A sharp weighted Wirtinger inequality

0.11 MB·English

by  Tonia Ricciardi

A sharp weighted Wirtinger inequality
Tonia Ricciardi∗
Dipartimento di Matematica e Applicazioni
5 Universit`a di Napoli Federico II
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Via Cintia, 80126 Naples, Italy
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2 fax: [phone]
[email protected]
n
a
J

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Abstract
]
P
We obtain a sharp estimate for the best constant C > 0 in the Wir-
A
tinger typeinequality
.
h
2π 2π
at γpw2 ≤C γqw′2
m Z0 Z0
[ where γ is bounded above and below away from zero, w is 2π-periodic
and such that 2πγpw = 0, and p+q ≥ 0. Our result generalizes an
1 0
inequality of Piccinini and Spagnolo.
v R
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Let C(a,b) > 0 denote the best constant in the following weighted Wirtinger
4
type inequality:
0
1
0 2π 2π
5 (1) aw2 C(a,b) bw′2,
≤
0 Z0 Z0
h/ where w H1 (R) is 2π-periodic and satisfies the constraint
∈ loc
t
a
2π
m
(2) aw=0,
v: Z0
Xi and a,b with
∈B
r
a = a L∞(R) : a is 2π periodic and infa>0 .
B { ∈ − }
Hereandinwhatfollows,foreverymeasurablefunctionawedenotebyinfaand
supatheessentiallowerboundandtheessentialupperboundofa,respectively.
For every L>1, we denote
(L)= a L (0,2π) : a is 2π periodic, infa=1 and supa=L .
∞
B { ∈ − }
Our aim in this note is to prove:
∗Supported in part by Regione Campania L.R. 5/02 and by the MIUR National Project
Variational Methods and Nonlinear DifferentialEquations.
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Theorem 1. Suppose a=γp and b=γq for some γ (M), M >1, and for
some p,q R such that p+q 0. Then ∈B
∈ ≥
1 2πγ(p q)/2 2
(3) C(γp,γq) 2π 0 − .
≤ π4 arctaRn M−(p+q)/4 !
(cid:0) (cid:1)
If p+q>0, then equality holds in (3) if and only if γ(θ)=γ¯ (θ+ϕ) for some
p,q
ϕ R, where
∈
1, if 0 θ <c π, π θ <π+c π
γ¯ (θ)= ≤ p,q2 ≤ p,q2 ,
p,q (M, if cp,qπ2 ≤θ <π, π+cp,qπ2 ≤θ <2π
with
2
c = .
p,q 1+M (p q)/2
− −
Furthermore,equalityholdsin (1)–(2)witha(θ)=γ¯p (θ+ϕ)andb(θ)=γ¯q (θ+
p,q p,q
ϕ) if and only if w(θ)=w¯ (θ+ϕ) where
p,q
w¯ (θ)=
p,q
sin √µ c−p,1qθ− π4 ,
if 0 θ <c π
 (cid:2) (cid:0) (cid:1)(cid:3) ≤ p,q2
=M−s−in(p+√q)µ/4cπo+s(cid:2)c√−p,µ1q((cid:0)θπ2−+πc)−p−,1qM54π(p−ifq,)c/p2,(qθπ2−≤cpθ,q<π2)π− 34π(cid:1)(cid:3), ,
if π θ <π+c π
(cid:2) (cid:0) (cid:1)(cid:3) ≤ p,q2
and µ=−(4M/π−)(pa+rcqt)/a4ncMos(cid:2)√(p+µq(cid:0))322π.+c−p,1qMi(fp−πq+)/2c(pθ,q−π2 π≤−θc<p,q2ππ2)− 74π(cid:1)(cid:3),
−
If p+q = 0, then (3) is an equality for any weight function γ. Equality is
(cid:0) (cid:1)
attained in (1)–(2) with a=γp and b=γ p if and only if
−
2π θ
w(θ)=Ccos γp+ϕ ,
02πγp Z0 !
for some C =0 and ϕ R. R
6 ∈
Note that when p = q = 0, Theorem 1 yields C(1,1) = 1 according to the
classicalWirtinger inequality. When p=q =0, the estimate (3) reduces to the
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estimate obtained by Piccinini and Spagnolo in [4]. More related results may
be found in [1, 2, 3] and in the references therein. We begin by recalling in the
following lemma the Wirtinger inequality of Piccinini and Spagnolo [4].
Lemma 1 ([4]). Suppose b=a (L). Then,
∈B
2
(4) C(a,a) 4 arctanL−1/2 − .
≤ π
(cid:18) (cid:19)
2
Equality holds in (4) if and only if a(θ)=a¯(θ+ϕ) for some ϕ R, where a¯ is
∈
defined by
1, if 0 θ < π, π θ < 3π
(5) a¯(θ)= ≤ 2 ≤ 2
(L, if π θ <π, 3π θ <2π
2 ≤ 2 ≤
and equality holds in (1)–(2) with a(θ)=b(θ)=a¯(θ+ϕ) if and only if w(θ)=
w¯(θ+ϕ), where
sin √λ θ π , if 0 θ < π
− 4 ≤ 2
(6) w¯(θ)=L−−s1ihn/2c√o(cid:0)λsh√θλ−(cid:0)(cid:1)5θi4π− 3,4π(cid:1)i, iiff ππ2 ≤≤θθ<<3π2π ,
where λ= 4π 1arcta−nLL−11h//22co2(cid:0)s.h√λ(cid:0)θ(cid:1)i− 74π(cid:1)i, if 32π ≤θ <2π
− −
In order(cid:0)to prove Theorem(cid:1)1, we need the following lemma, which yields an
estimate for C(a,b) for arbitrary weight functions a,b.
Lemma 2. Let a,b . The following estimate holds:
∈B
2
1 2π√ab 1
(7) C(a,b) 2π 0 − .
≤ 1/4
4 arctaRn infab
π supab 
 (cid:16) (cid:17) 
If √ab (L), L>1, then
∈B
(8)
2
1 C2π(a√,ba)b 1 2 =√a′bs′up(L) 1 C2π(a√′,ab′b) 1 2 =(cid:18)π4 arctanL−1/2(cid:19)−
2π 0 − ∈B 2π 0 ′ ′−
(cid:16) (cid:17) (cid:16) (cid:17)
R R
if and only if the following equation is satisfied:
(9) a(θ(τ))b(θ(τ)) =a¯2(τ +ϕ) a.e. τ (0,2π), for some ϕ R,
∈ ∈
where θ(τ) is the homeomorphism of R defined by
1 θ a(θ˜)
(10) τ(θ)= dθ˜,
c Z0 sb(θ˜)
c is defined by
1 2π a(θ˜)
(11) c= dθ˜,
2π Z0 sb(θ˜)
and a¯ is the function defined in Lemma 1.
2
If b = a 1, then C(a,a 1) = (2π) 1 2πa and equality is attained in
− − − 0
(1)–(2) with b=a−1 if and only if w(cid:16)(θ)=CRcos(2(cid:17)π( 02πa)−1 0θa+ϕ) for some
C =0 and ϕ R.
6 ∈ R R
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Proof. Under the change of variables θ = θ(τ) defined by (10)–(11), setting
α(τ)=a(θ(τ)), β(τ)=b(θ(τ)), ξ(τ)=w(θ(τ)), we obtain
αθ′ =c αβ, βθ′−1 =c−1 αβ,
and therefore: p p
2π 2π
aw2dθ = αθ ξ2dτ =c αβξ2dτ
′
Z0 Z0 Z
2π 2π p
awdθ = αθ ξdτ =c αβξdτ =0
′
Z0 Z0 Z
2π 2π p
bw′2dθ = βθ′−1ξ′2dτ =c−1 αβξ′2dτ.
Z0 Z0 Z
p
Upon substitution, (1)–(2) takes the form:
2π C(a,b) 2π
(12) Z0 αβξ2dτ ≤ 1 2π√ab 1 2 Z0 αβξ′2dτ,
p 2π 0 − p
with constraint (cid:16) R (cid:17)
2π
(13) αβξdτ =0.
Z0
p
If √ab (L), in view of Lemma 1 we obtain
∈B
2
−
C(a,b) 4 inf√αβ
(14) =C( αβ, αβ) arctan
1 2π√ab 1 2 ≤π ssup√αβ
2π 0 − p p
 
(cid:16) R (cid:17) 4 infab 1/4 −2
= arctan .
π supab
(cid:18) (cid:19) !
This yields (7). Moreover, we have C(√αβ,√αβ) = (4/π)arctanL−1/2 −2 if
and only if α(τ)β(τ) = a¯(τ +ϕ), for some ϕ R. That is, (8) holds if and
∈ (cid:0) (cid:1)
only if (9) holds.
p
If b=a 1, then (12)–(13) takes the form
−
2π C(a,a 1) 2π
ξ2dτ − ξ2dτ
Z0 ≤ 1 2πa 2 Z0 ′
2π 0
(cid:16) (cid:17)
with constraint R
2π
ξdτ =0.
Z0
Therefore, by the classical Wirtinger inequality,
1 2π 2
C(a,a−1)= a
2π
(cid:18) Z0 (cid:19)
andequality holdsin(1)–(2)withb=a 1 ifandonlyif ξ(τ)=Ccos(τ+ϕ)for
−
someC =0andϕ R,thatis,ifandonlyifw(θ)=Ccos(2π( 2πa) 1 θa+ϕ),
6 ∈ 0 − 0
as asserted.
R R
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Lemma 3. Suppose a,b satisfy √ab (L), L > 1, and (9), where θ(τ) is
∈ B
defined in (10) and c is defined by (11). Suppose
(15) a=γp, b=γq
for some γ (M), with M = L2/(p+q), and for some p,q R such that
p+q >0. T∈henBγ(θ)=γ¯ (θ+ϕ) for some ϕ R, where γ¯ ∈is the function
p,q p,q
∈
defined in Theorem 1.
Proof. When p+q > 0, we have γ(p+q)/2 (L). In view of (9) and (15) we
∈ B
have
γ(θ(τ))=a¯2/(p+q)(τ +ψ), τ R
∀ ∈
for some ψ R. It follows that
∈
τ b(θ(τ¯)) τ
(16) θ(τ)=c dτ¯=c a¯ (p q)/(p+q)(τ¯+ψ)dτ¯
− −
Z0 sa(θ(τ¯)) Z0
and, in view of the 2π-periodicity of a and b,
1
1 2π b(θ(τ¯)) − 1 2π −1
c= dτ¯ = a¯ (p q)/(p+q)(τ¯)dτ¯ .
− −
2π Z0 sa(θ(τ¯)) ! (cid:18)2π Z0 (cid:19)
Setting
τ
h (τ)=c a¯ (p q)/(p+q)(τ¯)dτ¯,
p,q − −
Z0
we have θ(τ ψ)=h (τ) h (ψ) for every τ R, and consequently τ(θ)=
p,q p,q
− − ∈
h 1(θ+h (ψ)) ψ. Inviewofthedefinitionofa¯ withL=M(p+q)/2,wehave:
−p,q p,q −
τ
a¯ (p q)/(p+q)(τ¯)dτ¯=
− −
Z0
τ, if 0 τ < π
≤ 2
π +M (p q)/2(τ π), if π τ <π
=2 − − − 2 2 ≤ .
π2(1+M−(p−q)/2)+τ −π, if π ≤τ < 32π
π(2+M (p q)/2)+M (p q)/2(τ 3π), if 3π τ <2π
2 − − − − − 2 2 ≤
In particular, we derive
2
c= =c .
1+M (p q)/2 p,q
− −
It follows that h (τ) is the piecewise linear homeomorphism of R defined in
p,q
[0,2π) by
c τ, if 0 τ < π
p,q ≤ 2
c π +M (p q)/2(τ π) , if π τ <π
h (τ)= p,q 2 − − − 2 2 ≤
p,q cp,q(cid:2)π2(1+M−(p−q)/2)+τ −(cid:3) π , if π ≤τ < 32π
c π(2+M (p q)/2)+M (p q)/2(τ 3π) , if 3π τ <2π
p,q(cid:2)2 − − − (cid:3)− − 2 2 ≤
 (cid:2) (cid:3)
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andby h (τ+2πn)=2πn+h (τ), for any τ [0,2π)and for any integern.
p,q p,q
∈
Inversion yields
c 1θ, if 0 θ <c π
−p,q ≤ p,q2
π +c 1M(p q)/2(θ c π), if c π θ <π
h−p,1q(θ)=π2+c−p−p,,1qq(θ−−π), − p,q2 if πp,≤q2θ≤<π+cp,qπ2 ,
3π +c 1M(p q)/2(θ π c π), if π+c π θ <2π
2 −p,q − − − p,q2 p,q2 ≤
for θ [0,2π) and h 1(θ + 2πn) = 2πn + h 1(θ) for any τ [0,2π) and
∈ −p,q −p,q ∈
for any integer n. Substitution yields γ(θ) = a¯2/(p+q) h 1(θ+h (ψ)) =
−p,q p,q
a¯2/(p+q) h 1(θ+ϕ) =γ¯ (θ+ϕ), with ϕ=h (ψ).
−p,q p,q p,q (cid:0) (cid:1)
Now(cid:0)we can prov(cid:1)e Theorem 1.
Proof of Theorem 1. Estimate (7) with a = γp and b = γq yields (3). Suppose
p+q >0. In view of Lemma 2 and Lemma 3 we have
1 C2(πγγp(,pγqq))/2 2 =(cid:18)π4 arctanM−(p+q)/4(cid:19)−2
2π 0 −
(cid:16) (cid:17)
R
if and only if γ(θ)=γ¯ (θ+ϕ) for some ϕ R. Equality is attainedin (1)–(2)
p,q
∈
witha(θ)=γ¯p (θ+ϕ)andb(θ)=γ¯q (θ+ϕ) ifandonlyif w(θ)=w¯ (θ+ϕ).
p,q p,q p,q
If p + q = 0, then the conclusion follows by Lemma 2 with a = γp and
b=γ p.
−
Acknowledgements
I am grateful to Professor Carlo Sbordone for many useful and stimulating
discussions.
References
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(1992), 29–50.
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