Markov-type inequalities and duality in weighted Sobolev spaces

0.22 MB·English

by  Francisco Marcellán

MARKOV-TYPE INEQUALITIES AND DUALITY IN WEIGHTED
SOBOLEV SPACES
5
1 FRANCISCOMARCELLA´N(1), YAMILETQUINTANA(1), ANDJOSE´ M. RODR´IGUEZ(2)
0
2
n Abstract. The aim of this paper is to provide Markov-type inequalities in the setting
a ofweightedSobolevspaceswhentheconsideredweightsaregeneralizedclassicalweights.
J Also, as results of independent interest, some basic facts about Sobolev spaces with re-

5 spect to certain vector measures are stated.
2
Key words and phrases: Extremal problems; Markov-type inequality; weighted Sobolev

] norm; weighted L2-norm; duality.
A
C
.
h
t
a
m 1. Introduction
[ Given a norm on the linear space P of polynomials with real coefficients, the so-called

1 Markov-type inequalities are estimates connecting the norm of derivatives of a polynomial
v
with the norm of the polynomial itself. These inequalities are interesting by themselves
3
1 and play a fundamental role in the proof of many inverse theorems in polynomial approx-
2 imation theory (cf. [23, 27] and the references therein).
6
0
It is well known that for every polynomial P of degree at most n, the Markov inequality
.
1
0 P′ L∞([−1,1]) n2 P L∞([−1,1])
5 k k ≤ k k
1 holds and it is optimal since you have equality for the Chebyshev polynomials of the first
:
v kind.
i
X
The above inequality has been extended to the p norm (p 1) in [14]. For every
r
≥
a polynomial P of degree at most n their result reads
P′ C(n,p)n2 P
Lp([−1,1]) Lp([−1,1])
k k ≤ k k
Date: January 27, 2015.
2010 AMS Subject Classification: 33C45, 41A17, 26C99.
(1) Supported in part by a grant from Ministerio de Econom´ıa y Competitividad (MTM2012-36732-
C03-01), Spain.
(2) SupportedinpartbyagrantfromMinisteriodeEconom´ıayCompetitividad(MTM2013-46374-P),
Spain.
1
2 F.MARCELLA´N,Y.QUINTANAANDJ.M.RODR´IGUEZ
where the value of C(n,p) is explicitly given in terms of p and n. Indeed, you have a
bound C(n,p) 6e1+1/e for n > 0 and p 1. In [11] admissible values for C(n,p) and
≤ ≥
some computational results for p = 2 are deduced. Notice that for any p > 1 and every
polynomial P of degree at most n
P′ Cn2 P ,
Lp([−1,1]) Lp([−1,1])
k k ≤ k k
where C is explicitly given and it is less than the constant C(n,p) in [14].
On the other hand, using matrix analysis, in [8] it is proved that the exact value of
C(n,2) is the greatest singular value of the matrix A = [a ] , where
n j,k 0≤j≤n−1,0≤k≤n
a = 1 p′(x)p (x)dx and p ∞ is the sequence of orthonormal Legendre polyno-
j,k −1 j k { n}n=0
mials. A simple proof of this result, with an interpretation of the the sharp constant
R
C(n,2) as the largest positive zero of a polynomial as well as an explicit expression of the
extremal polynomial (the polynomial such that the inequality becomes an equality) in the
L2- Markov inequality appears in [15].
If you consider weighted L2-spaces, the problem becomes more difficult. For instance,
let be a weighted L2-norm on P, given by
L2((a,b),w)
k · k
b 1/2
P = P(x)2w(x)dx ,
L2((a,b),w)
k k | |
(cid:18)Za (cid:19)
where w is an integrable function on (a,b), a < b , such that w > 0 a.e. on
−∞ ≤ ≤ ∞
(a,b) and all moments
b
r := xnw(x)dx, n 0,
n
≥
Za
are finite. It is clear that there exists a constant γ = γ (a,b,w) such that
n n
(1.1) P′ γ P , for all P P ,
L2((a,b),w) n L2((a,b),w) n
k k ≤ k k ∈
where P is the space of polynomials with real coefficients of degree at most n. Indeed,
n
the sharp constant is the greatest singular value of the matrix B = [b ] ,
n j,k 0≤j≤n−1,0≤k≤n
where b = 1 p′(x)p (x)w(x)dx and p ∞ is the orthonormal polynomial system
j,k −1 j k { n}n=0
with respect to the positive measure w(x)dx. Thus, from a computational point of view
R
you need to find the connection coefficients between the sequences p′ ∞ and p ∞
{ n}n=0 { n}n=0
in order to proceed with the computation of the matrix, and in a second step, to find
the greatest singular value of the matrix B . Notice that for classical weights (Jacobi,
n
Laguerre and Hermite), such connection coefficients can be found in a simple way.
Markov-typeinequalitiesinweighted Sobolevspaces 3
Mirsky[25]showedthatthebestconstantγn∗ := supP∈Pn{||P′||L2((a,b),w) :||P||L2((a,b),w) =
1 in (1.1) satisfies
}
n 1/2
(1.2) γ∗ ν p′ 2 .
n ≤ k νkL2((a,b),w)
!
ν=1
X
Notice that themain interest of the above resultis however qualitative, since the bound
specified by (1.2) can be very crude. In fact, when w(x) = e−x2 on ( , ), the estimate
−∞ ∞
(1.2) becomes
n 1/2
1
γ∗ 2ν2 = n(n+1)(2n+1) = O n3/2 .
n ≤ 3
ν=1 ! r
X (cid:0) (cid:1)
ThecontrastbetweenthisestimateandtheclassicresultofSchmidt[38],whichestablishes
γ∗ =√2n, is evident.
n
Also, when we consider the weighted L2-norm associated with the Laguerre weight
w(x) := xαe−x in [0, ), the results in [3] give the following inequality
∞
(1.3) P′ C n P , for all P P .
L2(w) α L2(w) n
k k ≤ k k ∈
Notice that the nature of the extremal problems associated to the inequalities (1.1) and
(1.3)isdifferent, sincein thefirstcase theconstant on theright-hand sideof (1.1)depends
onn, whileinthesecondonethemultiplicative constant C ontheright-handsideof (1.3)
α
is independent of n.
There exist a lot of results on Markov-type inequalities (see, e.g. [9, 10, 23], and the
references therein). In connection with the research in the field of the weighted approx-
imation by polynomials, Markov-type inequalities have been proved for various weights,
norms, sets over which the norm is taken (see, e.g. [22] and the references therein) and
more recently, the study of asymptotic behavior of the sharp constant involved in some
kind of these inequalities have been done in [3] for Hermite, Laguerre and Gegenbauer
weights and in [4] for Jacobi weights with parameters satisfying some constraints.
On the other hand, a similar problem connected with the Markov-Bernstein inequality
has been analyzed in [12] when you try to determine the sharp constant C(n,m;w) such
that
(1.4) Am/2P(m) C(n,m;w) P , for all P P .
L2((a,b),w) L2((a,b),w) n
k k ≤ k k ∈
Here w is a classical weight satisfying a Pearson equation (A(x)w(x))′ = B(x)w(x) and
A,B are polynomials of degree at most 2 and 1, respectively.
4 F.MARCELLA´N,Y.QUINTANAANDJ.M.RODR´IGUEZ
An analogue of the Markov-Bernstein inequality for linear operators T from P into P
n
has beenstudied in [17]in terms of singular values of matrices. Someillustrative examples
when T is either the derivative or the difference operator and you deal with some classical
weights (Laguerre, Gegenbauer in the first case, Charlier, Meixner in the second one) are
shown. Another recent application of Markov-Bernstein-type inequalities can be found in
[5].
With these ideas in mind, one of the authors of the present paper posed in 2008 during
a conference on Constructive Theory of Functions held in Campos do Jord˜ao, Brazil,
the following problem: Find the analogous of Markov-type inequalities in the setting of
weighted Sobolev spaces. A partial answer of this problem was given in [27], considering
an extremal problem with similar conditions to those given by Mirsky, and following the
scheme of Kwon and Lee [17], mainly.
The first part of this paper is devoted to provide another partial solution of the above
problem, which is based on the adequate use of inequalities of kind (1.3) [3, 9, 38], in the
setting of weighted Sobolev spaces, when the considered weights are generalized classical
weights. In the second part we study some basic facts about Sobolev spaces with respect
to measures: separability, reflexivity, uniform convexity and duality, which to the best of
our knowledge are not available in the current literature. These Sobolev spaces appear
in a natural way and are a very useful tool when we study the asymptotic behavior of
Sobolev orthogonal polynomials (see [7], [18], [19], [31], [32], [33], [35], [37]).
The outline of the paper is as follows. The first part of Section 2 provides some short
background about Markov-type inequalities in L2 spaces with classical weights and the
second one deals with a Markov-type inequality corresponding to each weighted Sobolev
norm with respect to these classical weights and to some generalized weights (see The-
orem 2.1). Section 3 contains definitions and a discussion about the appropriate vector
measures which we will need in order to get completeness of our Sobolev spaces with re-
spect to measures. Finally, Section 4 contains some basic results on Sobolev spaces with
respect to the vector measures defined in the previous section (see Theorems 4.2 and 4.3):
separability, reflexivity, uniform convexity and duality.
2. Markov-type inequalities in Sobolev spaces with weights
The following proposition summarizes the Markov-type inequalities in L2 spaces with
classical weights, which will be used in the sequel. Recall that we denote by P the linear
n
space of polynomials with real coefficients and degree less than or equal to n.
Proposition 2.1. The following inequalities are satisfied.
(1) Laguerre case [3]:
P′ C n P ,
L2(w) α L2(w)
k k ≤ k k
where w(x) := xαe−x in [0, ), α> 1 and P P .
n
∞ − ∈
Markov-typeinequalitiesinweighted Sobolevspaces 5
(2) Generalized Hermite case [38], [9]:
P′ √2n P ,
L2(w) L2(w)
k k ≤ k k
where w(x) := x αe−x2 in R, α 0 and P P .
n
| | ≥ ∈
(3) Jacobi case [38] (see also [10]):
P′ C n2 P ,
L2(w) α,β L2(w)
k k ≤ k k
where w(x) := (1 x)α(1+x)β in [ 1,1], α,β > 1 and P P .
n
− − − ∈
The multiplicative constants C and C are independent of n.
α α,β
InTheorem2.1belowweextendtheseresultstothecontextofweighted Sobolevspaces.
We want to remark that the proof provides explicit expressions for the involved constants.
Theorem 2.1. The following inequalities are satisfied.
(1) Laguerre-Sobolev case:
P′ C n P ,
k kWk,2(w,λ1w,...,λkw) ≤ α k kWk,2(w,λ1w,...,λkw)
where w(x) := xαe−x in [0, ), α > 1, λ ,...,λ 0, P P and C is the
1 k n α
∞ − ≥ ∈
same constant as in Proposition 2.1 (1).
(2) Generalized Hermite-Sobolev case:
P′ √2n P ,
k kWk,2(w,λ1w,...,λkw) ≤ k kWk,2(w,λ1w,...,λkw)
where w(x) := x αe−x2 in R, α 0, λ ,...,λ 0 and P P .
1 k n
| | ≥ ≥ ∈
(3) Jacobi-Sobolev case:
P′ C n2 P ,
k kWk,2(w,λ1w,...,λkw) ≤ α,β k kWk,2(w,λ1w,...,λkw)
where w(x) := (1 x)α(1+x)β in [ 1,1], α,β > 1, λ ,...,λ 0, P P and
1 k n
− − − ≥ ∈
C is the constant in Proposition 2.1 (3).
α,β
(4) Let us consider the generalized Jacobi weight w(x) := h(x)Πrj=1|x−cj|γj in [a,b]
with c ,...,c R, γ ,...,γ R, γ > 1 when c [a,b], and h a measurable
1 r 1 r j j
∈ ∈ − ∈
function satisfying 0 < m h M in [a,b] for some constants m,M. Then we
≤ ≤
have
P′ C (a,b,c ,...,c ,γ ,...,γ ,m,M)n2 P ,
k kWk,2(w,λ1w,...,λkw) ≤ 1 1 r 1 r k kWk,2(w,λ1w,...,λkw)
for every λ ,...,λ 0 and P P .
1 k n
≥ ∈
(5) Consider now the generalized Laguerre weight w(x) := h(x)Πrj=1|x −cj|γje−x in
[0, ) with c < < c , c 0, γ ,...,γ R, γ > 1 when c 0, and h
1 r r 1 r j j
∞ ··· ≥ ∈ − ≥
a measurable function satisfying 0 < m h M in [0, ) for some constants
≤ ≤ ∞
m,M.
(5.1) If r−1γ = 0, then
j=1 j
P′ P C (c ,...,c ,γ ,...,γ ,m,M)n2 P ,
k kWk,2(w,λ1w,...,λkw) ≤ 2 1 r 1 r k kWk,2(w,λ1w,...,λkw)
6 F.MARCELLA´N,Y.QUINTANAANDJ.M.RODR´IGUEZ
for every λ ,...,λ 0 and P P .
1 k n
(5.2) Assume that≥c < <∈c and r γ > 1. Let r := min 1 j
1 ··· r j=1 j − 0 { ≤ ≤
r c 0 , γ′ := γ′ := 0 and γ′ := γ for every r j r. Assume that
| j ≥ } r0−1 r+1 j Pj 0 ≤ ≤
max γ′,γ′ 1/2 for every r 1 j r. Then we have
{ j j+1} ≥ − 0− ≤ ≤
P′ C′(c ,...,c ,γ ,...,γ ,m,M)na′ P ,
k kWk,2(w,λ1w,...,λkw) ≤ 2 1 r 1 r k kWk,2(w,λ1w,...,λkw)
for every λ ,...,λ 0 and P P , where
1 k n
≥ ∈
b′+2
a′ := max 2, , b′ := max γ′ +γ′ + γ′ γ′ +2 .
2 r0−1≤j≤r j j+1 | j − j+1|
n o
(cid:0) (cid:1)
(6) Let usconsider the generalized Hermite weight w(x) := h(x)Πrj=1|x−cj|γje−x2 inR
with c < < c , γ ,...,γ > 1 with r γ 0 and h a measurable function
satisfy1ing ·0··< mr h1 M irn R−for some cjo=n1stajn≥ts m,M. Define γ := γ := 0
0 r+1
≤ ≤ P
and assume that max γ ,γ 1/2 for every 0 j r. Then we have
j j+1
{ } ≥ − ≤ ≤
P′ C (c ,...,c ,γ ,...,γ ,m,M)na P ,
k kWk,2(w,λ1w,...,λkw) ≤ 3 1 r 1 r k kWk,2(w,λ1w,...,λkw)
for every λ ,...,λ 0 and P P , where
1 k n
≥ ∈
b+1
a := max 2, , b := max γ +γ + γ γ +2 .
j j+1 j j+1
2 0≤j≤r | − |
n o
(cid:0) (cid:1)
In each case the multiplicative constants depend just on the specified parameters (in
particular, they do not depend on n).
Remark 2.1. Note that (4), (5), and (6) are new results in the classical (non-Sobolev)
context (taking λ = = λ = 0). In (5.2), there is no hypothesis on r−1γ .
1 ··· k j=1 j
Proof. First of all, note that if the inequality P
P′ C(n,w) P
L2(w) L2(w)
k k ≤ k k
holds for every polynomial P P and some fixed weight w, then we have
n
∈
(2.5) P(j+1) C(n,w) P(j)
L2(λw) L2(λw)
k k ≤ k k
for every polynomial P P and every λ 0. Consequently, for the weighted Sobolev
n
norm on P ∈ ≥
k
1/2
P := P 2 + P(j) 2 , λ ,...,λ 0,
k kWk,2(w,λ1w,...,λkw) L2(w) L2(λjw) 1 k ≥
(cid:16)(cid:13) (cid:13) Xj=1(cid:13) (cid:13) (cid:17)
(cid:13) (cid:13) (cid:13) (cid:13)
we have
(2.6) P′ C(n,w) P ,
k kWk,2(w,λ1w,...,λkw) ≤ k kWk,2(w,λ1w,...,λkw)
for every polynomial P P and every λ ,...,λ 0.
n 1 k
∈ ≥
Thus, (1), (2) and (3) hold.
Markov-typeinequalitiesinweighted Sobolevspaces 7
Inordertoprove(4),notethatusinganaffinetransformationoftheformTx = α x+α ,
1 2
we obtain from Proposition 2.1 (3)
P′ C(a ,a ,α,β)n2 P ,
L2(w) 1 2 L2(w)
k k ≤ k k
for the weight w(x) := (a x)α(x a )β in [a ,a ] and every polynomial P P .
2 1 1 2 n
− − ∈
Without loss of generality we can assume that a c < < c b, since otherwise
1 r
≤ ··· ≤
we can consider
w(x) = h˜(x) x c γj, h˜(x) := h(x) x c γj.
j j
| − | | − |
1≤j≤r 1≤j≤r
Y Y
cj∈[a,b] cj∈/[a,b]
If we define c := a, c := b and γ := γ := 0, then we can write w(x) =
0 r+1 0 r+1
h(x)Πrj=+01|x−cj|γj. Denote by hj the function
w(x)
h (x):= ,
j x cj γj x cj+1 γj+1
| − | | − |
for 0 j r. It is clear that there exist positive constants m ,M (depending just on
j j
≤ ≤
m,M,c ,...,c ,γ ,...,γ ), with m h (x) M for every x [c ,c ].
1 r 1 r j j j j j+1
Hence, for P P , we have ≤ ≤ ∈
n
∈
cj+1 1/2
P′ = P′(x)2h (x)x c γj x c γj+1dx
k kL2([cj,cj+1],w) | | j | − j| | − j+1|
(cid:16)Zcj (cid:17)
cj+1 1/2
M P′(x)2 x c γj x c γj+1dx
j j j+1
≤ | | | − | | − |
p (cid:16)Zcj cj+1 (cid:17) 1/2
M C(c ,c ,γ ,γ )n2 P(x)2 x c γj x c γj+1dx
j j j+1 j j+1 j j+1
≤ | | | − | | − |
p (cid:16)Zcjcj+1 h ((cid:17)x) 1/2
M C(c ,c ,γ ,γ )n2 P(x)2 x c γj x c γj+1 j dx
j j j+1 j j+1 j j+1
≤ | | | − | | − | m
p (cid:16)Zcj j (cid:17)
M
= j C(c ,c ,γ ,γ )n2 P .
smj j j+1 j j+1 k kL2([cj,cj+1],w)
Next, “pasting” several times this last inequality in each subinterval [c ,c ] [a,b],
j j+1
⊆
0 j r, we obtain
≤ ≤
P′ C (a,b,c ,...,c ,γ ,...,γ ,m,M)n2 P
L2(w) 1 1 r 1 r L2(w)
k k ≤ k k
for every polynomial P P , with
n
∈
M
j
C (a,b,c ,...,c ,γ ,...,γ ,m,M) := max C(c ,c ,γ ,γ ).
1 1 r 1 r j j+1 j j+1
0≤j≤rsmj
Hence, we obtain the case (4) by applying (2.6).
8 F.MARCELLA´N,Y.QUINTANAANDJ.M.RODR´IGUEZ
Similarly, for the case (5.1) we can write
r
w(x) =H (x) x c γj,
1 j
| − |
j=1
Y
where H (x) := h(x)e−x satisfies 0 < me−cr H M in [0,c ]. Then the case (4)
1 1 r
≤ ≤
provides a constant C , which just depends on the appropriate parameters, with
1
(2.7) P′ C n2 P ,
k kWk,2([0,cr],w,λ1w,...,λkw) ≤ 1 k kWk,2([0,cr],w,λ1w,...,λkw)
for every λ ,...,λ 0 and every polynomial P P .
1 k n
≥ ∈
Proposition 2.1 (1) gives
P′ C n P ,
k kL2(w1) ≤ α k kL2(w1)
where w (x) := xαe−x in [0, ), α > 1 and P P . Hence, replacing x by x c, we
1 n
∞ − ∈ −
obtain with the same constant C
α
P′ L2([c,∞),(x−c)αec−x) Cαn P L2([c,∞),(x−c)αec−x),
k k ≤ k k
for every c 0 and P P . Now, if w (x) := (x c)αe−x, then the previous inequality
n 2
≥ ∈ −
implies
P′ C n P ,
k kL2([c,∞),w2) ≤ α k kL2([c,∞),w2)
for every c 0 and P P , and (2.6) gives
n
≥ ∈
P′ C n P ,
k kWk,2([c,∞),w2,λ1w2,...,λkw2) ≤ α k kWk,2([c,∞),w2,λ1w2,...,λkw2)
for every c 0, λ ,...,λ 0 and P P .
1 k n
≥ ≥ ∈
We can write now
w(x) =H (x)(x c )γre−x,
2 r
−
where H2(x) := h(x)Πjr=−11|x−cj|γj and there exist constants m2,M2 with 0 < m2 ≤ H2 ≤
M in [c , ), since r−1γ = 0. Thus,
2 r ∞ j=1 j
P M
(2.8) P′ C 2 n P ,
k kWk,2([cr,∞),w,λ1w,...,λkw) ≤ γr m k kWk,2([cr,∞),w,λ1w,...,λkw)
r 2
for every λ ,...,λ 0 and P P .
1 k n
≥ ∈
If we define C := max C , C M /m , then (2.7) and (2.8) give
2 1 γr 2 2
P′ (cid:8) p C(cid:9)n2 P ,
k kWk,2(w,λ1w,...,λkw) ≤ 2 k kWk,2(w,λ1w,...,λkw)
for every λ ,...,λ 0 and P P .
1 k n
≥ ∈
Let us prove now (5.2). Define A := 1+c . We can write
r
r
w(x) =H (x) x c γj,
3 j
| − |
j=1
Y
Markov-typeinequalitiesinweighted Sobolevspaces 9
where H (x) := h(x)e−x satisfies 0 < me−A H M in [0,A]. Then the case (4)
3 3
≤ ≤
provides a constant C , which just depends on the appropriate parameters, with
1
(2.9) P′ C n2 P ,
k kWk,2([0,A],w,λ1w,...,λkw) ≤ 1 k kWk,2([0,A],w,λ1w,...,λkw)
for every λ ,...,λ 0 and every polynomial P P .
1 k n
≥ ∈
Proposition 2.1 (1) gives a constant C with
s
P′ 2 C2n2 P 2 ,
k kL2(w3) ≤ s k kL2(w3)
where w (x) := xse−x, s := r γ > 1 and P P .
3 j=1 j − ∈ n
We can write now
P
w(x) = H (x)xse−x = H (x)w (x),
4 4 3
where H4(x) := h(x)x−sΠrj=1|x−cj|γj, and there exist constants m4,M4 with 0 < m4 ≤
H M in [A, ), since s = r γ . Thus,
4 ≤ 4 ∞ j=1 j
P′ 2 PM P′ 2 C2n2M P 2
k kL2([A,∞),w) ≤ 4k kL2(w3) ≤ s 4k kL2(w3)
(2.10) M
C2n2 4 P 2 +C2n2M P 2 ,
≤ s m k kL2([A,∞),w) s 4k kL2([0,A],w3)
4
for every P P .
n
∈
Using Lupa¸s’ inequality [20] (see also [24, p.594]):
Γ(n+α+β +2) n+q+1 1
kPkL∞([−1,1]) ≤ s2α+β+1Γ(q+1)Γ(n+q′+1)(cid:18) n (cid:19)sZ−1|P(x)|2(1−x)α(1+x)βdx,
for every P P , where q = max(α,β) 1/2 and q′ = min(α,β), we obtain that
n
∈ ≥ −
Γ(n+α+β +2)Γ(n+q+2) 1
2 P 2 P(x)2(1 x)α(1+x)βdx.
k kL∞([−1,1]) ≤ 2α+βΓ(q+1)Γ(q+2)Γ(n+1)Γ(n+q′+1) | | −
Z−1
Now, taking into account that
Γ(n+x)
lim = 1, x,y R,
n→∞ Γ(n+y)nx−y ∈
we get
Γ(n+α+β +2)Γ(n+q+2) nα+β+1nq−q′+1 = nα+β+|α−β|+2.
Γ(n+1)Γ(n+q′+1) ∼
Consequently, there exists a constant k , which just depends on α and β, such that
1
1 1
P(x)2dx 2 P 2 k (α,β)nv(α,β) P(x)2(1 x)α(1+x)βdx,
| | ≤ k kL∞([−1,1]) ≤ 1 | | −
Z−1 Z−1
where v(α,β) =α+β + α β +2 and P P .
n
| − | ∈
Recall that max γ′,γ′ 1/2 for every r 1 j r and
{ j j+1} ≥ − 0− ≤ ≤
b′ := max γ′ +γ′ + γ′ γ′ +2 .
r0−1≤j≤r j j+1 | j − j+1|
(cid:0) (cid:1)
10 F.MARCELLA´N,Y.QUINTANAANDJ.M.RODR´IGUEZ
Therefore, a similar argument to the one in the proof of (4) gives
A A r+1 A r
P(x)2dx k2nb′ P(x)2 x cj γj′ dx = k2nb′ P(x)2 x cj γj dx,
| | ≤ | | | − | | | | − |
Z0 Z0 j=Yr0−1 Z0 jY=r0
foreverypolynomialP P andsomeconstantk whichjustdependsonc ,...,c ,γ ,...,γ .
∈ n 2 r0 r r0 r
Thus,
A A r
P(x)2dx k nb′ P(x)2 x c γj dx,
3 j
| | ≤ | | | − |
Z0 Z0 j=1
Y
foreverypolynomialP P andsomeconstantk whichjustdependsonc ,...,c ,γ ,...,γ .
n 3 1 r 1 r
∈
Hence,
A A
P 2 = P(x)2xse−xdx As P(x)2dx
k kL2([0,A],w3) | | ≤ | |
Z0 Z0
A r
k nb′As P(x)2 x c γj dx
3 j
≤ | | | − |
Z0 j=1
Y
1 k nb′AseA A P(x)2h(x) r x c γje−xdx
3 j
≤ m | | | − |
Z0 j=1
Y
1 k AseAnb′ P 2 ,
≤ m 3 k kL2(w)
for every polynomial P P .
n
∈
This inequality and (2.10) give
P′ 2 max C2 M4 n2, C2 M4 k AseAnb′+2 P 2
k kL2([A,∞),w) ≤ s m s m 3 k kL2(w)
4
k nbn′+2 P 2 , o
≤ 4 k kL2(w)
for every polynomial P P , where
n
∈
M M
k := max C2 4 ,C2 4 k AseA .
4 s m s m 3
4
n o
Hence,
P′ 2 k nb′+2 P 2 ,
k kWk,2([A,∞),w,λ1w,...,λkw) ≤ 4 k kWk,2(w,λ1w,...,λkw)
for every λ ,...,λ 0 and every polynomial P P , and (2.9) allows to deduce
1 k n
≥ ∈
P′ 2 C2n4+k nb′+2 P 2 ,
k kWk,2(w,λ1w,...,λkw) ≤ 1 4 k kWk,2(w,λ1w,...,λkw)
for every λ ,...,λ 0 and every po(cid:0)lynomial P P .(cid:1)If we define
1 k n
≥ ∈
k := C2+k 1/2,
5 1 4
(cid:0) (cid:1)