Journal of Inequalities in Pure and
Applied Mathematics
INEQUALITIES APPLICABLE TO CERTAIN PARTIAL
DIFFERENTIAL EQUATIONS
volume 5, issue 1, article 18,
2004.
J. LANG, O. MENDEZ AND A. NEKVINDA
Department of Mathematics,
100 Math Tower, 231 West 18th Ave.
Columbus, OH 43210-1174, USA.
EMail: lang@math.ohio-state.edu
W. University Ave., 124 Bell Hall,
University of Texas at El Paso
El, Paso, TX 79968, USA.
EMail: mendez@math.utep.edu
Faculty of Civil Engineering,
Czech Technical University,
Thakurova 7,
166 29 Prague 6,
Czech Republic.
EMail: nekvinda@fsv.cvut.cz
Received 17 December, 2003;
accepted 04 February, 2004.
Communicated by: D. Hinton
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
172-03
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Abstract
We consider the Hardy-type operator
Zx
(T f) (x) := v(x) u(t)f(t)dt,
x > a,
a
and establish properties of T as a map from Lp (a, b) into Lq (a, b) for 1 < p ≤
q ≤ 2, 2 ≤ p ≤ q < ∞ and 1 < p ≤ 2 ≤ q < ∞. The main result is that, with
appropriate assumptions on u and v, the approximation numbers an (T ) of T
satisfy the inequality
Zb
Zb
c1 |uv|r dt ≤ lim inf narn (T ) ≤ lim sup narn (T ) ≤ c2 |uv|r dt
n→∞
a
n→∞
a
when 1 < p ≤ q ≤ 2 or 2 ≤ p ≤ q < ∞, and in the case 1 < p ≤ 2 ≤ q < ∞ we
Zd
have
r
lim sup nan (T ) ≤ c3 |u(t)v(t)|r dt
n→∞
and
Z
c4
p0 q
p0 +q
0
0
d
|u(t)v(t)|r dt ≤ lim inf n(1/2−1/q)r+1 arn (T ),
n→∞
and constants c1 , c2 , c3 , c4 . Upper and lower estimates for the ls
where r =
and ls,k norms of {an (T )} are also given.
Inequalities Applicable To
Certain Partial Differential
Equations
J. Lang, O. Mendez and
A. Nekvinda
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2000 Mathematics Subject Classification: Primary 46E30; Secondary 47B38
Key words: Approximation numbers, Hardy operator, Voltera operator.
J. Lang wishes to thank the Leverhulme Foundation for its support and also to the
Ohio State University for support via SRA.
A. Nekvinda was supported by MSM 210000010.
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Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
Bounds for the Approximation Numbers . . . . . . . . . . . . . . . . .
4
Local Asymptotic Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
The Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
lr and Weak–lr Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References
4
8
30
42
52
62
Inequalities Applicable To
Certain Partial Differential
Equations
J. Lang, O. Mendez and
A. Nekvinda
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1.
Introduction
The operator T : Lp (a, b) → Lq (a, b) (where 0 ≤ a ≤ b ≤ d < ∞) defined by
Z x
(1.1)
T f (x) = v(x)
u(t)f (t)dt
0
was studied in [1] and [5], in the case 1 ≤ p ≤ q ≤ ∞, for real–valued functions
0
u ∈ Lp (0, c), v ∈ Lp (c, d), for any c ∈ (0, d) and p0 = p/(p − 1). In the
aformentioned works, the following estimates for the approximation numbers
an (T ) of T were obtained:
(1.2)
(1.3)
aN (ε)+3 ≤ σp ε,
1
1
aN (ε)−1 ≥ νq (N (ε) − 1) q − p ε,
J. Lang, O. Mendez and
A. Nekvinda
for p < q < ∞
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and
(1.4)
Inequalities Applicable To
Certain Partial Differential
Equations
aN (ε)/2−1 ≥ ε/2,
for p = q,
where σp , νq , are constants depending on q, and N (ε) is an ε-depending natural
number .
In the case p = q, these results are sharp and are used in [2] and [5] to obtain
asymptotic results for the approximation numbers.
Specifically, it was proved in [2] that for p = q = 2
Z
1 d
(1.5)
lim nan (T ) =
|u(t)v(t)|dt
n→∞
π 0
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and that for 1 < p = q < ∞,
Z d
1
(1.6)
αp
|u(t)v(t)|dt ≤ lim inf nan (T )
n→∞
4
0
Z
≤ lim sup nan (T ) ≤ αp
n→∞
d
|u(t)v(t)|dt.
0
The endpoint cases were studied in [5]: it was shown there that for p = q = ∞
(and similarly for p = q = 1)
Z
1 d
(1.7)
|u(t)vs (t)|dt ≤ lim inf nan (T )
n→∞
4 0
Z d
≤ lim sup nan (T ) ≤
|u(t)vs (t)|dt,
n→∞
J. Lang, O. Mendez and
A. Nekvinda
0
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where
vs (t) = lim kv χ(t−ε,t+ε) kL∞ .
ε→0+
If p < q, the estimates (1.2) and (1.3) are not sharp.
The estimates (1.2) and (1.3) were used in [7] to obtain the following asymptotic results for the approximation numbers in the case 1 < p < q < ∞:
Z d
r
(1.8)
lim sup nan (T ) ≤ cp,q
|u(t)v(t)|r dt
n→∞
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and
Z
(1.9)
Inequalities Applicable To
Certain Partial Differential
Equations
≤ dp,q
Page 5 of 78
d
r
|u(t)v(t)| dt ≤ lim inf
0
n→∞
1
1
n( p − q )r+1 arn (T )
J. Ineq. Pure and Appl. Math. 5(1) Art. 18, 2004
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where r = pq 0 /(q + p0 ).
Since the estimates upon which they are based are not sharp, these results
are not sharp either, in contrast to (1.5), (1.6). Our research is directed toward
finding alternative, refined versions of (1.2) and (1.3) in the case p < q, aiming
to get better asymptotic results than (1.8) and (1.9). In this paper, we succeed
in showing that for 1 ≤ p ≤ q ≤ ∞,
aN (ε)+1 ≤ 2ε,
(1.10)
Inequalities Applicable To
Certain Partial Differential
Equations
and for 1 ≤ p ≤ q ≤ 2 or 2 ≤ p ≤ q ≤ ∞
aN (ε)/4−1 ≥ cε,
(1.11)
J. Lang, O. Mendez and
A. Nekvinda
and for 1 < p ≤ 2 ≤ q < ∞
1
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1
aN (ε)/4−1 ≥ cεN (ε) 2 − q ,
(1.12)
Contents
where c is a constant independent of ε and N (ε). And under some condition on
u and v we show that for 1 ≤ p ≤ q ≤ 2 or 2 ≤ p ≤ q ≤ ∞
Z b
Z b
r
r
r
c1
|uv| ≤ lim inf nan (T ) ≤ lim inf nan (T ) ≤ c2
|uv|r ,
a
n→∞
n→∞
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and for 1 < p ≤ 2 ≤ q < ∞
lim sup narn (T ) ≤ cp,q
n→∞
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Z
d
|u(t)v(t)|r dt
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0
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and
Z
dp,q
0
d
1
1
|u(t)v(t)|r dt ≤ lim inf n( 2 − q )r+1 arn (T ),
n→∞
p0 q
where r = p0 +q . We also describe lr and lr,s norms of {an }∞
n=1 .
Under much stronger conditions on u and v in neighborhood of boundary
points of I this problem was also studied in [6] by using different techniques.
Inequalities Applicable To
Certain Partial Differential
Equations
J. Lang, O. Mendez and
A. Nekvinda
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2.
Preliminaries
Throughout this paper we will suppose that 1 < p ≤ q ≤ 2. In what follows we
shall be concerned with the operator T defined in (1.1) as a map from Lp (0, d)
into Lq (0, d) where 0 < d ≤ ∞. The functions u, v are subject to the following
restrictions: for all x ∈ (0, d)
0
u ∈ Lp (0, x),
(2.1)
and
Inequalities Applicable To
Certain Partial Differential
Equations
q
v ∈ L (x, d).
(2.2)
It is well-known that these assumptions guarantee that T is well defined (see
(1.9)). Moreover, the norm of this operator is equivalent to:
Z x
10 Z d
1q
p
p0
q
J := sup
|u(t)| dt
|v(t)| dt ,
x∈(0,d)
0
x
x > 0,
0
where I = (a, b) ⊂ (0, d), and the quantity
Z x
10 Z
p
p0
(2.4)
J(I) ≡ J(a, b) := sup
|u(t)| dt
a
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(see [4], [8] and [5]). We define the operator TI by
Z x
(2.3)
TI f (x) := v(x)χI (x)
u(t)f (t)χI (t)dt,
x∈I
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d
1q
|v(t)| dt .
q
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x
It is obvious that J(I) ≈ kTI kp→q , where the symbol ≈ indicates that the quotient of the two sides is bounded above and below by positive constants.
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Proposition 2.1. There are two positive constants K1 , K2 such that for any
I = (a, b) ⊂ (0, d) the inequality
K1 J(a, b) ≤ kTI k ≤ K2 J(a, b)
holds.
We start by proving an important continuity property of J:
Lemma 2.2. Suppose that (2.1) and (2.2) are satisfied. Then the function J(·, b)
is continuous and non–increasing on (0, b), for any b ≤ ∞.
Proof. It is easy to verify that J(·, b) is non-increasing on (0, b). To prove the
continuity of J, fix x ∈ (0, b) an ε > 0. By (2.1) and (2.2) there exists 0 < h0 <
min{x, b − x} such that
Z x
10
p
p0
|u(t)| dt
kvkq,(x−h0 ,x) < ε.
x−h0
J(x, b) ≤ J(x − h, b)
Z
= sup
10
p
kvkq,(z,b)
|u(t)| dt
z
p0
x−h
= max
sup
Z
x−h<z<x
Z
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10
p
|u(t)| dt
kvkq,(z,b) ,
z
p0
Z z
+
x−h
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x−h
x
sup
x<z<b
J. Lang, O. Mendez and
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Contents
It follows that for h, 0 < h < h0 ,
x−h<z<b
Inequalities Applicable To
Certain Partial Differential
Equations
x
10
p
kvkq,(z,b)
|u(t)| dt
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p0
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(2.5)
≤ max {ε, ε + J(x, d)} = ε + J(x, d),
which yields 0 < J(x − h, b) − J(x, b) < ε. The inequality 0 < J(x, b) − J(x +
h, b) < ε can be proved analogously.
For the sake of completeness, we include the following known result (see [4]
and [9]):
Proposition 2.3. The operator T defined by (1.1), with 1 < p < ∞ and u, v
satisfying (2.1), (2.2) and J < ∞ is a compact map from Lp (0, d) into Lq (0, d)
if and only if limc→0+ J(0, c) = limc→d− J(c, d) = 0.
In what follows A(I) is a function defined on all sub-intervals I = (a, b) ⊂
(0, d), defined by
(2.6)
A(I) = A(a, b) := sup
inf kT f − αvkp,I .
kf kp,I =1 α∈<
A similar function can be found in [5]. Next, we prove some basic properties
of A(I). Choosing α = 0 in (2.6) we immediately obtain for any I = (a, b),
0 ≤ a < b ≤ d,
A(I) ≤ kTI k.
(2.7)
Lemma 2.4. Let I = (a, b) and kukp0 ,I < ∞, kvkq,I < ∞. Set
e
A(I)
= sup
inf
kf kp,I =1 |α|≤2kukp0 ,I
e
Then A(I) = A(I).
kT f − αvkp,I .
Inequalities Applicable To
Certain Partial Differential
Equations
J. Lang, O. Mendez and
A. Nekvinda
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Proof. Hölder’s inequality yields
Z b Z
kTI k = sup
kf kp,I =1
a
a
|v(x)|q
≤ sup
Z
≤
a
b
q
x
Z
pq Z
p
|f (t)| dt
a
Z
|v(x)|
a
q 1
q
f (t)u(t)dt dx
b
Z
kf kp,I =1
x
b
q0 ! 1q
p
dx
|u(t)| dt
p0
a
q
p0
|u(t)| dt
dx
p0
x
! 1q
= kukp0 ,I kvkq,I .
a
e
If kvkq,I = 0 then A(I) = A(I)
= 0. Assume kvkq,I > 0. Let kf kp,I = 1 and
kTI k
suppose that |α| > 2kukp0 ,I . Then |α| ≥ 2 kvk
and using the trivial inequality
q,I
q
1−q
q
q
|a − b| ≥ 2 |a| − |b| valid for any real numbers a, b we obtain for each
α∈<
q
Z b Z x
α−
dx
v(x)
f
(t)u(t)dt
a
a
Z x
q
Z b
|αv(x)| − dx
≥
f
(t)u(t)dt
a
a
q
Z b
Z b
Z x
1−q
q
q
dx
≥ 2 |α|
|v(x)| dx −
v(x)
f
(t)u(t)dt
a
a
a
q Z b
kTI k
|v(x)|q dx − kTI kq = kTI kq .
> 21−q 2
kvkq,I
a
Inequalities Applicable To
Certain Partial Differential
Equations
J. Lang, O. Mendez and
A. Nekvinda
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In conjunction with (2.7), the above yields
kTI k ≥ A(I)
(
= sup min
inf
|α|≤2kukp0 ,I
kf kp,I =1
Z b Z
α−
a
inf
|α|>2kukp0 ,I
=
inf
|α|≤2kukp0 ,I
Z b Z
α−
a
a
x
a
Z b Z
α−
a
a
q 1q
f (t)u(t)dt v(x)
,
x
q 1q )
f (t)u(t)dt v(x)
q 1q
x
e
= A(I),
f (t)u(t)dt v(x)
J. Lang, O. Mendez and
A. Nekvinda
which finishes the proof.
Lemma 2.5. Let u and v satisfy (2.1) and (2.2) respectively. Then A(I1 ) ≤
A(I2 ), provided I1 ⊂ I2 . Moreover, given 0 < b < d the function A(·, b) is
continuous on (0, b).
Proof. Let 0 ≤ a1 ≤ a2 < b2 ≤ b1 ≤ d, I1 = (a1 , b1 ), I2 = (a2 , b2 ). Then
Z b1 Z x
q 1q
v(x)
dx
A(I1 ) = sup inf
(f
(t)u(t)dt
−
α)
kf kp,I1 =1 α∈<
a1
a1
Z
≥
sup
kf χI2 kp,I1
inf
=1 α∈<
sup
inf
kf kp,I2 =1 α∈<
Z x
q 1q
b1 v(x)
(f (t)u(t)dt − α) dx
a1
Z
≥
Inequalities Applicable To
Certain Partial Differential
Equations
a1
Z x
q 1q
b2 v(x)
dx
(f
(t)u(t)dt
−
α)
= A(I2 )
a2
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a2
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which proves the first part of Lemma 2.5.
For the remaining statement, fix b ∈ (0, d) and 0 < y < b. Let ε > 0. By
(2.1) and (2.2) there exists 0 < h0 such that 0 < y − h0 and
Z y
Z y
p0
|u| < ε and
|v|q < ε.
y−h0
y−h0
Set Dh = 2kukp0 ,(y−h,b) for any 0 ≤ h < y. Recall that by (2.1), one has
1
1
1
Dh < ∞ for 0 ≤ h < d. Using the trivial inequality (a + b) q ≤ a q + b q , the
triangle inequality and the Hölder inequality, it follows that
A(y, b) ≤ A(y − h, b)
q 1q
Z x
b α−
=
sup
inf
f (t)u(t)dt v(x) dx
kf kp,(y−h,b) =1 α∈<
y−h
y−h
q
Z y Z x
α−
dx
=
sup
inf
f
(t)u(t)dt
v(x)
Inequalities Applicable To
Certain Partial Differential
Equations
J. Lang, O. Mendez and
A. Nekvinda
Z
kf kp,(y−h,b) =1 |α|≤Dh
Z b Z
+
y
y−h
y−h
y
y−h
Z
f (t)u(t)dt +
("Z
x
y
q 1q
f (t)u(t)dt − α v(x) dx
y
≤
sup
Z
x
×
y−h
Z
q0
p
|u(t)| dt
x
p0
|v(x)|
inf
kf kp,(y−h,b) =1 |α|≤Dh
q
y−h
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y−h
pq # 1q Z
p
q
|f (t)| dt dx + |α|
y
|v(x)|q dx
1q
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"Z
b
|v(x)|q dx
+
y
y
≤
ε
1+ p10
p0
y−h
Z b Z
+
v(x)
(
q0 Z
p
|u(t)| dt
y
Z
y
x
y
pq # 1q
|f (t)|p dt
y−h
q 1q )
f (t)u(t)dt − α dx
1
1
+ Dh ε q + kvkq,(y,b) ε p0
+
sup
kf kp,(y−h,b) =1
Z b Z
inf
|α|≤Dh
y
y
x
q 1q )
.
f (t)u(t)dt − α v(x) dx
Since D0 ≤ Dh ≤ Dh0 we have by Lemma 2.4
Z b Z
inf
|α|≤Dh
y
y
x
y
x
y
J. Lang, O. Mendez and
A. Nekvinda
Title Page
q 1q
f (t)u(t)dt − α v(x) dx
Z b Z
≤ inf
|α|≤D0
Inequalities Applicable To
Certain Partial Differential
Equations
Contents
q 1q
f (t)u(t)dt − α v(x) dx
= A(y, b)
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and thus
q−1
A(y, b) ≤ A(y − h, b) ≤ 2
ε
1+ p10
1
q
+ Dh0 ε + kvkq,(y,b) ε
1/p0
+ A(y, b)
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which proves that
lim A(y − h, b) = A(y, b).
h→0+
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Analogously,
lim A(y + h, b) = A(y, b),
h→0+
which finishes the proof of our lemma.
Lemma 2.6. Suppose u, v > 0 satisfy (2.1) and (2.2) and that T : Lp (a, b) →
Lq (a, b) is compact. Let I1 = (c, d) and I2 = (c0 , d0 ) be subintervals of (a, b),
Rb
with I2 ⊂ I1 , |I2 | > 0, |I1 − I2 | > 0, a v q (x)dx < ∞. Then 0 < A(I2 ) <
A(I1 ).
p
Proof. Let 0 ≤ f ∈ L (I2 ), 0 < kf kp,I2 ≤ kf kp,I1 ≤ 1 with supp f ⊂ I2 . Let
y ∈ I2 then
kT(c0 ,y) kp,I2 > 0
and
kT(y,d0 ) kp,I2 > 0
and then by simple modification of [5, Lemma 3.5] for the case p < q we have
min{kT(c0 ,y) kq,I2 , kT(y,d0 ) kq,I2 } ≤ min kTx,J kq,I2
x∈J
which means A(I2 ) > 0.
Next, suppose that c = c0 < d0 < d. A slight modification of [5, Theorem
3.8] for p < q, yields x0 ∈ I2 and x1 ∈ I1 such that A(I2 ) = kTx0 ,I2 kq,I2
and A(I1 ) = kTx1 ,I1 kq,I1 . Since u, v > 0 on I1 , it is then quite easy to see that
x0 ∈ I2o and x1 ∈ I1o .
If x0 = x1 , then, since u, v > 0 on I1 , we get
Inequalities Applicable To
Certain Partial Differential
Equations
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A(I1 ) = kTx1 ,I1 kq,I1 > kTx1 ,I1 kq,I2 = kTx1 ,I2 kq,I2 = A(I2 ).
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On the other hand, if x0 6= x1 , then
A(I1 ) = kTx1 ,I1 kq,I1 ≥ kTx1 ,I1 kq,I2 ≥ kTx1 ,I2 kq,I2 > kTx0 ,I2 kq,I2 = A(I2 ).
The case c < c0 < d0 = d could be proved similarly and the case c < c0 < d0 < d
follows from previous cases and the monotonicity of A(I1 ).
Let I = (a, b) ⊂
S (0, d) and Ii = (ai , bi ) ⊂ I, i = 1, 2, . . . , k. Say that
{Ii }ki=1 ∈ P(I) if ki=1 Ii ⊃ I and assume the intervals {Ii }ki=1 to be nonoverlapping.
Now, for any interval I ⊆ (0, d) and ε > 0, we define the numbers M and
N , as follows:
(2.8)
M (I, ε) := inf{n : J(Ii ) ≤ ε, {Ii }ni=1 ∈ P(I)}
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and
(2.9)
Inequalities Applicable To
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N (I, ε) := inf{n; A(Ii ) ≤ ε, {Ii }ni=1 ∈ P(I)}.
Since by Proposition 2.1, A(I) ≤ kTI k ≤ K2 J(I), we have
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N (I, ε) ≤ M (I, K2 ε).
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Put N (ε) = N ((0, d), ε) and M (ε) = M ((0, d), ε). From Proposition 2.3 and
the definition of J(I) one gets the following:
Close
Remark 2.1. Suppose that (2.1) and (2.2) are satisfied. Then T : Lp (0, d) →
Lq (0, d) is compact if and only if M (ε) < ∞ for each ε > 0.
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(2.10)
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Lemma 2.7. Let T be a compact operator. Then
lim A(0, x) = 0 and lim A(x, d) = 0.
x→0+
x→d−
Lemma 2.8. Suppose that T is a compact operator, ε > 0 and I = (a, b) ⊂
(0, d). Let m = N (I, ε). Then there exists a sequence of non-overlapping
intervals {Ii }m
i=1 covering I, such that A(Ii ) = ε for i ∈ {2, . . . , m − 1},
A(I1 ) ≤ ε, and A(Im ) ≤ ε.
Proof. From Remark 2.7 and (2.10), one has m < ∞. Define a system S =
{Ij }j∈J , Ij ⊂ I, of intervals as follows: Set b1 = inf{x ∈ I; A(x, b) ≤ ε}.
By Lemma 2.7 we have a ≤ b1 < b. Put I1 = [b1 , b]. Then A(I1 ) ≤ ε. If
a = b1 write S = {I1 }, otherwise set b2 = inf{x ∈ I; A(x, b1 ) ≤ ε} and
I2 = [b2 , b1 ]. Observe that by Lemma 2.5 we have A(I2 ) = ε. We can now
proceed by mathematical induction to construct a (finite or infinite) system of
intervals S = {Ij }αj=1 . Note that we have only A(Iα ) ≤ ε (not A(Iα ) = ε)
provided α < ∞ and A(Iβ ) = ε for β < α. Writing b0 = b we can set
Ij = [bj , bj−1 ], 1 ≤ j ≤ α.
Our next step is to show that α = m. By the definition of m one has α ≥ m
and a finite sequence of numbers a = am < am−1 < . . . a0 = b and intervals
Ji = [ai , ai−1 ], i = 1, 2, . . . , m such that A(Ji ) ≤ ε. Notice that b1 ≤ a1 , for
if not, we can take λ : 0 < λ < b1 , which, from Lemma 2.5 and the definition
of I, would yield ε < A(λ, b0 ) ≤ A(J1 ) ≤ ε, which is a contradiction. Assume
now that for some α > 1, bk > ak . If bk−1 ≤ ak−1 , then talking ak < λ < bk ,
Lemma 2.5 and the definition of Ik yield ε < A(λ, bk−1 ) ≤ A(Jk ) ≤ ε, which
is a contradiction, so that ak−1 ≤ bk−1 . Repeating this reasoning, one arrives at
Inequalities Applicable To
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b1 > a1 , which is again a contradiction. Thus, bk ≤ ak for all k = 1, 2, . . . m.
Choosing k = m we have bm = a and consequently, α = m and S covers I
which finishes the proof.
For future reference (see the proof of (1.11) in the next section) we include
the following lemmas and remarks.
Let X be a Banach space and M ⊂ X. Recall the definition of the distance
function dist(·, M ),
dist(x, M ) = inf{kx − yk; y ∈ M }, x ∈ X.
Lemma 2.9. Let T be a compact operator, u, v > 0, ε > 0, I = (a, b) ⊂ (0, d)
and m = N (I, ε).
(i) Then there exists 0 < ε1 < ε and a sequence of non-overlapping intervals
{Ii }m
i=1 covering I, such that A(Ii ) = ε1 for i ∈ {1, . . . , m}.
(ii) There exists ε2 : 0 < ε2 < ε such that m + 1 = N (I, ε2 ).
Proof. The proof follows from the strict monotonicity and the continuity of
A(I).
Lemma 2.10. Let H be an infinite dimensional separable Hilbert space. Let
Y = {u1 , . . . , u2n } be any orthonormal set with 2n vectors and let X be any
m-dimensional subspace of H with m ≤ n. Then there exists an integer j,
1 ≤ j ≤ 2n, such that
1
dist(uj , X) ≥ √ .
2
Inequalities Applicable To
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Proof. Denote the inner product in H by (u, v). Extend Y to an orthonormal topological basis {ui }∞
i=1 of H. Choose an orthonormal basis of X, say
v1 , . . . , vm . Denote by P the orthogonal projection of H into X. Then
m
X
Pu =
(u, vj )vj
for any u ∈ H.
j=1
Since P is a self-adjoint projection we obtain
2n
X
2
kuk − P uk k =
k=1
2n
X
(1 − 2(uk , P uk ) + (P uk , P uk ))
k=1
= 2n −
= 2n −
2n
X
(uk , P uk )
k=1
2n X
m
X
(uk , vj )2
k=1 j=1
= 2n −
m X
2n
X
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(uk , vj )2 .
j=1 k=1
The Parseval identity yields
∞
X
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2
2
(uk , vj ) = kvj k = 1,
k=1
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which implies
2n
X
k=1
2
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(uk , vj ) ≤ 1.
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Consequently,
2n
X
kuk − P uk k2 ≥ 2n − m ≥ n,
k=1
which guarantees the existence of an integer j, 1 ≤ j ≤ 2n, with kuj −P uj k2 ≥
1
. Then
2
1
dist(uj , X) = kuj − P uj k ≥ √ ,
2
which finishes the proof.
Lemma 2.11. Let 1 ≤ p ≤ 2 and X be any n-dimensional subspace of lp . Set
ej ∈ lp , ej = {δij }∞
i=1 where δij is Kronecker’s symbol. Then there exists an
integer j, 1 ≤ j ≤ 2n, such that
1
distp (ej , X) ≥ √ .
2
Proof. Denote by k · kp the norm and by distp the distance function in lp . Since
k · kl2 ≤ k · klp we can consider X as an n-dimensional subspace of l2 . Thus,
using the previous lemma there is j, 1 ≤ j ≤ 2n with dist2 (ej , X) ≥ √12 from
which immediately follows that
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distp (ej , X) = inf{kej − xkp ; x ∈ X}
≥ inf{kej − xk2 ; x ∈ X}
1
= dist2 (ej , X) ≥ √ .
2
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Lemma 2.12. Let 2 < p ≤ ∞, n ∈ N and X be any n-dimensional subspace
of lp . Set ej = {δij }∞
i=1 ∈ lp where δij is the Kronecker’s symbol. Then there is
j, 1 ≤ j ≤ 2n such that
(2.11)
1
1
1
distp (ej , X) ≥ 2 p −1 n p − 2 .
Proof. Let R : lp → lp be the restriction operator given by
R(a) = (a1 , a2 , . . . , a2n , 0, 0, . . . ),
where a = (a1 , a2 , . . . ) ∈ lp . Choose ui ∈ X such that distp (ei , X) = kei −ui k.
Using the well-known inequality
kR(a)k2 ≤ (2n)
1
− p1
2
kR(a)kp for all a ∈ l
p
it follows that for each 1 ≤ i ≤ 2n,
≥ (2n)
1
− p1
2
1
− p1
2
kR(ei ) − R(ui )k2
dist2 (ei , R(X)).
Since R(X) is a linear subspace of l2 , by Lemma 2.10 there exists j with
1
dist2 (ej , X) ≥ √ ,
2
which finishes the proof of the lemma.
J. Lang, O. Mendez and
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distp (ei , X) = kei − ui kp
≥ kR(ei ) − R(ui )kp
≥ (2n)
Inequalities Applicable To
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It is shown in the appendix that the power of n in (2.11) is the best possible
if 2 < p ≤ ∞.
With the aid of the last lemmas we can get a modified version Lemma 2.10
with H replaced by Lp (0, d) .
We start by recalling some lemmas referring to the properties of the map
taking x ∈ X to its nearest element MA (x) ∈ A ⊂ X.
Lemma 2.13. Assume that X is a strictly convex Banach space, V ⊂ X is a
finite dimensional subspace of X and x0 ∈ X. Set A = {x0 + v; v ∈ V }. Then
for any x ∈ X there exists a unique element v such that
Inequalities Applicable To
Certain Partial Differential
Equations
kx − vk = inf{kx − yk; y ∈ A}.
Denote by MA the mapping which assigns to x ∈ X the nearest element of A.
Lemma 2.14. For any α ∈ R, x ∈ X and v ∈ V , one has
(2.12)
(2.13)
MV (αx) = αMV (x),
MV (x + v) = MV (x) + v
and
(2.14)
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1
kx − vk ≥ kMV (x) − vk.
2
The proof of these last two lemmas can be found in [10].
Recall that P : X → X is called a projection if P is linear, P 2 = P and
kP k < ∞.
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Lemma 2.15. Let√
X be a strictly convex Banach space and V ⊂ X be a subspace, dim(V ) = n is finite.
Then there exists a projection P : X → V which
√
is onto such that kP k ≤ n.
For a proof of the above lemma, see [10, III. B, Theorem 10].
The following lemma, whose proof is included for the sake of completeness,
plays a critical role in the sequel, since it provides an approximation to the map
MA above by a linear operator of at most one dimensional range. The proof can
also be found in [5].
R
Lemma 2.16. Let I ⊂ (0, d), 1 ≤ q ≤ ∞ and let I |g(t)v(t)|q dt < ∞. Set
R

0
if
|v(t)|q dt = 0;

I
R


R
g(t)|v(t)|q dt
IR
ωI (g) =
if 0 < I |v(t)|q dt < ∞;
q

|v(t)| dt

I
R

0
if
|v(t)|q dt = ∞.
I
Then
(2.15)
inf k(g − α)vkq,I ≤ k(g − ωI (g))vkq,I ≤ 2 inf k(g − α)vkq,I .
α∈<
α∈<
R
Proof. It suffices to prove the second inequality. Fix g such that I g(t)|v(t)|q dt <
∞.
R
Assume first that I |v(t)|q dt = 0. Then v(t) = 0 almost everywhere in I
and all Rmembers in (2.15) are equal zero.
Let I |v(t)|q dt = ∞. We claim that kαvkq,I ≤ k(α − g)vkq,I . If α = 0
the inequality is clear. Let α 6= 0, otherwise kαvkq,I = ∞ and by the triangle
Inequalities Applicable To
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inequality, it follows that k(α − g)vkq,I ≥ kαvkq,I − kgvkq,I = ∞ and hence
the claim. Thus, for each α ∈ R
k(g − ωI (g))vkq,I = k(g − α + α)vkq,I ≤ 2k(g − α)vkq,I
which gives
k(g − ωI (g))vkq,I ≤ 2 inf k(g − α)vkq,I .
α∈<
R
Assume now 0 < I |v(t)|q dt < ∞. By the Hölder’s inequality, we obtain,
for any α ∈ <
k(α − wI (g))vkqq,I
R
q
Z q
g(t)|v(t)|
dt
I
dx
= α − R
v(x)
q dt
|v(t)|
I
I
R
Z
I (α − g(t))|v(t)|q dt q
q
dx
R
= |v(x)| q dt
|v(t)|
I
I
q
Z
Z
|v(t)|q
q
(α − g(t))|v(t)| dt dx
R
=
q
q I
I ( I |v(t)| dt)
q
Z
1−q Z
q
q−1
(α − g(t))|v(t)||v(t)| dt
=
|v(t)| dx
I
I
Z
1−q Z
Z
q0
q
q
q
q 0 (q−1)
≤
|v(x)| dx
|(α − g(t))v(t)| dt
|v(t)|
dt
I
I
Z I
= |(α − g(t))v(t)|q dt = k(α − g)vkqq,I
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which proves k(α − wI (g))vkq,I ≤ k(α − g)vkq,I .
Now, using this inequality, for any real α one has:
k(g − wI (g))vkq,I ≤ k(g − α)vkq,I + k(α − wI (g))vkq,I
≤ 2k(α − wI (g))vk.
The lemma follows by taking the infimum over α on the right hand side.
Lemma 2.17. Let X = Lp (0, d), p > 1. Let v1 , v2 , . . . , vn be functions in X
with pairwise disjoint supports and kvi kp = 1 for i = 1, 2, . . . , n. Set V =
span{v1 , v2 , . . . , vn }. Then there is a projection PV with rank Pv ≤ n, such
that
kf − MV (f )kp,(0,d) ≤ kf − PV (f )kp,(0,d) ≤ 2kf − MV (f )kp,(0,d) ,
where MV is as defined in Lemma 2.13.
kf − MV (f )kp,(0,d) ≤ kf − PV (f )kp,(0,d) ,
kf −
=
≤
i=1
n
X
i=1
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which is the first inequality. Also
n
X
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Proof. Denote Si = supp vi , Vi = span{vi }. Given any f ∈P
X, with supp f ⊂
Si , let Mi (f ) = Mvi (f ). Put Pi f = ωi (f χSi )χSi , and P f = ni=1 Pi (f χSi )χSi .
From the definition of Mv and Pv we have
MV (f )kpp
Inequalities Applicable To
Certain Partial Differential
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kf χSi + Mv (f )χSi kpp,Si
kf χSi −
Mi (f χSi )χSi kpp,Si
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1
n
X
1
i=1
p
n
X
Pi (f χSi )χSi f −
≤ 2− p
= 2− p
kf χSi − Pi (f χSi )χSi kpp,Si
i=1
− p1
≤2
p
kf − P (f )χSi kpp ,
which gives the second inequality and completes the proof.
Lemma 2.18. Let 1 < p ≤ 2 and let u1 , ..., u2n be a system of functions from
Lp (0, d) with disjoint supports. Let X ⊂ Lp (0, d) be a subspace, dim X ≤ n.
Then there exists an integer j, 1 ≤ j ≤ 2n, such that
1
distp (uj , X) ≥ √ kuj kp .
3 2
Proof. If kui kp = 0 for some i, it suffices to choose j = i. Let kui kp > 0 for
all 1 ≤ i ≤ 2n. Set vi = kuuiikp . Let V = span{v1 , v2 , . . . , v2n } and let PV
be the projection from the previous lemma. Let Y = PV (X). Then Y ⊂ V ,
dim Y ≤ n. Denote by Z the subspace of lp consisting of all sequences {ai }∞
i=1
such that ak = 0 for all k > 2n. Let ei be the canonical basis of Z. Define a
linear mapping I : Y → Z by
!
2n
2n
X
X
I
αi vi =
αi ei .
i=1
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i=1
Since kvi k = 1 and the functions vi have pairwise disjoint supports, it follows
that I is an isometry between Y and Z. According to Lemma 2.11 there exists
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1 ≤ j ≤ 2n such that
(2.16)
1
distp (ej , I(Y )) ≥ √ ,
2
and from Lemma 2.13 there is a unique x ∈ X with
(2.17)
distp (vj , X) = kvj − xkp .
By the definition of PV and MV , we have
1
1
kx − M (x)kp ≤ kx − PV (x)kp ≤ kx − MV (x)kp ≤ kvj − xkp
2
2
Inequalities Applicable To
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J. Lang, O. Mendez and
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which yields, with the triangle inequality,
kPV (x) − vj kp ≤ kPV (x) − xkp + kx − vj kp
≤ 2kx − vj kp
≤ 2kx − vj kp + kx − vj kp ≤ 3kx − vj kp .
This together with (2.16) and (2.17), gives
distp (vj , X) = kvj − xkp
1
≥ kvj − PV (x)kp
3
1
1
1
≥ distp (vj , Y ) = distp (ej , I(Y )) ≥ √ .
3
3
3 2
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Denoting by M1 the mapping which assigns to any f ∈ Lp (0, d) the element of
X nearest to f and using (2.12) we can rewrite the previous inequality as
distp (uj , X) = kuj − M1 (uj )kp
= kuj kp kvj − M1 (vj )kp
1
= kuj kp distp (vj , X) ≥ √ kuj kp
3 2
which yields the claim.
Lemma 2.19. Let 2 < p ≤ ∞ and let u1 , ..., u2n be a system of functions from
Lp (0, d) with disjoint supports. Let X ⊂ Lp (0, d) be a subspace, dim X ≤ n.
Then there exists an integer j, 1 ≤ j ≤ 2n, such that
1
1
1
distp (uj , X) ≥ √ kuj kp n p − 2 .
2 2
Proof. Let V, MV , PV , Y, Z and I have the same meanings as in Lemma 2.18.
Proceeding as before, Lemma 2.12 yields j : 1 ≤ j ≤ 2n such that
1 1 1
distp (ej , I(Y )) ≥ n p − 2 .
2
Let x ∈ X be the element given by Lemma 2.13 so that
dist(vj , X) = kvj − xkp .
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In exactly the same way as in Lemma 2.18, one gets
1 1 1
distp (vj , X) ≥ n p − 2 ,
3
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which can be written as
1
1
1
distp (uj , X) ≥ kuj kp n p − 2 ,
3
and the proof is complete.
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3.
Bounds for the Approximation Numbers
We recall that, given any m ∈ N, the mth approximation number aM (S) of a
bounded operator S from Lp into Lq , is defined by
am (S) := inf kS − F kp→q ,
F
where the infimum is taken over all bounded linear maps F : Lp (0, d) →
Lq (0, d) with rank less than m. Futher discussions on approximation numbers
may be found in [3]. An operator S is compact if and only if am (S) → 0 as
m → ∞. The first two lemmas of this section provide estimates for am (T ) with
T as in (1.1), which are analogous to those obtained in [1] and [5]. Hereafter,
we shall always assume (2.1) and (2.2).
Inequalities Applicable To
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Lemma 3.1. Let 1 ≤ p ≤ q ≤ ∞ and suppose that T : Lp (0, d) → Lq (0, d)
is bounded. Let ε > 0 and suppose that there exist N ∈ N and numbers ck ,
k = 0, 1, . . . , N , with 0 = c0 < c1 < · · · < cN = d, such that A(Ik ) ≤ ε for
k = 0, 1, . . . , N − 1, where Ik = (ck , ck+1 ). Then aN +1 (T ) ≤ 2ε.
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Proof. Consider for f ∈ Lp (a, b) and 0 ≤ k ≤ N − 1 one-dimensional linear
operators given by
Z x
Z x
PIk f (x) := χIk (x)v(x)
uf dt + ωIk
uf dt
,
ck
ck
where ωIk is the functional from Lemma 2.16. We claim that Pk is bounded
from Lp (0, d) into Lq (0, d) for each k.
Assume first that either 0 = kvkq,Ik or kvkq,Ik = ∞. Then Pk = 0 and
consequently, it is bounded.
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Assume now 0 < kvkq,Ik < ∞ and fix f, kf kp,(0,d) = 1. Then using Hölder’s
inequality, we obtain
Z x
ωI
u(t)f (t)dt k
ck
R R x
q
u(t)f
(t)dt|v(x)|
dx
I c
= k kR
q
|v(x)| dx
Ik
R
Rx
|v(x) ck u(t)f (t)dt| |v(x)|q−1 dx
I
R
≤ k
|v(x)|q dx
Ik
R
1q R
10
Rx
q
(q−1)q 0
q
|v(x)|
dx
|v(x)
u(t)f
(t)dt|
dx
Ik
ck
Ik
R
≤
|v(x)|q dx
Ik
kTIk f kq
≤
kvkq,Ik
and consequently,
Z d
Z
q
|(Pk f )(x)| dx =
≤
0
kT k
kvkq,Ik
Z x
Z x
q
v(x)
uf dt + ωIk
uf dt dx
Ik
c
q ck Z x
Z k Z x
q
q−1
v(x)
≤2
uf dt + ωIk
uf dtdx
Ik
ck
ck
kT k
q−1
≤2
kTk f kq +
kvkq,Ik
1
≤ kT k 1 +
.
kvkq,Ik
Inequalities Applicable To
Certain Partial Differential
Equations
J. Lang, O. Mendez and
A. Nekvinda
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PN −1
p
Set P =
k=0 Pk . Then P is a linear bounded operator from L (0, d) into
Lq (0, d). Moreover, we have by Lemma 2.16 and the well-known inequality
∞
X
! 1q
|ak |q
≤
∞
X
k=1
kT f −
P f kqq
=
=
N
−1
X
,
kT f − PIk f kqq,Ik
v(x)
≤ 2q−1
≤ 2q
|ak |p
k=1
k=0
N
−1 X
k=0
! p1
N
−1
X
k=0
N
−1
X
Z
Inequalities Applicable To
Certain Partial Differential
Equations
x
Z
x
uf dt − ωIk
ck
ck
q
uf dt q,Ik ,µ
inf kTIk f − αf kqq,Ik
Title Page
α∈<
Contents
Aq (Ik )kf kqp,Ik
k=0
N
−1
X
q
≤ (2ε)
JJ
J
kf kqp,Ik
N
−1
X
Close
! pq
kf kpp,Ik
II
I
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k=0
≤ (2ε)q
J. Lang, O. Mendez and
A. Nekvinda
≤ (2ε)q
k=0
Quit
Page 32 of 78
by Lemma 2.5. Since rank P ≤ N , the proof of the lemma is complete.
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Lemma 3.2. Let 1 < p ≤ q < ∞, T be bounded from Lp (0, d) to Lq (0, d),
0 ≤ a < b < c < d and denote I = [a, b], and J = [b, c]. Further, let
f, g ∈ Lp (0, d) with supp f ⊂ I, supp g ⊂ J, kf kp = kgkp = 1.
Let r, s be real numbers and set
Z d
h(x) = v(x)
u(t)(rf (t) + sg(t))dt.
0
Assume
Rc
a
u(t)h(x) = 0. Then
khkq ≥
q
q
q
|r| inf kTI f − αvk + |s| inf kTJ g − αvk
α∈<
α∈<
q
Inequalities Applicable To
Certain Partial Differential
Equations
1q
.
Proof. Since supp f ⊂ I and supp g ⊂ J we have
q
Z a
Z x
dx = 0.
(3.1)
v(x)
u(t)(rf
(t)
+
sg(t))dt
0
(3.2)
c
d
Z
v(x)
0
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0
If x ∈ (c, d) we have (recall that
Z
J. Lang, O. Mendez and
A. Nekvinda
x
Rc
a
u(t)h(x) = 0) that
q
u(t)(rf (t) + sg(t))dt dx
q
Z d
Z c
v(x)
dx = 0.
=
u(t)h(t)dt
c
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a
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Assume first s 6= 0. Then it follows from (3.1) and (3.2) that
q
Z d
Z x
q
dx
khkq =
v(x)
u(t)(rf
(t)
+
sg(t))dt
0
0
Z a Z b Z c Z d
=
+
+
+
0
a
b
c
q
Z Z x
= v(x)
u(t)(rf (t) + sg(t))dt dx
I
q
Z 0 Z x
+ v(x)
u(t)(rf (t) + sg(t))dt dx
q
Z J Z x 0
= v(x)
u(t)rf (t)dt dx
I
0
Z b
q
Z Z x
+ v(x)
u(t)rf (t)dt +
u(t)sg(t))dt dx
b
q
Z J
Z x0
u(t)f (t)dt dx
= |r|q v(x)
I
a
q
Z
Z x
Z r
q
dx
f
(t)dt
+
u(t)g(t))dt
+ |s|
v(x)
u(t)
s
b
Z Ix
Z J
q
≥ |r|q inf v(x)
u(t)f (t)dt − α dx
α∈< I
q
Z a Z x
q
v(x)
dx
+ |s| inf
u(t)g(t)dt
−
α
α∈<
q
J
= |r| inf kTI f −
α∈<
b
αvkqq,I
q
+ |s| inf kTJ g −
α∈<
Inequalities Applicable To
Certain Partial Differential
Equations
J. Lang, O. Mendez and
A. Nekvinda
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αvkqq,J .
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Assume now s = 0. Then
q
Z d
Z x
q
khkq =
u(t)rf (t)dt dx
v(x)
0
q
Z 0Z x
q
v(x)
= |r|
u(t)f (t)dt dx
I
q
Z a Z x
q
≥ |r| inf v(x)
u(t)f (t)dt − α dx
α∈<
q
I
= |r| inf kTI f −
α∈<
a
αvkqq,I
which finishes the proof of the lemma.
Lemma 3.3. Let 1 < p ≤ q ≤ 2, T be bounded from Lp (0, d) to Lq (0, d),
ε > 0, N ∈ N and 0 ≤ d0 < d1 < · · · < d4N < d. Set Ik = (dk , dk+1 ) and
1
1
3
assume that A(Ik ) ≥ ε for k = 0, 1, . . . , 4N − 1. Then aN (T ) ≥ 2 q − p − 2 ε.
Proof. Let 0 < γ < 1. Then there exist functions fk ∈ Lp (Ik ) such that
kfk kp,Ik = 1 and
(3.3)
inf kT fk − αvkq,Ik ≥ γA(Ik ) ≥ γε.
α∈<
Inequalities Applicable To
Certain Partial Differential
Equations
J. Lang, O. Mendez and
A. Nekvinda
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JJ
J
II
I
By definition of the approximation numbers, there is a bounded linear mapping
with rank P ≤ N such that
Go Back
aN +1 (T ) ≥ γkT − P kp→q .
Quit
P
p
Then P = N
i=1 Pi , where Pi are one-dimensional operators from L (0, d) into
Lq (0, d). Thus, we can write (Pi f )(x) = φi (x)Ri (f ), where φi ∈ Lq (0, d)
Page 35 of 78
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0
and Ri ∈ (Lp (0, d))∗ . Since (Lp (0, d))∗ = Lp (0, d), it follows that Ri f (x) =
Rd
0
ψi (t)f (t)dt and there are functions ψi ∈ Lp (0, d) such that
0
(P f )(x) =
N
X
Z
φi (x)
d
ψi (t)f (t)dt.
0
i=1
Denote by X the range of P . Notice that dim(X) ≤ N .
Define Ji := I2i ∪I2i+1 for i = 0, 1, . . . , 2N −1. For each i ∈ {0, 1, . . . , 2N −
1} let (ri , si ) be orthogonal to the 2-dim vector such that
Z
Z
p
p
(3.4)
|ri | + |si | > 0 and ri
uf2i + si
uf2i+1 = 0.
I2i
I2i+1
Rx
Set gi (t) = ri f2i + si f2i+1 and hi (x) = v(x) 0 u(t)gi (t)dt. From kfi k = 1 for
each i: 0 ≤ i ≤ 2N − 1 and by (3.2), one has
p1
Z
Z
p
p
p
p
kgi kp = |ri |
|f2i (t)| dt + |si |
|f2i+1 (t)| dt
I2i
I2i+1
1
= (|ri |p + |si |p ) p .
Rd
R
Consequently, khi kq = kT gi kq < ∞. Moreover, 0 hi (t)dt = Ji hi (t)dt = 0,
whence
supp hi ⊂ Ji for all i = 0, 1, . . . , 2N − 1.
Thus, using Lemma 2.18 one finds that there exists an integer k, 0 ≤ k ≤
2N − 1, such that
1
distq (hk , X) ≥ √ khk kq ,
2 2
Inequalities Applicable To
Certain Partial Differential
Equations
J. Lang, O. Mendez and
A. Nekvinda
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from which it follows that
aN +1 (T ) ≥ γkT − P kp→q
γkT f − P f kq
≥
sup
kf kp
f ∈Lp ,supp f ⊂Jk
γkT gk − P gk kq
≥
kgk kp
γkhk − P gk kq
=
kgk kp
distq (hk , X)
γ khk kq
≥γ
≥ √
.
kgk kp
2 2 kgk kp
Inequalities Applicable To
Certain Partial Differential
Equations
J. Lang, O. Mendez and
A. Nekvinda
Using Lemma 3.2, (3.3) and the inequality
1
1
1
1
(|rk |p + |sk |p ) p ≤ 2 p − q (|rk |p + |sk |p ) p ,
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we obtain
Contents
(|rk |q inf α∈< kTI2k f − αvkqq + |sk |q inf α∈< kTI2k+1 − αvkqq )
khk kq
≥
1
kgk kp
(|rk |p + |sk |p ) p
1
q
JJ
J
II
I
1
≥ γε
(|rk |q + |sk |q ) q
|p
(|rk + |sk
|p )
1
p
1
Go Back
1
≥ γ ε 2q−p
Close
which, together with the previous estimate gives
1
1
aN +1 (T ) ≥ γ 2 2 q − p −3/2 .
The proof is complete.
Quit
Page 37 of 78
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Using the properties of approximation numbers on dual operators we can
extend the previous result.
Lemma 3.4. Let 2 ≤ p ≤ q ≤ ∞ and suppose that T : Lp (0, d) → Lq (0, d)
is bounded. Let ε > 0 and suppose that there exist N ∈ N and numbers
dk , k = 0, 1, . . . , 4N with 0 = d0 < d1 < · · · < d4N < d such that A(Ik ) ≥ ε
for k = 0, 1, . . . , N − 1, where Ik = (dk , dk+1 ). Then aN (T ) ≥ cε where c is
positive and depends only on p, d.
0
0
Proof. The adjoint of T , T 0 , is bounded from Lq into Lp . It is easy to see that
Lemma 3.2 holds for T replaced by T 0 . Then the proof follows immediately
from Lemma 2.5 and Remark 2.6 in [3].
Lemma 3.5. Let 1 ≤ p ≤ 2 ≤ q ≤ ∞ and suppose that T : Lp (0, d) →
Lq (0, d) is bounded. Let ε > 0 and suppose that there exists N ∈ N and
numbers dk , k = 0, 1, . . . , 4N with 0 = d0 < d1 < · · · < d4N < d such that
A(Ik ) ≥ ε for k = 0, 1, . . . , N − 1, where Ik = (dk , dk+1 ). Then aN (T ) ≥
1
1
cεn q − 2 where c is positive and depends only on p, d.
Proof. Let 0 < γ < 1. Then there exist functions fk ∈ Lp (Ik ) such that
kfk kp,Ik = 1 and
Inequalities Applicable To
Certain Partial Differential
Equations
J. Lang, O. Mendez and
A. Nekvinda
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(3.5)
inf kT fk − αvkq,Ik ≥ γA(Ik ) ≥ γε.
α∈<
By definition of the approximation numbers there is a bounded linear mapping
with rank P ≤ N such that
aN +1 (T ) ≥ γkT − P kp→q .
Close
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P
Write P = N
i=1 Pi and let Ji be as in the proof of Lemma 3.3. InRthe notation
d
Rof Lemma 3.3, in this case we also have khi kq = kT gi kq < ∞ and 0 hi (t)dt =
h t)dt, so that
Ji (
supp hi ⊂ Ji for all i = 0, 1, . . . , 2N − 1,
whence, by Lemma 2.18, there exists an integer k, 0 ≤ k ≤ 2N − 1, such that
1 1 1
distq (hk , X) ≥ √ n q − 2 khk kq ,
3 2
Inequalities Applicable To
Certain Partial Differential
Equations
which gives
J. Lang, O. Mendez and
A. Nekvinda
aN +1 (T ) ≥ γkT − P kp→q
γkT f − P f kq
≥
sup
kf kp
f ∈Lp ,supp f ⊂Jk
γkhk − P gk kq
γkT gk − P gk kq
≥
=
kgk kp
kgk kp
γ
khk kq 1q − 12
distq (hk , X)
≥γ
≥ √ ·
n
.
kgk kp
3 2 kgk kp
Using Lemma 3.2, (3.5) and the inequality
1
1
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I
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1
1
(|rk |p + |sk |p ) p ≤ 2 p − q (|rk |p + |sk |p ) p
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we obtain
1
(|rk |q inf α∈< kTI2k f − αvkqq + |sk |q inf α∈< kTI2k+1 − αvkqq ) q
khk kq
≥
1
kgk kp
(|rk |p + |sk |p ) p
1
≥ γε
(|rk |q + |sk |q ) q
(|rk |p + |sk |p )
1
p
1
1
≥ γ ε 2q−p ,
which gives with the previous estimate
1
1
aN +1 (T ) ≥ γ 2 cεn q − 2
Inequalities Applicable To
Certain Partial Differential
Equations
for fixed c > 0 and completes the proof.
The following theorem follows immediately from the previous lemmas. It
improves results from [1] and [5].
Theorem 3.6. Suppose that T is compact (see Proposition 2.3 and Remark 2.1).
Then, for small ε > 0, 1 ≤ p ≤ q ≤ ∞
aN (ε)+1 (T ) ≤ 2ε,
J. Lang, O. Mendez and
A. Nekvinda
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JJ
J
for 1 ≤ p ≤ q ≤ 2 or 2 ≤ p ≤ q ≤ ∞
a[ N (ε) ]−1 (T ) > cε,
4
II
I
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and for 1 ≤ p ≤ 2 ≤ q ≤ ∞
Close
a[ N (ε) ]−1 (T ) > cεN (ε)
1
− 12
q
.
Quit
4
Here N (ε) ≡ N ((0, d), ε) is defined in (2.9) and [x] denotes the integer part of
x.
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Proof. The first inequality is an immediate consequence of Lemma 3.1 and
definition of N (ε). The second inequality follows from Lemmas 2.4, 3.1 and
3.2.
Inequalities Applicable To
Certain Partial Differential
Equations
J. Lang, O. Mendez and
A. Nekvinda
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4.
Local Asymptotic Result
The first part of this section is devoted to proving lemmas that will be needed in
the proof of our local asymptotic results, which we present in the second part.
Lemma 4.1. Let u and v be constant functions on the interval I = (a, b) ⊂
(0, d) and let 1 ≤ p ≤ q ≤ ∞. Then
1
A(I) := A(I, u, v) = |u||v||I| p0
+ 1q
A((0, 1), 1, 1).
Proof. If u = 0 then A(I, u, v) = 0 and the assertion is trivial. Assume that
u 6= 0. Using the substitutions y = x−a
and t = a + s(b − a), we obtain
b−a
Z x
A(I, u, v) = sup inf v
u f (t)dt − α kf kp,I =1 α∈<
a
q,I
Z
= |v||u| sup inf α∈< kf kp,I ≤1
= sup
x
a
1
inf (b − a)1− q
kf kp,I =1 α∈<
f (t)dt − α
q,I
Z y
f
(a
+
s(b
−
a))ds
−
α
0
.
Writing g(s) = f (a + s(b − a)) we have kgkp,(0,1) = (b − a)− p kf kp,(a,b) and
thus
Z x
1+ 1q
A(I, u, v) = |v||u||I|
g(t)dt
−
α
sup
kgkp,(0,1) =(b−a)
1
−p
a
q,(0,1)
J. Lang, O. Mendez and
A. Nekvinda
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q,(0,1)
1
Inequalities Applicable To
Certain Partial Differential
Equations
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= |v||u||I|
1
+ 1q
p0
sup
kgkp,(0,1) =1
= |v||u||I|
1
+ 1q
p0
Z
x
a
g(t)dt − α
q,(0,1)
A((0, 1), 1, 1).
The proof is complete.
0
Lemma 4.2. Let I = (a, b) ⊂ (0, d), 1 ≤ p ≤ q ≤ ∞, u1 , u2 ∈ Lp (I) and
v ∈ Lq (I). Then
Inequalities Applicable To
Certain Partial Differential
Equations
|A(I, u1 , v) − A(I, u2 , v)| ≤ kvkq,I ku1 − u2 kp0 ,I
Proof. Suppose first that A(I, u1 , v) ≥ A(I, u2 , v). Then
A(I, u1 , v) − A(I, u2 , v)
Z x
= sup inf v(x)
(u1 (t) − u2 (t) + u2 (t))f (t)dt − α − A(I, u2 , v)
kf kp,I =1 α∈<
a
q,I
"
Z x
≤ sup inf v(x)
(u
(t)
−
u
(t))f
(t)dt
1
2
kf kp,I =1 α∈<
a
Z
+
v(x)
a
x
q,I
#
− A(I, u2 , v)
u2 (t)f (t)dt − α inf kvkq,I ku1 − u2 k
kf kp,I =1 α∈<
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JJ
J
II
I
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q,I
"
≤ sup
J. Lang, O. Mendez and
A. Nekvinda
p0 ,I
Z x
#
+
v(x)
u
(t)f
(t)dt
−
α
2
a
− A(I, u2 , v)
≤ kvkq,I ku1 − u2 kp0 ,I + A(I, u2 , v) − A(I, u2 , v).
Close
Quit
q,I
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The remaining case can be proved analogously.
0
Lemma 4.3. Let I = (a, b) ⊂ (0, d), 1 ≤ p ≤ q ≤ ∞, u ∈ Lp (I), and
v1 , v2 ∈ Lq (I). Then
|A(I, u, v1 ) − A(I, u, v2 )| ≤ 3kv1 − v2 kq,I kukp0 ,I
Proof. If A(I, u, v1 ) ≥ A(I, u, v2 ) then by Lemma 2.4 we have
A(I, u, v1 ) − A(I, u, v2 )
Z x
− A(I, u, v2 )
= sup inf v
(x)
u(t)f
(t)dt
−
α
1
kf kp,I =1 α∈<
a
q,I
Z x
v1 (x)
= sup
inf
u(t)f (t)dt − α − A(I, u, v2 )
kf kp,I =1 |α|≤2kukp0 ,I
a
kf kp,I =1 |α|≤2kukp0 ,I
a
x
J. Lang, O. Mendez and
A. Nekvinda
q,I
"
Z x
≤ sup
inf
(v
(x)
−
v
(x))
u(t)f
(t)dt
−
α
1
2
|α|≤2kuk
0
p ,I
kf kp,I =1
a
q,I
Z x
#
− A(I, u, v2 )
+
v
(x)
u(t)f
(t)dt
−
α
2
a
q,I
h
≤ sup
inf
k(v1 (x) − v2 (x))kq,I kukp0 ,I kf kp,I + k(v1 − v2 )αkq,I
Z
+
v2
Inequalities Applicable To
Certain Partial Differential
Equations
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Close
#
u(t)f (t)dt − α − A(I, u, v2 )
q,I
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≤ 3kv1 − v2 kq,I kukp0 ,I
inf
+ sup
kf kp,I =1 |α|≤kukp0 ,I
Z
v2 (x)
a
x
u(t)f (t)dt − α − A(I, u, v2 )
q,I
= 3kv1 − v2 kq,I kukp0 ,I .
Now we prove a local asymptotic result which in some sense extends those
in [2] and [5]:
Lemma 4.4. Let I = (a, b) ⊂ (0, d), |I| < ∞ and 1 < p ≤ q ≤ ∞. Assume
0q
0
that u ∈ Lp (I) and v ∈ Lq (I). Set r = pp0 +q
. Then
Z
Z
r
r
r
c1 αp,q |uv| ≤ lim inf ε N (ε, I) ≤ lim sup ε N (ε, I) ≤ c2 αp,q |uv|r ,
ε→0+
I
ε→0+
I
where αp,q = A((0, 1), 1, 1).
Proof. Set s =
p0
q
Let l ∈ N be fixed. Then by the absolute convergence of the Lebesque integral
and the Luzin Theorem there exists m := m(l) ∈ N, {Wj }m
j=1 ∈ P and real
numbers ξj , ηj such that setting
m
X
j=1
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JJ
J
rs = p0 , rs0 = q.
ul =
J. Lang, O. Mendez and
A. Nekvinda
Contents
+ 1. Clearly,
(4.1)
Inequalities Applicable To
Certain Partial Differential
Equations
ξj χWj ,
vl =
m
X
j=1
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I
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ηj χ W j ,
J. Ineq. Pure and Appl. Math. 5(1) Art. 18, 2004
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we have
and
1
ku − ul kp0 ,I < ,
l
1
k |u|r − |ul |r ks,I < ,
l
1
kv − vl kq,I < .
l
1
k |v|r − |vl |r ks0 ,I < .
l
Consequently,
Z
Z
r
r
r
r
|u| |v| − |ul | |vl | I
ZI
Z
r r
r
≤ |u| |vl | − |v| + |vl |r |ul |r − |u|r I
I r
r
≤ (kukp0 ,I |vl | − |v| + |ul |r − |u|r k |vl |r kq,I )
s0 ,I
J. Lang, O. Mendez and
A. Nekvinda
s,I
1
kukp0 + kvl kq
l
1
≤ kukp0 + kv − vl kq + kvl kq
l
1 1
≤
+ kukp0 ,I + kvkq .
l l
≤
Inequalities Applicable To
Certain Partial Differential
Equations
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Let ε > 0. Put N (ε) = N (ε, I). According to Lemma 2.8 there is a system of
N (ε)
intervals {Ij }j=1 ∈ P such that
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A(I1 ) ≤ ε, A(IN (ε) ) ≤ ε and A(Ii ) = ε for 2 ≤ i ≤ N (ε).
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Define,
Ji = I2i ∪ I2i+1 , i = 1, 2, . . . , N (ε)/2, for even N (ε)
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and
Ji = I2i ∪ I2i+1 , i = 1, 2, . . . , (N (ε) − 3)/2,
J(N (ε)−1)/2 = JN (ε)−2 ∪ JN (ε)−1 ∪ JN (ε) for odd N (ε).
[ N (ε) ]
In both cases {Ji }j=12 ∈ P and according to the definition of N (ε), A(Ji ) > ε
i
h
for all 1 ≤ i ≤ N2(ε) . Let Wi = [di−1 , di ], where a = d0 < d1 < d2 < · · · <
dm = b. Set
K = {Ji ; 1 ≤ i ≤ [
n(ε)
] and there exists j ∈ {1, 2, . . . , m} such that Ji ⊂ Wj }.
2
If Ji ∈
/ K, there exists k ∈ {1, 2, . . . , m−1} such that dk ∈ int(J
h i ). The
i number
N (ε)
of such intervals Ji can be estimate by m − 1. Then #K ≥
− m + 1.
2
Using Lemmas 4.1, 4.2 and 4.3 one sees that
N (ε)
− m − 1 εr
2
X
≤
Ar (Ik ; u, v)
k∈K
X
≤
A(Ik ; ul , vl ) + A(Ik ; u, v) − A(Ik ; ul , v)
k∈K
r
+ A(Ik ; ul , v) − A(Ik ; ul , vl )
Inequalities Applicable To
Certain Partial Differential
Equations
J. Lang, O. Mendez and
A. Nekvinda
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≤ max(1, 3r−1 )
X
r
Ar (Ik ; ul , vl ) + A(Ik ; u, v) − A(Ik ; ul , v)
k∈K
r + A(Ik ; ul , v) − A(Ik ; ul , vl )
"
m
X
r−1
r
|ξj |r |ηj |r |W (j)|
≤ max(1, 3 ) αp,q
j=1
+
m
X
ku − ul krp0 ,W (j) kvkrq,W (j)
Inequalities Applicable To
Certain Partial Differential
Equations
j
+
m
X
#
kv − vl krq,W (j) kukrp0 ,W (j) .
J. Lang, O. Mendez and
A. Nekvinda
j=1
Using the discrete version of Hölder’s inequality
m
X
ai b i ≤
i=1
m
X
! 1s
asi
i=1
m
X
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! 10
s
bsi
0
i=1
and (4.1) we obtain
N (ε)
− m + 1 εr
2
r−1
≤ max(1, 3
)
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αp,q
m
X
j=1
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r
r
|ξj | |ηj | |Wj |
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+
! 1s
m
X
ku −
m
X
0
ul kpp0 ,Wj
j=1
+
s
j=1
m
X
! 10
m
X
s
kv − vl kqq,Wj
! 1s 
0
kukpp0 ,Wj

j=1
j=1
! 10
kvkqq,Wj
1 1
≤ max(1, 3 )
|uv| +
+ kukp0 ,I + kvkq,I
l l
I
1
r
r
+ r kukp0 ,I + kvkq,I
l
Z
1 1
r−1
r
r
≤ max(1, 3 ) αp,q |uv| +
+ kukp0 ,I + kvkq,I
l l
I
1
r
r
+ r kukp0 ,I + kvkq,I .
l
r−1
r
αp,q
Z
r
Inequalities Applicable To
Certain Partial Differential
Equations
J. Lang, O. Mendez and
A. Nekvinda
Title Page
Contents
Thus, there is a constant c1 > 0 independent of ε and l such that
Z
1
N (ε)
1
r
r
(4.2)
− m + 1 ε ≤ c1
|uv| + + r
2
l
l
I
Let Ii = [ci−1 , ci ], i = 1, 2, . . . , N (ε). Thus, a = c0 < c1 < · · · < cN (ε) = b.
Let D = {ek : 1 ≤ k ≤ M } stand for the set-theoretic union of {ci : 1 ≤
i ≤ N (ε)} and {dj : 1 ≤ j ≤ m}, so that a = e1 < e2 < · · · < eM = b
and write Lk = [ek−1 , ek ]. Then {Lk }M
k=1 ∈ P and for each 1 ≤ k ≤ M there
exists i, 1 ≤ i ≤ N (ε) such that Lk ⊂ Ii and, consequently, by Lemma 2.5 it is
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A(Lk ) ≤ A(Ii ) ≤ ε. Thus,
Z
r
αp,q |uv|r
I
Z
Z
Z
r−1
r
r
r
r
r
r
≤ max(1, 3 )αp,q
|ul vl | + |u − ul | |v| + |ul | |v − vl |
I
r
≤ max(1, 3r−1 )αp,q
m
X
I
I
|ξj |r |ηj |r |Wj |
j=1
+
m
X
!1/s
0
ku − ul kpp0 ,Wj
j=1
+
m
X
m
X
Inequalities Applicable To
Certain Partial Differential
Equations
!1/s0
kvkqq,Wj
J. Lang, O. Mendez and
A. Nekvinda
j=1
!1/s0
kv −
vl kqq,Wj
m
X
!1/s 
0
kukpp0 ,Wj

Title Page
j=1
j=1

m
X

1

|ξj | |ηj | |Lk | + r (kukrp0 ,I + kvkrq,I )
≤ max(1, 3
l
j=1 {k;Lk ⊂Wj }


m
r
X
X
αp,q
≤ max(1, 3r−1 ) 
Ar (Lk , ξj , ηj ) + r (kukrp0 ,I + kvkrq,I )
l
j=1 {k;Lk ⊂Wj }
r
αp,q
r−1
r
r
r
≤ max(1, 3 ) (N (ε) + m)ε + r (kukp0 ,I + kvkq,I ) .
l
r−1
r
)αp,q
X
r
r
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Thus, there exists c2 > 0, independent of ε and l such that
Z
1
r
r
|uv| ≤ c2 (N (ε) + m)ε + r .
l
I
Letting ε → 0+ here and in (4.2) we obtain for each l
Z
1
1
r
r
|uv| + + r
lim sup ε N (ε) ≤ 2c1
l
l
ε→0+
I
and
Z
|uv|r ≤ c2 lim inf εr N (ε) +
ε→0+
I
1
lr
Inequalities Applicable To
Certain Partial Differential
Equations
.
J. Lang, O. Mendez and
A. Nekvinda
The lemma follows letting l → ∞.
The latter lemma coupled with Theorem 3.6 yields the following theorem:
Theorem 4.5. Let 1 < p ≤ q ≤ 2 or 2 < p ≤ q < ∞, kvkq < ∞, kukp0 < ∞
and u, v > 0. Then
Z d
Z d
r
r
r
c1
|uv| ≤ lim inf nan (T ) ≤ lim sup nan (T ) ≤ c2
|uv|r .
n→∞
0
n→∞
0
Let 1 < p < 2 < q < ∞ kvkq < ∞, kukp0 < ∞ and u, v > 0. Then
Z d
Z d
( 12 − 1q )r+1 r
r
r
c3
|uv| ≤ lim inf n
an (T ) ≤ lim sup nan (T ) ≤ c4
|uv|r .
n→∞
0
where r =
p0 q
.
p0 +q
n→∞
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5.
The Main Result
Rd
0
Throughout this section we assume that 0 |u(t)|p dt = ∞. Furthermore, we
Rx
0
set U (x) := 0 |u(t)|p dt. Let {ξk }∞
k=−∞ , be a sequence satisfiyng
U (ξk ) = 2
(5.1)
kp0
q
,
and
σk := 2k/q kvkq,Zk ,
(5.2)
Inequalities Applicable To
Certain Partial Differential
Equations
Zk = [ξk , ξk+1 ].
The sequence {σk } is the analogue of the sequence defined in [2] and [5], which
in turn, was motivated by a similar sequence introduced in [8].
The following technical lemmas play a central role in this section.
1
Lemma 5.1. Let r > 0, k0 , k1 ∈ Z with k0 ≤ k1 . Let I = (a, b) ⊂ ∪kk=k
Zk .
0
Then
r
J r (I) ≤ 4 q max σkr .
k0 ≤k≤k1
Proof. Let x ∈ (a, b). Then there exists n ∈ Z, k0 ≤ n ≤ k1 such that x ∈ Zn .
Clearly,
Z
a
x
0
|u|p
r0
p
kvχ(x,b) krq ≤
Z
ξn+1
0
|u|p
pr0
0
r
≤ 2(n+1) q
k1
X
kvχ(ξn ,ξk1 +1 ) krq
! rq
kvχ(ξi ,ξi+1 ) kqq
J. Lang, O. Mendez and
A. Nekvinda
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i=n
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(n+1) rq
=2
k1
X
σq
i=n
! rq
(n+1) rq
i
2i
≤2
max
i=n,...,k1
so that
σiq
rq
r
r
2(1−n) q = 4 q max σir ,
i=n,...,k1
r
J r (I) ≤ 4 q max σkr .
k0 ≤k≤n1
0
q
Lemma 5.2. Let r ≥ pp0 +q
, Ii = (ai , bi ), 1 ≤ i ≤ l and bi ≤ ai+1 , 1 ≤ l − 1.
l
Let k ∈ Z be such that ∪i=1 Ii ⊂ Zk . Then
l
X
0
r
J r (Ii ) ≤ (2p /q − 1) p0 σkr .
J. Lang, O. Mendez and
A. Nekvinda
i=1
Proof. Set s = (p0 + q)/p0 . Thus s > 1 and p0 /s0 = q/s = p0 q/(p0 + q). Fix
0q
xi ∈ (ai , bi ). According to the assumption r ≥ pp0 +q
we have r ≥ p0 /s0 , r ≥ q/s
and
r0
l Z xi
l
X
X
p
p0
|u|
kvχ(xi ,bi ) krq ≤
kuχIi krp0 kvχIi krq
i=1
ai
i=1
≤
l
X
! s10
kuχIi krs
p0
0
i=1
≤
l
X
i=1
Inequalities Applicable To
Certain Partial Differential
Equations
l
X
! 1s
kvχIi krs
q
i=1
! pr0
0
kuχIi kpp0
l
X
! rq
kvχIi kqq
≤ kuχZk krp0 kvχZk krq = (2
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i=1
p0 /q
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r
p0
− 1) σkr .
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Thus,
l
X
r
J (Ii ) =
i=1
l
X
i=1
Z
xi
p0
r0
p
|u|
sup
xi ∈Ii
r
ai
1
Lemma 5.3. Let ∪li=1 Ii ⊂ ∪kk=k
Zk and r ≥
0
l
X
0
kvχ(xi ,bi ) krq ≤ (2p /q − 1) p0 σkr .
p0 q
.
p0 +q
Then
k1
0
X
r
r
J r (Ii ) ≤ (2p /q − 1) p0 + 21+2 q
σkr .
i=1
k=k0
Proof. Let
Inequalities Applicable To
Certain Partial Differential
Equations
J. Lang, O. Mendez and
A. Nekvinda
A = {i ∈ {1, 2, . . . , l} : there exists k ∈ Z such that ξk ∈ int Ii },
B = {i ∈ {1, 2, . . . , l} : there exists k ∈ Z such that Ii ⊂ Zk }.
Clearly, A ∩ B = ∅, A ∪ B = {1, 2, . . . , l}. By Lemma 5.2 we obtain
X
(5.3)
p0 /q
r
J (Ii ) ≤ (2
− 1)
i∈B
r
p0
k1
X
k=k0
Set Ai = {k ∈ Z; int(Ii ∩ Zk ) 6= ∅} for i ∈ A. Let A = {Ai ; i ∈ A}. Since
each k belongs at most to two elements of A, Lemma 5.1 yields
X
i∈A
r
J r (Ii ) ≤ 4 q
X
i∈A
r
max σkr ≤ 4 q 2
k∈Ai
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J
σkr .
k1
X
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σkr ,
k=k0
which coupled, with (5.3) yields the assertion of this lemma.
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Lemma 5.4. Let K1 , K2 be the constants from Proposition 2.1. Then
1
K1 sup σk ≤ kT k ≤ 4 q K2 sup σk .
k∈Z
k∈Z
Moreover, T is compact if and only if
lim sup σk = lim sup σk = 0.
n→∞ k≥n
n→−∞ k≤n
Inequalities Applicable To
Certain Partial Differential
Equations
Proof. Let (a, b) ⊂ (0, d). Set
a(ε) = a + ε, b(ε) =
(
b−ε
if b < ∞,
if b = ∞.
1
ε
Define a function f (ε, x) by
J. Lang, O. Mendez and
A. Nekvinda
Title Page
Z
x
p0
10
Z
p
b(ε)
|u|
f (ε, x) =
! 1q
|v|q
Contents
.
x
a(ε)
Since f (ε, x) % f (0, x) for ε → 0+ and any fixed x we have
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J
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J(a(ε), b(ε)) =
sup
f (ε, x) % sup f (0, x) = J(a, b).
a≤x≤b
a(ε)≤x≤b(ε)
Choosing a = 0, b = d we have by Lemma 5.1
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1
q
J(a(ε), b(ε)) ≤ 4 sup σk
k∈Z
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and consequently,
1
J(a, b) ≤ 4 q sup σk .
k∈Z
By the definition of σk it is easy to see that σk ≤ J(0, d) for each k ∈ Z which
implies
sup σk ≤ J(a, b).
k∈Z
Now, the first part of our lemma follows by applying Lemma 2.1.
The second part can be proved analogously by using Proposition 2.3.
Lemma 5.5. Let I 0 = [a0 , b0 ] ⊂ I = [a, b] ⊂ [0, d] and let ε > 0. Let
N (I,ε)
{Ii }i=1 ∈ P(I) and A(Ii ) ≤ ε. Set K = {i; Ii ⊂ I 0 }, K = #K. Then
Inequalities Applicable To
Certain Partial Differential
Equations
J. Lang, O. Mendez and
A. Nekvinda
K − 2 ≤ N (I, ε) ≤ K + 2.
N (I 0 ,ε)
Proof. Let {Ii0 }i=1 ∈ P(I 0 ), A(Ii0 ) ≤ ε. Let Ii = [ai , ai+1 ], i = 1, 2, . . . , N (I, ε),
and Ij0 = [a0j , a0j+1 ], j = 1, 2, . . . , N (I 0 , ε) and put k0 = min K and k1 =
max K. Write
(
(
{[a0 , ak0 ]} if a0 < ak0 ,
{[ak1 +1 , b0 ]} if ak1 +1 < b0 ,
S1 =
S
=
2
∅
if a0 = ak0 ,
∅
if ak1 +1 = b0 .
e ≤ ε for each Ie ∈ S1 ∪ S2 . Take a system of
Remark that by Lemma 2.5, A(I)
e ≤ ε for Ie ∈ L.
intervals L = S1 ∪ S2 ∪ {Ii ; i ∈ K} so that L ∈ P(I 0 ) and A(I)
Thus, by the definition of N (I 0 , ε) one has
N (I 0 , ε) ≤ #L ≤ #K + 2 = K + 2.
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To prove the inequality K − 2 ≤ N (I 0 , ε) set
(
(
0
0
{[a
,
a
]}
if
a
<
a
,
{[b0 , ak1 +2 ]}
k
−1
k
−1
0
0
0
S10 =
S
=
2
∅
if ak0 −1 = a0 ,
∅
if b0 < ak1 +2 ,
if b0 = ak1 +2 .
e ≤ ε for Ie ∈ S 0 ∪ S 0 . Denote N0 = {Ii ; Ii ⊂ [a, a0 ]}, N1 =
Clearly, A(I)
1
2
{Ii ; Ii ⊂ [b0 , b]} and set n0 = #N0 , n1 = #N1 . Take a system of intervals
L0 = S10 ∪ S20 ∪ N0 ∪ N1 ∪ {Ij0 ; j = 1, 2, . . . , N (I 0 , ε)}.
e ≤ ε for any Ie ∈ L0 and by definition of N (I, ε), N (I, ε) ≤ #L0 .
Since, A(I)
Moreover, since
Inequalities Applicable To
Certain Partial Differential
Equations
J. Lang, O. Mendez and
A. Nekvinda
n0 + n1 + K ≤ N (I, ε) ≤ n0 + n1 + K + 2
Title Page
and
n0 + n1 + N (I 0 , ε) ≤ #L0 ≤ n0 + n1 + N (I 0 , ε) + 2
JJ
J
we obtain
0
n0 + n1 + K ≤ n0 + n1 + N (I , ε) + 2,
which finishes the proof.
r
i∈Z σi
P
Lemma 5.6. Let 1 < p ≤ q < ∞, r =
Let
< ∞. Then T is
Rd
r
compact, 0 |uv| < ∞ and there are positive constants c1 , c2 such that
d
r
r
r
Z
|uv| ≤ lim inf ε N (ε) ≤ lim sup ε N (ε) ≤ c2
c1
0
ε→0+
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p0 q
.
p0 +q
Z
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ε→0+
d
|uv|r .
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Proof. By Lemma 5.4, T is compact. Let k ∈ Z and set s = p0 /q + 1. It follows
that rs = p0 , rs0 = q and using Hölder’s inequality, we obtain
Z ξk+1
pr0 Z ξk+1
rq
Z
p0
r
q
|uv| ≤
|u|
|v|
.
Zk
ξk
ξk
Moreover by the definition of ξk one has
Z
Z ξk
p10
1
0
0
p /q
p
0
p
(2
− 1)
|u|
=
0
ξk+1
p0
p10
|u|
ξk
and consequently,
Z
(5.4)
0
r
|uv|r ≤ (2p /q − 1) p0 σkr .
Inequalities Applicable To
Certain Partial Differential
Equations
J. Lang, O. Mendez and
A. Nekvinda
Zk
Rd
This proves 0 |uv|r < ∞.
Fix δ > 0. Take k0 , k1 ∈ Z such that
−1
0
X
X
r
r
σir ≤ (2p /q − 1) p0 + 21+2 q
δ.
σir +
i≤k0 −1
i≥k1
N (ε)
Let ε > 0. Let {Ij }j=1 ∈ P(0, d), A(Ij ) ≤ ε. Remark that according to the
definition of N (ε), A(Ij ∪Ij+1 ) > ε for j = 1, 2, . . . , N (ε)−1. Set I = [ξk0 , ξk1 ]
and
N0 = {Ij ; Ij ⊂ [0, ξk0 ]},
N1 = {Ij ; Ij ⊂ [ξk1 , d]},
e = {Ij ; Ij ⊂ I},
N
n0 (ε) = #N0 ,
n1 (ε) = #N1 ,
e.
n
e(ε) = #N
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Then N (ε) ≤ n
e(ε) + n0(ε)+ n1(ε) + 2. By Lemma 5.5, n
e(ε) − 2 ≤ N (I, ε) ≤
n
n
e(ε) + 2. Since n ≤ 2 2 + 1 for any positive integer n, we obtain
εr (N (ε) − N (I, ε)) ≤ εr (N (ε) − n
e(ε) + 2)
r
≤ ε (n0 (ε) + n1 (ε) + 4)
n0 (ε)
n1 (ε)
r
≤ 2ε
+
+3 .
2
2
For j0 = min{j; Ij ∈ N1 (ε)}, one has
h
n0 (ε)
i
2
X
1 r
ε (N (ε) − N (I, ε) − 6) ≤
εr +
2
j=1
h
n0 (ε)
2
n1 (ε)
X2
Inequalities Applicable To
Certain Partial Differential
Equations
i
εr
J. Lang, O. Mendez and
A. Nekvinda
j=j0
i
X
≤
j0 +
h
j0 +
h
i
n1 (ε)
2
X
Ar (Ij ∪ Ij+1 ) +
j=1
Ar (Ij ∪ Ij+1 ).
Contents
j=j0
JJ
J
Since A(I, ε) ≤ J(I, ε) for I ⊂ J and according to Lemma 5.5 we have
h
n0 (ε)
2
i
j0 +
X
1 r
ε (N (ε) − N (I, ε) − 6) ≤
J r (Ij ∪ Ij+1 ) +
2
j=1
0
r
r
≤ (2p /q − 1) q + 21+2 q
h
n1 (ε)
2
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i
X
J r (Ij ∪ Ij+1 )
!
i≤k0 −1
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j=j0
X
II
I
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X
i≥k1
σir
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≤δ
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which gives
εr N (ε) ≤ 2δ + εr N (I, ε) + 6εr
and consequently,
lim sup εr N (ε) ≤ 2δ + lim sup εr N (I, ε).
(5.5)
ε→0+
ε→0+
e + 2 ≤ N (ε) + 2 and thus
Again, Lemma 5.5 gives N (I, ε) ≤ n
lim sup εr N (I, ε) ≤ lim sup εr N (ε).
(5.6)
ε→0+
ε→0+
Inequalities Applicable To
Certain Partial Differential
Equations
By (5.4) we have
Z
(5.7)
d
r
Z
|uv| −
0
I
r
0
|uv| ≤ (2p ,q − 1) p0 δ.
J. Lang, O. Mendez and
A. Nekvinda
r
Using Lemma 4.4 one easily sees that
Z
Z
r
r
r
c1 αp,q |uv| ≤ lim inf ε N (I, ε) ≤ lim sup ε N (I, ε) ≤ c2 αp,q |uv|r
ε→0+
I
ε→0+
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J
which yields with (5.5), (5.6) and (5.7) that for any δ > 0,
Z d
r
r
p0 ,q
0
p
c1 αp,q
|uv| − (2 − 1) δ ≤ lim inf εr N (ε)
0
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ε→0+
r
≤ lim sup ε N (ε)
Close
ε→0+
Z
≤ c2 αp,q
r
|uv|
0
Letting δ → 0+ we obtain our lemma.
d
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+ 2δ.
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0
q
Theorem 5.7. Suppose that (2.1) and (2.2) are satisfied and let r = pp0 +q
and
P∞
r
i=−∞ σi < ∞.
Let 1 < p ≤ q ≤ 2 or 2 ≤ p ≤ q < ∞. Then
Z d
(5.8)
c1
|u(t)v(t)|r dt ≤ lim inf narn (T )
n→∞
0
Z d
r
|u(t)v(t)|r dt.
≤ lim sup nan (T ) ≤ c2
n→∞
0
Let 1 < p ≤ 2 ≤ q < ∞. Then
Z d
1
1
c3
|u(t)v(t)|r dt ≤ lim inf n( 2 − q )r+1 arn (T )
(5.9)
n→∞
0
Z d
r
≤ lim sup nan (T ) ≤ c4
|u(t)v(t)|r dt.
n→∞
0
Inequalities Applicable To
Certain Partial Differential
Equations
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6.
lr and Weak–lr Estimates
In this section we show that the Lr (Lr,∞ )-norms
of {an (T )}n∈N , and {σn }n∈Z
are equivalent for r ≥ mins≥1 max
p0 q
,
s0 s
.
Lemma 6.1. Let I = [a, b] and ε > 0. Set
σ(ε) := {k ∈ Z : Zk ⊂ I, σk > ε} .
Suppose that σk contains at least four elements. Then
A(I) >
ε
1
.
4q
Proof. Let Zki , i = 1, 2, 3, 4, k1 < k2 < k3 < k4 , be 4 distinct members of
σ(ε), and set I1 = (ξk1 , ξk2 ), I2 = (ξk2 +1 , ξk4 ). Then, with f0 = χI1 + χI2 ,
Z x
A(I) ≥ inf v(x)
|u(t)|f
(t)dt
−
α
0
α c
q,I
Z
≥ inf max kvkq,Zk2 |u(t)f (t)|dt − α ;
α
Z I1
kvkq,Zk4 |u(t)f (t)|dt − α
I1 ∪I2
n
= inf max kvkq,Zk2 |2k2 /q − 2k1 /q − α|;
α
k /q
o
k1 /q
k4 /q
(k2 +1)/q
2
kvkq,Zk4 2
−2
+2
−2
− α
Inequalities Applicable To
Certain Partial Differential
Equations
J. Lang, O. Mendez and
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≥ inf max
α
ε
n
2(k2 +1)/q
ε
|2k2 /q − 2k1 /q − α|;
k /q
o
2 2 − 2k1 /q + 2k4 /q − 2(k2 +1)/q − α
2(k4 +1)/q
1
ε
ε
· 1 2k4 − 2k2 +1 ≥ 1 .
≥ k4 /q
2
+ 1 2q
4q
1
Lemma 6.2. Let ε > 0. Let K = {k ∈ Z; σk > 2 q ε}. Then
#K ≤ 4N (ε) − 1.
Proof. Let Ii = [ci−1 , ci ] and i = 1, . . . , N (ε). Divide K into two disjoint sets
Z1 and Z2 by
Z1 = {k ∈ K; there exists j ∈ {1, . . . , N (ε)} such that cj ∈ Zk },
Inequalities Applicable To
Certain Partial Differential
Equations
J. Lang, O. Mendez and
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Z2 = {k ∈ K; there exists j ∈ {1, . . . , N (ε)} such that Zk ∈ Ij },
Clearly, #Z1 ≤ N (ε) − 1.
Say that k1 , k2 ∈ Z2 are equivalent if there exists j such that Zk1 ∪ Zk2 ⊂ Ij .
Denote the equivalence classes in Z2 by Y1 and Y2 . Assume #Yi ≥ 4 for some
i. Then there are k1 , k2 , k3 , k4 and j such that Zk1 ∪ Zk2 ∪ Zk3 ∪ Zk4 ⊂ Ij . Using
1
Lemma 6.1 with 2 q ε instead of ε, we have A(I) > ε which contradicts the
definition of A(I). Then #Yi ≤ 3 for any i ∈ Z2 . Consequently, the mapping
P defined by
P (i) = j if Zi ⊂ Ij for any i ∈ Z2
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is an injection and, therefore,
#Z2 ≤ 3N (ε).
Thus,
#K = #Z1 + #Z2 ≤ 4N (ε) − 1
which completes the proof.
Lemma 6.3. Let 1 < p ≤ q ≤ 2 or 2 ≤ p ≤ q < ∞. Then there are positive
constants c1 , c2 , c3 depending on p and q such that the inequality
#{k; σk > t} ≤ c1 #{k; ak (T ) ≥ c2 t} + c3
holds for all t > 0.
Inequalities Applicable To
Certain Partial Differential
Equations
J. Lang, O. Mendez and
A. Nekvinda
Proof. According to Lemma 3.4 there are two positive constants c1 , c2 depending on p, q such that
a[c1 N (ε)]−1 (T ) > c2 ε.
Then
#{k; ak (T ) > c2 ε} ≥ c1 N (ε) − 2
and, according to Lemma 6.2, we have
t
# {k; σk > t} ≤ 4N
−1
1
q
2
t
4
4
=
c1 N
−2 + −1
1
c1
c
2q
1
4
c2
≤ # k; ak (T ) > 1 t .
c1
2q
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The lemma follows by writing c1 , c2 and c3 instead of
2 c1
, 1
c1
2q
and
4
c1
− 1.
We recall the following well-known fact: given a countable set S we have
for any p, 1 ≤ p < ∞
Z ∞
X
p
|ak | = p
tp−1 #{k ∈ S; |ak | > t}dt.
0
k∈S
It is easy to see that also
Z
X
p
|ak | = p
k∈S
∞
tp−1 #{k ∈ S; |ak | ≥ t}dt.
0
Lemma 6.4. Let r > 0. There are constants c1 ≥ 0 and c2 ≥ 0 such that
Inequalities Applicable To
Certain Partial Differential
Equations
J. Lang, O. Mendez and
A. Nekvinda
k{σ}krlr (Z) ≤ c1 k{ak (T )}krlr (N) + c2 k{σ}krl∞ (Z) .
Proof. Set λ = k{σ}kl∞ (Z) . By Lemma 6.3 we have,
Z λ
r
k{σk }klr (Z) = r
tr−1 #{k ∈ Z; σk > t}dt
0
Z λ
≤r
tr−1 (c1 #{k; ak (T ) > c2 t} + c3 dt
0
Z λ
c1
= r+1 q
tr−1 #{k; ak (T ) > t}dt + c3 λr
c2
0
c1
= r+1 k{ak (T )}krlr (N) + c3 λr ,
c2
and hence the proof is complete.
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Lemma 6.5. Let r > 0. Then there is a positive constant c such that
k{σ}klr (Z) ≤ ck{ak (T )}klr (N) .
Proof. By Lemma 5.5,
k {σk } kl∞ (Z) ≤ CkT k = Ca1 (T )
≤ Ck {ak (T )} klr (N) .
Inequalities Applicable To
Certain Partial Differential
Equations
The result then follows from Lemma 6.4.
Now, we tackle the remaining inequality:
Lemma 6.6. Let 1 < p ≤ q ≤ 2 or 2 ≤ p ≤ q < ∞ and s > r =
p0 q
.
p0 +q
Then
k{an (T )}kls ≤ ck{σk }kls .
Proof. Let Ii , i = 1, 2, . . . , N (ε), be the collection of intervals given by (2.8)
with I = (a, b) and N (ε) ≡ N ((a, b), ε): note that in view of Lemma 2.2, we
have J(Ii ) = ε for 1 ≤ i < N (ε). We group the intervals Ii into families
Fj , j = 1, 2, . . . such that each Fj consists of the maximal number of those
intervals IK−1 in the collection, which satisfy the hypothesis of Lemma 5.1
and Lemma 5.2: Ik1 ⊂ (ξk0 , ξk2 +1 ), for some k0 , k2 ,, and the next interval Ik
intersects Zk2 +1 (This construction is based on our construction from [2], for
more see Lemma 5.1. and Section 6 in [2]). Hence, by Lemma 5.1 and Lemma
5.2, there is a positive constant c such that
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εr #Fj ≤ c max σnr = cσkrj .
k0 ≤n≤k2
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It follows that, with nj = [cσkrj /εr ],
N (ε) =
X
#Fj
j
≤
nj
XX
1=
1
n=1 j:nj ≥n
n=1
j
∞ X
X
∞
X
cσkrj
=
# j: r ≥n
ε
n=1
∞
X
nεr
r
≤
# k : σk ≥
.
c
n=1
(6.1)
Inequalities Applicable To
Certain Partial Differential
Equations
J. Lang, O. Mendez and
A. Nekvinda
Thus, if {σk } ∈ ls (Z) for some s ∈ (r, ∞),
Z
s
∞
ts−1 N (t)dt ≤ s
0
Z
∞
∞X
0
= scs/r
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ts−1 # k : σkr >
n=1
∞
∞X
Z
nt
c
r
dt
n−s/r z s−1 # {k : σk > z} dz
0
(6.2)
n=1
s
k {σk } kls (Z) ,
where stands for less than or equal to a positive constant multiple of the right
hand side. From the inequality N (ε) ≤ M (ε) and Theorem 3.6, aN (ε)+1 (T ) ≤
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2ε and therefore
t
# {k ∈ N : ak (T ) > t} ≤ N
+1
2
t
≤M
+ 1.
2
This yields
k {ak (T )} ksls (N)
Z
∞
ts−1 # {k ∈ N : ak (T ) > t} dt
0
Z kT k
t
s−1
≤s
t
N ( ) + 1 dt
2
0
s
k {σk } kls (Z) + kT ks
=s
k {σk } ksls (Z)
Inequalities Applicable To
Certain Partial Differential
Equations
J. Lang, O. Mendez and
A. Nekvinda
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by (6.2) and then, in virtue of Lemma 5.1 and Lemma 5.5,
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kT k k {σk (T )} kl∞ (Z) ≤ k {σk } klq (Z) .
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Lemmas 6.4 and 6.5 imply the following theorem:
Theorem 6.7. Let 1 < p ≤ q ≤ 2 and 2 ≤ p ≤ q < ∞, r =
(i) Then there exists a positive constant c1 such that
p0 q
p0 +q
Close
and k > 0.
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k{σk }klk (Z) ≤ c1 k{ak (T )}klk (N) .
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(ii) Let s > r. Then there is a positive constant c2 such that
k{ak }kls (N) ≤ c2 k{σk }kls (Z) .
(iii) Let 1 ≤ j ≤ ∞. Then there exists a positive constant c1 such that
k{σk }klk,j (Z) ≤ c1 k{ak (T )}klk,j (N) .
(iv) Let s > r and 1 ≤ j ≤ ∞. Then there is a positive constant c2 such that
k{ak }kls,j (N) ≤ c2 k{σk }kls,j (Z) .
Proof. Claims (i) and (ii) follow from Lemma 6.4 and Lemma 6.5. The assertions (iii) and (iv) can be obtained from (i) and (ii), by using real interpolation
on the scale lp,q .
Inequalities Applicable To
Certain Partial Differential
Equations
J. Lang, O. Mendez and
A. Nekvinda
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Appendix
In this section we show that the power of n in (2.11) is the best possible for
2 < p ≤ ∞. Given a square matrix of a dimension L,

(6.3)


A=

a11
a21
..
.
aL1
a12 . . .
a22 . . .
..
..
.
.
aL2 . . .

a1L
a2L 



aLL
we will denote, for 1 ≤ I ≤ L, the i-th column of A by ui (A) and the i-th row
of A by vi (A) , i.e.
ci (A) = (a1i , a2i , . . . , aLi )
and
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ri (A) = (ai1 , ai2 , . . . , aiL ).
By h(A) denote the rank of A and by u·v the canonical scalar product of vectors
u and v, i. e.
L
X
u.v =
ui vi
i=1
where u = (u1 , u2 , . . . , uL ) and v = (v1 , v2 , . . . , vL ).
Lemma 6.8. Let m ∈ N and L = 2m . Then there exists a square matrix A
given by (6.3) such that
(6.4)
Inequalities Applicable To
Certain Partial Differential
Equations
|aij | = 1 for ≤ i, j ≤ L
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and
ui (A) · uj (A) = 0 for ≤ i, j ≤ L, i 6= j.
(6.5)
Proof. We use mathematical induction with respect to m. If m = 1 it suffices
to take
1
1
A=
.
1 −1
Assume that the matrix A given by (6.3) with L = 2m satisfies (6.4) and (6.5).
Let B be a square matrix of dimension 2L = 2m+1 given by

a11 a12 . . . a1L
a21 a22 . . . a2L
..
.. . .
..
.
.
.
.
aL1 aL2 . . . aLL








B=


 a11 a12 . . . a1L

 a21 a22 . . . a2L

..
.. . .
..

.
.
.
.
aL1 aL2 . . . aLL
a11
a21
..
.
a12 . . .
a22 . . .
.. . .
.
.
aL2 . . .
a1L
a2L
..
.






aL1
aLL 

A
A

.
 :=
A −A


−a11 −a12 . . . −a1L 

−a21 −a22 . . . −a2L 
..
.. . .
.. 
.
.
.
. 
−aL1 −aL2 . . . −aLL
It is easy to see that B satisfies (6.4) and (6.5).
Lemma 6.9. Let n ∈ N and set K = 2n , L = K 2 . Then there exists a square
Inequalities Applicable To
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Equations
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matrix of dimension 2L,



M =

m11
m21
..
.
mL1
m12 . . .
m22 . . .
..
...
.
mL2 . . .

m1L
m2L 

,

mLL
such that
(6.6)
h(M ) ≤ L,
(6.7)
mii = K for 1 ≤ i, j ≤ 2L,
J. Lang, O. Mendez and
A. Nekvinda
|mij | ≤ 1 for 1 ≤ i, j ≤ 2L, i 6= j.
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Inequalities Applicable To
Certain Partial Differential
Equations
and
(6.8)
Proof. Since L = 2n we have by Lemma 6.8 a matrix A,


a11 a12 . . . a1L
 a21 a22 . . . a2L 


A =  ..
..
..  ,
.
.
 .
.
.
. 
aL1 aL2 . . . aLL
which satisfies (6.4) and (6.5). For 1 ≤ i ≤ L, set

for 1 ≤ j ≤ L, i 6= j,
 0
K
for j = i,
(6.9)
mij :=

ai,j−L for L + 1 ≤ j ≤ 2L
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and let r1 , r2 , . . . , rL be 2L-dimensional vectors, ri = (mi1 , mi2 , . . . , mi,2L ).
Set for 1 ≤ i ≤ L
ri+L
(6.10)
L
1 X
=
aji rj .
K j=1
Let M be the matrix consisting of the rows r1 , r2 , . . . , r2L , i.e. vi (M ) = ri .
Denote the elements of M by mij , so that


m11
m12
. . . m1,2L
 m21
m22
. . . m2,2L 


M =  ..
.
..
.
.

 .
.
.
m2L,1 m2L,2 . . . m2L,2L
We claim that M satisfies (6.6), (6.7) and (6.8).
Let L+1 ≤ i ≤ 2L. Then ri is by (6.10) a linear combination of u1 , u2 , . . . , uL
and then h(M ) ≤ L.
Next, we calculate mii . If 1 ≤ i ≤ L, mii = K by (6.9). Let L + 1 ≤ i ≤ 2L
and write s = i − L. Then by (6.4) and (6.10) we have
mii = ms+L,s+L
=
1
K
j=1
ajs ajs =
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L
1 X
mj,s+L mj,s+L
=
K j=1
L
X
Inequalities Applicable To
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Equations
1
1
kus (A)k2 = L = K.
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K
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We now consider (6.8). Calculate mij , i 6= j. We have four posibilities:
(i) If 1 ≤ i, j ≤ L then by (6.9) we have mij = 0 and thus, mij = 0 satisfies
(6.8).
(ii) If 1 ≤ i ≤ L, L + 1 ≤ j ≤ 2L then mij = ai,j−L and due to (6.4) it is
|mij | ≤ 1.
(iii) If L + 1 ≤ i ≤ 2L, 1 ≤ j ≤ L then setting s = i − L we have by (6.9)
and (6.10)
mij = ms+L,j
L
1
1 X
=
aks mkj = ajs mjj = ajs
K k=1
K
Inequalities Applicable To
Certain Partial Differential
Equations
J. Lang, O. Mendez and
A. Nekvinda
which gives by (6.4) |mij | ≤ 1.
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(iv) If L + 1 ≤ i ≤ 2L, L + 1 ≤ j ≤ 2L denote s = i − L, t = j − L. By
(6.9) and (6.10) we obtain
mij = ms+L,j
L
L
1 X
1 X
1
=
aks mkj =
aks akt = us (A) ut (A),
K k=1
K k=1
K
which gives with (6.5) that mij = 0 and proves (6.8).
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Let ei | denote the sequence which has 1 on i-th coordinate and 0 on other.
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Lemma 6.10. Let 2 < p ≤ ∞ and n ∈ N. Set K = 2n and L = K 2 . Then
there exists a subspace X of lp , dim X ≤ L such that for each i, 1 ≤ i ≤ 2L.
1
distp (ei , X) ≤
2p
K 1−2/p
.
Proof. Let M be the matrix of rank 2L from Lemma 6.9. Set for 1 ≤ i ≤ 2L
xi = (mi1 , mi2 , . . . , mi,2L , 0, 0, . . . ),
Inequalities Applicable To
Certain Partial Differential
Equations
and
X = lin{x1 , x2 , . . . , x2L }.
By (6.6), dim X ≤ L.
Next, we estimate distp (ek , X) for 1 ≤ k ≤ 2L.
Assume first p < ∞. Then
1
distpp (ek , X) ≤ kek − xk kpp
K
p
1
1
1
1
=
mk1 , . . . , mk,k−1 , 0, mk,k+1 , . . . , mk,2L , 0, 0, . . . K
K
K
K
p
≤
2L−1
X
i=1
2L
X 1
1
2L
2
≤
= p = p−2 .
p
p
K
K
K
K
i=1
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This gives distp (ek , X) ≤
1
2p
1−2/p
K
.
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Next, assume p = ∞, so that
1 dist∞ (ek , X) ≤ ek − xk K ∞
1
1
1
1
=
m
,
.
.
.
,
m
,
0,
m
,
.
.
.
,
m
,
0,
0,
.
.
.
k1
k,k−1
k,k+1
k,2L
K
K
K
K
∞
1
≤
K
This concludes the proof.
Inequalities Applicable To
Certain Partial Differential
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References
[1] D.E. EDMUNDS, W.D. EVANS AND D.J. HARRIS, Approximation numbers of certain Volterra integral operators, J. London Math. Soc., (2) 37
(1988), 471–489.
[2] D.E. EDMUNDS, W.D. EVANS AND D.J. HARRIS, Two–sided estimates
of the approximation numbers of certain Volterra integral operators. Studia
Math., 124(1) (1997), 59–80.
[3] D.E. EDMUNDS AND W.D. EVANS, Spectral Theory and Differential
Operators, Oxford Univ. Press, Oxford, 1987.
[4] D.E. EDMUNDS, P. GURKA AND L. PICK, Compactness of Hardy–
type integral operators in weighted Banach function spaces, Studia Math.,
109(1) (1994), 73–90.
[5] W.D. EVANS, D.J. HARRIS AND J. LANG, Two-sided estimates for the
approximation numbers of Hardy-type operators in L∞ and L1 , Studia
Math., 130(2) (1998), 171–192.
[6] M.A. LIFSHITS AND W. LINDE, Approximation and entropy numbers
of Volterra operators with application to Brownian motion, Mem. Amer.
Math. Soc., 157 (2002), 745.
[7] E. LOMAKINA AND V. STEPANOV, On asymptotic behavior of the Approximation numbers and estimates of Shatten - Von Neumann norms of
the Hardy-type integral operators, preprint
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A. Nekvinda
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[8] J. NEWMAN AND M. SOLOMYAK, Two–sided estimates of singular values for a class of integral operators on the semi–axis, Integral Equations
Operator Theory, 20 (1994), 335–349.
[9] B. OPIC AND A. KUFNER, Hardy–type Inequalities, Pitman Res. Notes
Math. Ser., 219, Longman Sci. & Tech., Harlow, 1990.
[10] I. SINGER, Best Approximation in Normed Linear Spaces by Elements of
Linear Subspace, Springer-Verlag, New York 1970.
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