Journal of Inequalities in Pure and
Applied Mathematics
http://jipam.vu.edu.au/
Volume 1, Issue 1, Article 7, 2000
REVERSE WEIGHTED Lp –NORM INEQUALITIES IN CONVOLUTIONS
´ AND MASAHIRO YAMAMOTO
SABUROU SAITOH, VŨ KIM TUÂN,
D EPARTMENT OF M ATHEMATICS , FACULTY OF E NGINEERING , G UNMA U NIVERSITY, K IRYU 376-8515,
JAPAN
ssaitoh@eg.gunma-u.ac.jp
D EPARTMENT OF M ATHEMATICS AND C OMPUTER S CIENCE , FACULTY OF S CIENCE , K UWAIT U NIVERSITY,
P.O. B OX 5969, S AFAT 13060, K UWAIT
vu@sci.kuniv.edu.kw
D EPARTMENT OF M ATHEMATICAL S CIENCES , T HE U NIVERSITY OF T OKYO , 3-8-1 KOMABA , T OKYO
153-8914, JAPAN
myama@ms.u-tokyo.ac.jp
Received 14 December, 1999; accepted 14 December, 1999
Communicated by J.E. Pec̆arić
A BSTRACT. Various weighted Lp (p > 1)–norm inequalities in convolutions were derived by
using Hölder’s inequality. Therefore, by using reverse Hölder inequalities one can obtain reverse
weighted Lp –norm inequalities. These inequalities are important in studying stability of some
inverse problems.
Key words and phrases: Convolution, weighted Lp inequality, reverse Hölder inequality, inverse problems, Green’s function,
integral transform, stability.
2000 Mathematics Subject Classification. 44A35, 26D20.
1. I NTRODUCTION
For the Fourier convolution
Z
∞
(f ∗ g)(x) =
f (x − ξ)g(ξ) dξ,
−∞
the Young’s inequality
(1.1) kf ∗ gkr ≤ kf kp kgkq ,
f ∈ Lp (R), g ∈ Lq (R), r−1 = p−1 + q −1 − 1
(p, q, r > 0),
is fundamental. Note, however, that for the typical case of f, g ∈ L2 (Rn ), the inequality (1.1)
does not hold. In a series of papers [4, 5, 6, 7] (see also [1]) the first author obtained the
following weighted Lp (p > 1) inequality for convolution.
ISSN (electronic): 1443-5756
c 2000 Victoria University. All rights reserved.
The authors wish to express their sincere thanks to Professor Josip Pec̆arić for his valuable information on the reverse Hölder inequality.
The work of the second named author was supported by Kuwait University Research Administration under project SM 187.
018-99
2
S ABUROU S AITOH
AND
´ AND M ASAHIRO YAMAMOTO
V Ũ K IM T U ÂN
Proposition 1.1. ([7]). For two nonvanishing functions ρj ∈ L1 (R) (j = 1, 2) the following
Lp (p > 1) weighted convolution inequality
1
−1 (1.2)
((F1 ρ1 ) ∗ (F2 ρ2 )) (ρ1 ∗ ρ2 ) p ≤ kF1 kLp (R,|ρ1 |) kF2 kLp (R,|ρ2 |)
p
holds for Fj ∈ Lp (R, |ρj |)
(j = 1, 2). Equality holds if and only if
Fj (x) = Cj eαx ,
(1.3)
where α is a constant such that eαx ∈ Lp (R, |ρj |)
(j = 1, 2).
Here
Z
∞
p
kF kLp (R;ρ) =
p1
|F (x)| ρ(x) dx
.
−∞
Unlike the Young’s inequality, the inequality (1.2) holds also in case p = 2.
In many cases of interest, the convolution is given in the form
ρ2 (x) ≡ 1,
(1.4)
F2 (x) = G(x),
where G(x − ξ) is some Green’s function. Then the inequality (1.2) takes the form
1− p1
k (F ρ) ∗ Gkp ≤ kρkp
(1.5)
kGkp kF kLp (R,|ρ|) ,
where ρ, F , and G are such that the right hand side of (1.5) is finite.
The inequality (1.5) enables us to estimate the output function
Z ∞
(1.6)
F (ξ)ρ(ξ)G(x − ξ) dξ
−∞
in terms of the input function F . In this paper we are interested in the reverse type inequality for
(1.5), namely, we wish to estimate the input function F by means of the output (1.6). This kind
of estimate is important in inverse problems. Our estimate is based on the following version of
the reverse Hölder inequality
Proposition 1.2. ([2], see also [3], pages 125-126). For two positive functions f and g satisfying
0<m≤
(1.7)
f
≤M <∞
g
on the set X, and for p, q > 0, p−1 + q −1 = 1,
p1 Z
Z
1q
mZ
1
1
gdµ
≤ Ap,q
f p g q dµ,
M
X
X
f dµ
(1.8)
X
if the right hand side integral converges. Here
Ap,q (t) = p
1
− p1 − 1q − pq
J. Ineq. Pure and Appl. Math., 1(1) Art. 7, 2000
q
t
1
1
1 −p
1 −q
p
q
(1 − t) 1 − t
1−t
.
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R EVERSE W EIGHTED Lp –N ORM I NEQUALITIES IN C ONVOLUTIONS
2. A GENERAL REVERSE
WEIGHTED
Lp
3
CONVOLUTION INEQUALITY
Our main result is the following
Theorem 2.1. Let F1 and F2 be positive functions satisfying
1
(2.1)
1
1
0 < m1p ≤ F1 (x) ≤ M1p < ∞,
1
0 < m2p ≤ F2 (x) ≤ M2p < ∞,
p > 1,
x ∈ R.
Then for any positive functions ρ1 and ρ2 we have the reverse Lp –weighted convolution inequality
−1
1
m
m
1
2
−1
kF1 kLp (R,ρ1 ) kF2 kLp (R,ρ2 ) .
(2.2) ((F1 ρ1 ) ∗ (F2 ρ2 )) (ρ1 ∗ ρ2 ) p ≥ Ap,q
M1 M2
p
Inequality (2.2) and others should be understood in the sense that if the left hand side is finite,
then so is the right hand side, and in this case the inequality holds.
Proof. Let
f (ξ) = F1p (ξ)F2p (x − ξ)ρ1 (ξ)ρ2 (x − ξ),
g(ξ) = ρ1 (ξ)ρ2 (x − ξ).
Then condition (2.1) implies
m1 m2 ≤
f (ξ)
≤ M1 M2 ,
g(ξ)
ξ ∈ R.
Hence, one can apply the reverse Hölder inequality (1.8) for f and g to get
Z ∞
m1 m2
Ap,q
F1 (ξ)ρ1 (ξ)F2 (x − ξ)ρ2 (x − ξ) dξ
M1 M2
−∞
Z ∞
p1 Z ∞
1− p1
p
p
≥
F1 (ξ)F2 (x − ξ)ρ1 (ξ)ρ2 (x − ξ)dξ
ρ1 (ξ)ρ2 (x − ξ)dξ
.
−∞
−∞
Hence,
Z
p Z
∞
F1 (ξ)ρ1 (ξ)F2 (x − ξ)ρ2 (x − ξ) dξ
(2.3)
−∞
m1 m2
≥ Ap,q
M1 M2
1−p
∞
ρ1 (ξ)ρ2 (x − ξ)dξ
−∞
−p Z ∞
F1p (ξ)F2p (x − ξ)ρ1 (ξ)ρ2 (x − ξ)dξ.
−∞
Taking integration of both sides of (2.3) with respect to x from −∞ to ∞ we obtain the inequality
Z
∞
Z
p Z
∞
F1 (ξ)ρ1 (ξ)F2 (x − ξ)ρ2 (x − ξ) dξ
(2.4)
−∞
−∞
1−p
∞
ρ1 (ξ)ρ2 (x − ξ)dξ
dx
−∞
−p Z ∞
Z ∞
m1 m2
p
≥ Ap,q
F1 (ξ)ρ1 (ξ)dξ
F2p (x)ρ2 (x)dx.
M1 M2
−∞
−∞
Raising both sides of the inequality (2.4) to power
1
p
yields the inequality (2.2).
Inequality (1.8) reverses the sign if 0 < p < 1. Hence, inequality (2.2) reverses the sign if
0 < p < 1.
J. Ineq. Pure and Appl. Math., 1(1) Art. 7, 2000
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4
S ABUROU S AITOH
AND
´ AND M ASAHIRO YAMAMOTO
V Ũ K IM T U ÂN
In formula (2.3) replacing ρ2 by 1, and F2 (x − ξ) by G(x − ξ), and taking integration with
respect to x from c to d we arrive at the following inequality
p
Z d Z ∞
(2.5)
F (ξ) ρ(ξ) G(x − ξ) dξ dx
c
−∞
p−1 Z ∞
Z d−ξ
n
m o−p Z ∞
p
ρ(ξ) dξ
F (ξ) ρ(ξ) dξ
Gp (x)dx,
≥ Ap,q
M
−∞
−∞
c−ξ
valid if positive continuous functions ρ, F , and G satisfy
1
1
0 < m p ≤ F (ξ)G(x − ξ) ≤ M p ,
(2.6)
x ∈ [c, d],
ξ ∈ R.
Inequality (2.5) is especially important when G(x − ξ) is a Green’s function. See examples in
the next section.
3. E XAMPLES
3.1. The first order differential equation. The solution y(x) of the first order differential
equation
y 0 (x) + λ y(x) = F (x), y(0) = 0,
is represented in the form
Z
x
F (t)e−λ(x−t) dt.
y(x) =
0
So we shall consider the integral transform
Z x
f (x) =
F (t) ρ(t)e−λ(x−t) dt,
λ > 0.
0
Take
G(x) =
e−λx , x > 0
.
0,
x<0
The condition (2.6) reads
1
1
0 < m p ≤ F (t)e−λ(x−t) ≤ M p .
(3.1)
It will be satisfied for 0 ≤ t ≤ x ≤ d < ∞, if we have
1
1
0 < m p eλd−λt ≤ F (t) ≤ M p ,
(3.2)
Notice that
Z
(
d−ξ
p
G (x) dx =
c−ξ
0<d<
e−λpc −e−λpd λpξ
e ,
λp
1−eλpξ−λpd
,
λp
1
M
log .
pλ
m
ξ<c
.
c<ξ<d
Thus the inequality (2.5) yields
Z x
1−p
Z d
p
(3.3)
f (x)
ρ(t) dt
dx
c
0
Z c
n
m o−p 1 −λpc
−λpd
≥ Ap,q
(e
−e
)
F p (ξ)ρ(ξ)eλpξ dξ
M
λp
0
Z d
p
−λpd λpξ
+
F (ξ)ρ(ξ)(1 − e
e )dξ .
c
Here we assume that ρ is a positive continuous function on [0, d], and F satisfies (3.2).
J. Ineq. Pure and Appl. Math., 1(1) Art. 7, 2000
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R EVERSE W EIGHTED Lp –N ORM I NEQUALITIES IN C ONVOLUTIONS
5
3.2. Picard transform. Note that 12 e−|x−t| is the Green’s function for the boundary value problem
00
y − y = 0,
lim y(x) = 0.
x→±∞
So, we shall consider the Picard transform
Z
1 ∞
f (x) =
F (t)ρ(t)e−|x−t| dt.
2 −∞
Take G(x) = e−|x| . Since
e−a e|t| ≤ e|x−t| ≤ ea e|t| ,
|x| ≤ a,
we see that the condition (2.6)
1
1
0 < m p ≤ F (t)e−|x−t| ≤ M p ,
(3.4)
holds if
(3.5)
1
1
0 < m p ea e|t| ≤ F (t) ≤ M p e−a e|t| ,
t ∈ R,
0<a<
M
1
log .
2p
m
We have
Z
d−t
p
Z
d−t
G (x) dx =
c−t
−p|x|
e
dx =
c−t





ept
e−pc − e−pd
p
pd
−pt
e
e − epc ,
p
2−epc−pt −ept−pd
,
p
Thus, for −a ≤ c, d ≤ a the inequality (2.5) yields
p−1
Z d
m o−p Z ∞
1 n
p
(3.6)
f (x) dx ≥ p Ap,q
ρ(t)dt
2p
M
c
−∞
Z
Z
c p
−pc
−pd
pt
pd
pc
e −e
F (t)ρ(t)e dt + e − e
−∞
, t<c
t>d
.
c<t<d
∞
F p (t)ρ(t)e−pt dt
d
Z
d
p
pc−pt
F (t)ρ(t) 2 − e
+
pt−pd
−e
dt ,
c
if ρ is positive continuous, and F satisfies (3.5).
3.3. Poisson integrals. Consider the Poisson integral
Z
1 ∞
y
(3.7)
u(x, y) =
F (ξ)ρ(ξ)
dξ.
π −∞
(x − ξ)2 + y 2
Take
y
G(x) = 2
.
x + y2
Let
ξ ∈ [a, b], x ∈ [c, d].
Denote
α = max{|a − c|, |a − d|, |b − c|, |b − d|}.
We have
y
y
1
≤
≤
.
α2 + y 2
(x − ξ)2 + y 2
y
Thus,
p
p
Z d−ξ
Z d−ξ y
y
p
G (x) dx =
dx ≥ (d − c)
.
x2 + y 2
α2 + y 2
c−ξ
c−ξ
J. Ineq. Pure and Appl. Math., 1(1) Art. 7, 2000
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6
S ABUROU S AITOH
AND
´ AND M ASAHIRO YAMAMOTO
V Ũ K IM T U ÂN
Hence, for a function F satisfying
1
α2 + y 2 p1
m ≤ F (ξ) ≤ y M p ,
y
and for a positive continuous function ρ on [a, b] we obtain
p n
Z d
m o−p
(d − c)
y
p
(3.8)
u (x, y) dx ≥
Ap,q
πp
α2 + y 2
M
c
Z b
p−1 Z b
ρ(ξ) dξ
F p (ξ)ρ(ξ)dξ.
a
a
Consider now the conjugate Poisson integral
Z
1 ∞
x−ξ
dξ.
(3.9)
v(x, y) =
F (ξ)ρ(ξ)
π −∞
(x − ξ)2 + y 2
Take
G(x) =
x2
x
.
+ y2
For
ξ ∈ [a, b],
we have
x ∈ [c, d],
(b < c),
c−b
x−ξ
d−a
≤
≤
.
(d − a)2 + y 2
(x − ξ)2 + y 2
(c − b)2 + y 2
Thus,
Z
d−ξ
p
Z
d−ξ
G (x) dx =
c−ξ
c−ξ
x
2
x + y2
p
dx ≥ (d − c)
c−b
(d − a)2 + y 2
p
.
Hence, for a function F satisfying
1
(d − a)2 + y 2 p1
(c − b)2 + y 2
m ≤ F (ξ) ≤
M p,
c−b
d−a
and for a positive continuous function ρ on [a, b] we obtain
p n
Z d
m o−p
(d − c)
c−b
p
(3.10)
v (x, y) dx ≥
A
p,q
πp
(d − a)2 + y 2
M
c
Z b
p−1 Z b
ρ(ξ) dξ
F p (ξ)ρ(ξ)dξ.
a
a
3.4. Heat equation. We consider the Weierstrass transform
Z ∞
1
(x − ξ)2
(3.11)
u(x, t) = √
F (ξ)ρ(ξ) exp −
dξ,
4t
4πt −∞
which gives the formal solution u(x, t) of the heat equation
ut = ∆u
on R+ × R,
subject to the initial condition
u(x, 0) = F (x)ρ(x)
Take
on
R.
x2
G(x) = e− 4t .
J. Ineq. Pure and Appl. Math., 1(1) Art. 7, 2000
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R EVERSE W EIGHTED Lp –N ORM I NEQUALITIES IN C ONVOLUTIONS
Let
s
x ∈ [−a, a],
From
ξ ∈ [−b, b],
4t
M
log .
p
m
(a + b)2
1 ≤ exp
≤ exp
,
4t
2
1
1
(x
−
ξ)
0 < m p ≤ F (ξ) exp −
≤ M p,
4t
we have
(x − ξ)2
4t
a+b≤
7
if
(3.12)
1
p
m exp
(a + b)2
4t
1
≤ F (ξ) ≤ M p ,
ξ ∈ [−b, b].
It is easy to see that
r √
√
Z d−ξ
2
p (d − ξ)
p (c − ξ)
πt
− px
√
√
e 4t dx =
erf
− erf
,
p
2 t
2 t
c−ξ
where
Z x
2
2
erf (x) = √
e−t dt
π 0
is the error function. Therefore, for −a ≤ c < d ≤ a, the inequality (2.5) yields
p−1
Z d
n
m o−p Z b
1
p
(3.13)
u(x, t) dx ≥ p
ρ(ξ)dξ
√ Ap,q
2 (πt)(p−1)/2 p
M
c
−b
√
√
Z b
p (d − ξ)
p (c − ξ)
p
√
√
F (ξ)ρ(ξ) erf
− erf
dξ,
2 t
2 t
−b
where ρ is a positive continuous function on [−b, b], and F satisfies (3.12).
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Kluwer Academic Publishers, Dordrecht, 1993.
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J. Ineq. Pure and Appl. Math., 1(1) Art. 7, 2000
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