Journal of Inequalities in Pure and
Applied Mathematics
REVERSE WEIGHTED LP –NORM INEQUALITIES IN CONVOLUTIONS
´ AND MASAHIRO YAMAMOTO
SABUROU SAITOH, VŨ KIM TUÂN
Department of Mathematics
Faculty of Engineering
Gunma University
Kiryu 376-8515, JAPAN.
EMail: ssaitoh@eg.gunma-u.ac.jp
Department of Mathematics and Computer Science
Faculty of Science
Kuwait University
P.O. Box 5969, Safat 13060, KUWAIT.
EMail: vu@sci.kuniv.edu.kw
Department of Mathematical Sciences
The University of Tokyo
3-8-1 Komaba
Tokyo 153-8914, JAPAN.
EMail: myama@ms.u-tokyo.ac.jp
volume 1, issue 1, article 7,
2000.
Received and acepted
14 December, 1999.
Communicated by: J.E. Pec̆arić
Abstract
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School of Communications and Informatics,Victoria University of Technology
ISSN (electronic): 1443-5756
018-99
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Abstract
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.
2000 Mathematics Subject Classification: 44A35, 26D20
Key words: Convolution, weighted Lp inequality, reverse Hölder inequality, inverse
problems, Green’s function, integral transform, stability.
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.
Reverse Weighted Lp –Norm
Inequalities in Convolutions
Saburou Saitoh, Vũ Kim Tuấn and
Masahiro Yamamoto
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Contents
1
2
3
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
A general reverse weighted Lp convolution inequality . . . . . . 6
Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.1
The first order differential equation . . . . . . . . . . . . . . . 9
3.2
Picard transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.3
Poisson integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.4
Heat equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
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1.
Introduction
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),
n
is fundamental. Note, however, that for the typical case of f, g ∈ L2 (R ),
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.
Proposition 1.1. ([7]). For two nonvanishing functions ρj ∈ L1 (R) (j = 1, 2)
the following Lp (p > 1) weighted convolution inequality
1
−1 p
(1.2)
((F1 ρ1 ) ∗ (F2 ρ2 )) (ρ1 ∗ ρ2 ) ≤ kF1 kLp (R,|ρ1 |) kF2 kLp (R,|ρ2 |)
p
holds for Fj ∈ Lp (R, |ρj |)
(1.3)
(j = 1, 2). Equality holds if and only if
αx
Fj (x) = Cj e ,
where α is a constant such that eαx ∈ Lp (R, |ρj |)
Reverse Weighted Lp –Norm
Inequalities in Convolutions
Saburou Saitoh, Vũ Kim Tuấn and
Masahiro Yamamoto
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Here
Z
∞
kF kLp (R;ρ) =
−∞
|F (x)|p ρ(x) dx
p1
.
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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
(1.4)
ρ2 (x) ≡ 1,
F2 (x) = G(x),
where G(x − ξ) is some Green’s function. Then the inequality (1.2) takes the
form
(1.5)
1− p1
k (F ρ) ∗ Gkp ≤ kρkp
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
(1.7)
0<m≤
f
≤M <∞
g
Reverse Weighted Lp –Norm
Inequalities in Convolutions
Saburou Saitoh, Vũ Kim Tuấn and
Masahiro Yamamoto
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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
q
t
1
1
1 −p
1 −q
p
q
1−t
.
(1 − t) 1 − t
Reverse Weighted Lp –Norm
Inequalities in Convolutions
Saburou Saitoh, Vũ Kim Tuấn and
Masahiro Yamamoto
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2.
A general reverse weighted Lp convolution inequality
Our main result is the following
Theorem 2.1. Let F1 and F2 be positive functions satisfying
(2.1)
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
(2.2)
−1
1
m1 m2
−1 p
kF1 kLp (R,ρ1 ) kF2 kLp (R,ρ2 ) .
((F1 ρ1 ) ∗ (F2 ρ2 )) (ρ1 ∗ ρ2 ) ≥ 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) =
Reverse Weighted Lp –Norm
Inequalities in Convolutions
Saburou Saitoh, Vũ Kim Tuấn and
Masahiro Yamamoto
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F1p (ξ)F2p (x
− ξ)ρ1 (ξ)ρ2 (x − ξ),
g(ξ) = ρ1 (ξ)ρ2 (x − ξ).
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Then condition (2.1) implies
f (ξ)
≤ M 1 M2 ,
m1 m2 ≤
g(ξ)
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ξ ∈ R.
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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,
(2.3)
Z
Reverse Weighted Lp –Norm
Inequalities in Convolutions
p Z
∞
1−p
∞
Saburou Saitoh, Vũ Kim Tuấn and
Masahiro Yamamoto
F1 (ξ)ρ1 (ξ)F2 (x − ξ)ρ2 (x − ξ) dξ
ρ1 (ξ)ρ2 (x − ξ)dξ
−∞
−∞
−p Z ∞
m1 m2
F1p (ξ)F2p (x − ξ)ρ1 (ξ)ρ2 (x − ξ)dξ.
≥ Ap,q
M1 M2
−∞
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Taking integration of both sides of (2.3) with respect to x from −∞ to ∞ we
obtain the inequality
(2.4)
Z ∞ Z
−∞
∞
p Z
∞
1−p
F1 (ξ)ρ1 (ξ)F2 (x − ξ)ρ2 (x − ξ) dξ
ρ1 (ξ)ρ2 (x − ξ)dξ
−∞
−∞
−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
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yields the inequality (2.2).
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Inequality (1.8) reverses the sign if 0 < p < 1. Hence, inequality (2.2)
reverses the sign if 0 < p < 1.
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
Z
d
Z
p
∞
F (ξ) ρ(ξ) G(x − ξ) dξ
(2.5)
c
dx
−∞
p−1 Z ∞
Z d−ξ
m o−p Z ∞
n
p
≥ Ap,q
ρ(ξ) dξ
F (ξ) ρ(ξ) dξ
Gp (x)dx,
M
−∞
−∞
c−ξ
Saburou Saitoh, Vũ Kim Tuấn and
Masahiro Yamamoto
valid if positive continuous functions ρ, F , and G satisfy
(2.6)
1
1
0 < m p ≤ F (ξ)G(x − ξ) ≤ M p ,
x ∈ [c, d],
Reverse Weighted Lp –Norm
Inequalities in Convolutions
ξ ∈ R.
Inequality (2.5) is especially important when G(x − ξ) is a Green’s function.
See examples in the next section.
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3.
3.1.
Examples
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
Reverse Weighted Lp –Norm
Inequalities in Convolutions
x
y(x) =
−λ(x−t)
F (t)e
dt.
0
Saburou Saitoh, Vũ Kim Tuấn and
Masahiro Yamamoto
So we shall consider the integral transform
Z x
f (x) =
F (t) ρ(t)e−λ(x−t) dt,
λ > 0.
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0
Take
G(x) =
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e−λx , x > 0
.
0,
x<0
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The condition (2.6) reads
1
Close
1
0 < m p ≤ F (t)e−λ(x−t) ≤ M p .
(3.1)
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It will be satisfied for 0 ≤ t ≤ x ≤ d < ∞, if we have
(3.2)
1
1
0 < m p eλd−λt ≤ F (t) ≤ M p ,
0<d<
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1
M
log .
pλ
m
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Notice that
Z
(
d−ξ
p
G (x) dx =
c−ξ
e−λpc −e−λpd λpξ
e ,
λp
1−eλpξ−λpd
,
λp
ξ < c,
c < ξ < d.
Thus the inequality (2.5) yields
1−p
dx
f (x)
ρ(t) dt
c
0
Z c
n
m o−p 1 −λpc
−λpd
(e
−e
)
F p (ξ)ρ(ξ)eλpξ dξ
≥ Ap,q
M
λp
0
Z d
p
−λpd λpξ
+
F (ξ)ρ(ξ)(1 − e
e )dξ .
Z
(3.3)
d
p
Z
x
c
Here we assume that ρ is a positive continuous function on [0, d], and F satisfies
(3.2).
3.2.
Note that 12 e−|x−t| is the Green’s function for the boundary value problem
00
Saburou Saitoh, Vũ Kim Tuấn and
Masahiro Yamamoto
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Picard transform
y − y = 0,
Reverse Weighted Lp –Norm
Inequalities in Convolutions
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 −∞
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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
1
1
(3.5) 0 < m p ea e|t| ≤ F (t) ≤ M p e−a e|t| ,
t ∈ R,
0<a<
1
M
log .
2p
m
Saburou Saitoh, Vũ Kim Tuấn and
Masahiro Yamamoto
We have
Z
d−t
Z
p
d−t
G (x) dx =
c−t
−p|x|
e
c−t
dx =





ept
e−pc − e−pd ,
p
pd
−pt
e
e − epc ,
p
1
2 − epc−pt − ept−pd ,
p
t < c,
t > d,
c < t < d.
Thus, for −a ≤ c, d ≤ a the inequality (2.5) yields
p−1
m o−p Z ∞
1 n
(3.6)
f (x) dx ≥ p Ap,q
ρ(t)dt
2p
M
c
−∞
Z
Z
c p
∞ p
−pc
−pd
pt
pd
pc
e −e
F (t)ρ(t)e dt + e − e
F (t)ρ(t)e−pt dt
−∞
d
Z d
p
pc−pt
pt−pd
+
F (t)ρ(t) 2 − e
−e
dt ,
Z
Reverse Weighted Lp –Norm
Inequalities in Convolutions
d
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c
if ρ is positive continuous, and F satisfies (3.5).
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3.3.
Poisson integrals
Consider the Poisson integral
1
u(x, y) =
π
(3.7)
Z
∞
F (ξ)ρ(ξ)
−∞
Take
G(x) =
y
dξ.
(x − ξ)2 + y 2
y
.
x2 + y 2
Reverse Weighted Lp –Norm
Inequalities in Convolutions
Let
ξ ∈ [a, b],
x ∈ [c, d].
Saburou Saitoh, Vũ Kim Tuấn and
Masahiro Yamamoto
Denote
α = max{|a − c|, |a − d|, |b − c|, |b − d|}.
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We have
y
y
1
≤
≤ .
2
2
2
+y
(x − ξ) + y
y
α2
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Thus,
Z
d−ξ
p
Z
d−ξ
G (x) dx =
c−ξ
c−ξ
y
2
x + y2
p
dx ≥ (d − c)
Hence, for a function F satisfying
1
α2 + y 2 p1
m ≤ F (ξ) ≤ y M p ,
y
y
2
α + y2
p
.
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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
(3.9)
v(x, y) =
F (ξ)ρ(ξ)
dξ.
π −∞
(x − ξ)2 + y 2
Take
G(x) =
x2
Reverse Weighted Lp –Norm
Inequalities in Convolutions
Saburou Saitoh, Vũ Kim Tuấn and
Masahiro Yamamoto
x
.
+ y2
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For
ξ ∈ [a, b],
we have
Thus,
Z d−ξ
c−ξ
x ∈ [c, d],
(b < c),
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x−ξ
d−a
c−b
≤
≤
.
2
2
2
2
(d − a) + y
(x − ξ) + y
(c − b)2 + y 2
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Gp (x) dx =
Z
d−ξ c−ξ
x2
x
+ y2
p
dx ≥ (d − c)
c−b
(d − a)2 + y 2
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
p
.
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and for a positive continuous function ρ on [a, b] we obtain
Z
(3.10)
c
d
(d − c)
v (x, y) dx ≥
πp
p
p n
m o−p
c−b
A
p,q
(d − a)2 + y 2
M
Z b
p−1 Z b
ρ(ξ) dξ
F p (ξ)ρ(ξ)dξ.
a
3.4.
a
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
Reverse Weighted Lp –Norm
Inequalities in Convolutions
Saburou Saitoh, Vũ Kim Tuấn and
Masahiro Yamamoto
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ut = ∆u
on R+ × R,
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subject to the initial condition
u(x, 0) = F (x)ρ(x) on
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R.
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Take
2
− x4t
G(x) = e
.
Let
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s
x ∈ [−a, a],
ξ ∈ [−b, b],
a+b≤
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4t
M
log .
p
m
J. Ineq. Pure and Appl. Math. 1(1) Art. 7, 2000
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From
1 ≤ exp
we have
(x − ξ)2
4t
≤ exp
(a + b)2
4t
(x − ξ)2
0 < m ≤ F (ξ) exp −
4t
1
p
,
1
≤ M p,
if
1
p
m exp
(3.12)
(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
√
√
erf
− erf
,
e 4t dx =
p
2 t
2 t
c−ξ
Reverse Weighted Lp –Norm
Inequalities in Convolutions
Saburou Saitoh, Vũ Kim Tuấn and
Masahiro Yamamoto
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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
Z
(3.13)
c
d
p−1
n
m o−p Z b
1
p
u(x, t) dx ≥ p
ρ(ξ)dξ
√ Ap,q
2 (πt)(p−1)/2 p
M
−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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References
[1] M. CWICKEL AND R. KERMAN, On a convolution inequality of Saitoh,
Proc. Amer. Math. Soc., 124 (1996), 773-777.
[2] L. XIAO-HUA, On the inverse of Hölder inequality, Math. Practice and
Theory, 1 (1990), 84-88.
[3] D.S. MITRINOVIĆ, J.E. PEC̆ARIĆ AND A.M. FINK, Classical and New
Inequalities in Analysis, Kluwer Academic Publishers, Dordrecht, 1993.
[4] S. SAITOH, A fundamental inequality in the convolution of L2 functions
on the half line, Proc. Amer. Math. Soc., 91 (1984), 285-286.
[5] S. SAITOH, Inequalities in the most simple Sobolev space and convolution
of L2 functions with weights, Proc. Amer. Math. Soc., 118 (1993), 515520.
[6] S. SAITOH, Various operators in Hilbert spaces introduced by transforms,
Intern. J. Appl. Math., 1 (1999), 111-126.
[7] S. SAITOH, Weighted Lp -norm inequalities in convolutions, Handbook
on Classical Inequalities, Kluwer Academic Publishers, Dordrecht (to appear).
[8] H.M. SRIVASTAVA AND R.G. BUSCHMAN, Theory and Applications of
Convolution Integral Equations, Kluwer Academic Publishers, Dordrecht,
1992.
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Inequalities in Convolutions
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