and to q í c uniformly on each compact interval â
we can
c
assume that ´ c DF)m H I3 .
]q H c ACB3I . Now, part (b) of
If lsQTn!ï0 , let us consider any compact interval >
Corollary 3.3 gives
«
ÃË#% ß H c I œ… ë H c I # ž Í Hµð¾I !
#` ß H ‡ I œ… ë H ‡ I #ž Ÿ H ¡¢ £ ì I W
L‡ ¬¢^
c  ExB(½>@ , there exists a function ñ c DÙEUB(?>@ such that
As ( ß H I '
«
ÃË#` ß H c I œPñ=c_# ž Í Hµð¾I !
#% ß H ‡ I œP´š‡a#ž Ÿ H ¡C £ ì I W
‡L¬C^
q c
Since we can take as > q any compact interval contained in H q A¢B½I , we obtain that the function
ñ=c canq be extended to H c A¢B½I and we have in fact ñ"cDtEUJLB K , H c ACB3I3 . Itc is obvious that ´c
measure), since ß H I converges to ñ"c in
ñ=c in q H c c I (except for at most a set of zero q Lebesgue
JExµB K , H I7 and to ´c locally uniformly in H c I . Let us consider a set Y which concentrates
theq mass of
to show ñ"cV´c
Y™z\l ; we can take ñ"cs´c in Y . We only
q c [c
af Y, with
q c ), need
:
ó
l
s
‹
in c ‹p H IxòÂ
(recall
that
by
hypothesis
in
but
this
c
q
< q c 53: c implies the convergenceisinimmediate
E
EUJµB K , q c .
since : c DÙ; < c and the convergence in
q
c
Therefore we can assume that ´ c DtE JLB K , H A¢B½I .
q c
c
Inq fact, we have seen that ( ß H I ' converges to ´(c in EiJL* K , H I3 (if lo!nŒQ&0 ) and in
ExJµB K , H c A¢B½I7 (if l™Q…nˆ!S0 ).
q c
Let us see now that ´1c ô õ´caæ in the interior of H I for l€!n…Q0 . Let us consider
q Bc
a connected component ö of int H I . Given ÷Dd) , * ö , let us consider
convex hull
áq H c Ithe
in ö
. The uniform
â of supp ÷ . We chave that â is a compact interval contained
caæ
convergence of ß H I ' in â and the ExB convergence of ß H B3I ' in â gives that
Å ÷ ô ´ c ùߚøµû úL¯ Å ÷ ô ß H c I œüߚøLû úµ¯ Å ÷m ß H caæ B3I kœ Å ÷+´ caæ B W
* ã
* ã
ã
ã
q
q
a
c
æ
c
c
c
L
J
K
,
H
3
B
I
H
I
Then ´caæ S´ ^
q in int H I3 and ´ ^ DÙYh) intc H I33 for lm!€npQŽq 0 . In order to see
c B
that ´ ^ H I DÙYh) JµK , H c I3 , it is enough to recall that 9´ ^ H I ½ô Ž´caæ DFEUJLB K , H c I3 .
B
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GENERALIZED WEIGHTED SOBOLEV SPACES
93
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