Journal of Inequalities in Pure and
Applied Mathematics
REVERSE CONVOLUTION INEQUALITIES AND APPLICATIONS
TO INVERSE HEAT SOURCE PROBLEMS
´ AND MASAHIRO YAMAMOTO
SABUROU SAITOH, VŨ KIM TUÂN
Department of Mathematics
Faculty of Engineering
Gunma University
Kiryu 376-8515, JAPAN.
EMail: ssaitoh@math.sci.gunma-u.ac.jp
Department of Mathematics and Computer Science
Faculty of Science
Kuwait University
P.O. Box 5969, Safat 13060, KUWAIT.
EMail: vu@mcc.sci.kuniv.edu.kw
Graduate School of Mathematical Sciences
The University of Tokyo
3-8-1 Komaba
Tokyo 153-8914, JAPAN.
EMail: myama@ms.u-tokyo.ac.jp
volume 3, issue 5, article 80,
2002.
Received 2 April, 2002;
accepted 30 October, 2002.
Communicated by: S.S. Dragomir
Abstract
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2000
School of Communications and Informatics,Victoria University of Technology
ISSN (electronic): 1443-5756
029-02
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Abstract
We introduce reverse convolution inequalities obtained recently and at the same
time, we give new type reverse convolution inequalities and their important applications to inverse source problems. We consider the inverse problem of determining f(t), 0 < t < T , in the heat source of the heat equation ∂t u(x, t) =
∆u(x, t) + f(t)ϕ(x), x ∈ Rn , t > 0 from the observation u(x0 , t), 0 < t < T , at a
remote point x0 away from the support of ϕ. Under an a priori assumption that
f changes the signs at most N-times, we give a conditional stability of Hölder
type, as an example of applications.
2000 Mathematics Subject Classification: Primary 44A35; Secondary 26D20.
Key words: Convolution, Heat source, Weighted convolution inequalities, Young’s inequality, Hölder’s inequality, Reverse Hölder’s inequality, Green’s function, Stability in inverse problems, Volterra’s equation, Conditional stability of Hölder type, Analytic semigroup, Interpolation inequality, Sobolev
inequality.
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
Remarks for Reverse Hölder Inequalities . . . . . . . . . . . . . . . . . 7
3
New Reverse Convolution Inequalities . . . . . . . . . . . . . . . . . . . 9
4
Applications to Inverse Source Heat Problems and Results . . 11
5
Proof of Theorem 4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
References
Reverse Convolution
Inequalities and Applications to
Inverse Heat Source Problems
Saburou Saitoh, Vũ Kim Tuấn and
Masahiro Yamamoto
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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
Reverse Convolution
Inequalities and Applications to
Inverse Heat Source Problems
(p, q, r > 0),
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 [12] – [16] (see also [5]) we
obtained the following weighted Lp (p > 1) norm inequality for convolution
Proposition 1.1. ([15]). For two nonvanishing functions ρj ∈ L1 (R) (j = 1, 2),
the 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 here if and only if
(1.3)
kF kLp (R,|ρ|) =
p
|F (x)| |ρ(x)| dx
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where α is a constant such that eαx ∈ Lp (R, |ρj |) (j = 1, 2). Here
∞
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Fj (x) = Cj eαx ,
Z
Saburou Saitoh, Vũ Kim Tuấn and
Masahiro Yamamoto
p1
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.
−∞
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Unlike the Young’s inequality, inequality (1.2) holds also in case p = 2.
Note that the proof of Proposition 1.1 is direct and fairly elementary. Indeed,
we use only Hölder’s inequality and Fubini’s theorem for exchanging the orders
of integrals for the proof. So, for various type convolutions, we can also obtain
similar type convolution inequalities, see [17] for various convolutions.
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 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.
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 the related differential equation. We are also
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 estimates is
important in inverse problems. One estimate is obtained by using the following
famous reverse Hölder inequality
Proposition 1.2. ([18], see also [10, p. 125–126]). For two positive functions
f and g satisfying
(1.7)
f
0<m≤ ≤M <∞
g
Reverse Convolution
Inequalities and Applications to
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on the set X, and for p, q > 1, p−1 + q −1 = 1,
p1 Z
Z
f dµ
(1.8)
X
1q
mZ
1
1
gdµ
≤ Ap,q
f p g q dµ,
M
X
X
if the right hand side integral converges. Here
1
Ap,q (t) = p
− p1 − 1q
q
t− pq (1 − t)
p1 1q .
1
1
1 − tq
1 − tp
Reverse Convolution
Inequalities and Applications to
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Then, by using Proposition 1.2 we obtain, as in the proof of Proposition 1.1,
the following
Proposition 1.3. ([16]). Let F1 and F2 be positive functions satisfying
(1.9)
1
p
1
p
1
p
1
p
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0 < m1 ≤ F1 (x) ≤ M1 < ∞,
0 < m2 ≤ F2 (x) ≤ M2 < ∞,
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p > 1,
x ∈ R.
Then for any positive continuous functions ρ1 and ρ2 , we have the reverse Lp –
weighted convolution inequality
(1.10)
Saburou Saitoh, Vũ Kim Tuấn and
Masahiro Yamamoto
−1
m1 m2
kF1 kLp (R,ρ1 ) kF2 kLp (R,ρ2 )
Ap,q
M1 M2
1
−1 p
≤ ((F1 ρ1 ) ∗ (F2 ρ2 )) (ρ1 ∗ ρ2 ) .
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Inequality (1.10) should be understood in the sense that if the right hand side
is finite, then so is the left hand side, and in this case the inequality holds.
In formula (1.10) replacing ρ2 by 1, and F2 (x − ξ) by G(x − ξ), and integrating with respect to x from c to d we arrive at the following inequality
(1.11)
p−1Z ∞
Z d−ξ
n
m o−pZ ∞
p
Ap,q
ρ(ξ) dξ
F (ξ) ρ(ξ) dξ
Gp (x)dx
M
−∞
−∞
c−ξ
p
Z dZ ∞
≤
F (ξ) ρ(ξ) G(x − ξ) dξ dx,
c
−∞
Saburou Saitoh, Vũ Kim Tuấn and
Masahiro Yamamoto
if positive continuous functions ρ, F , and G satisfy
(1.12)
1
1
0 < m p ≤ F (ξ)G(x − ξ) ≤ M p ,
Reverse Convolution
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x ∈ [c, d],
ξ ∈ R.
Inequality (1.11) is especially important when G(x − ξ) is a Green’s function.
We gave various concrete examples in [16] from the viewpoint of stability in
inverse problems.
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2.
Remarks for Reverse Hölder Inequalities
In connection with Proposition 1.2 which gives Proposition 1.3, Izumino and
Tominaga [8] consider the upper bound of
X
apk
p1 X
bqk
1q
−λ
X
ak b k
for λ > 0, for p, q > 1 satisfying p1 + 1q = 1 and for positive numbers {ak }nk=1
and {bk }nk=1 , in detail. In their different approach, they showed that the constant
Ap,q (t) in Proposition 1.2 is best possible in a sense. Note that the proof of
Proposition 1.2 is quite involved. In connection with Proposition 1.2 we note
that the following version whose proof is surprisingly simple
Theorem 2.1. In Proposition 1.2, replacing f and g by f p and g q , respectively,
we obtain the reverse Hölder type inequality
1
q
Z
m − pq1 Z
q
f dµ
g dµ
≤
f gdµ.
M
X
X
X
Z
(2.1)
Proof. Since
1
p
fp
gq
p
1
p
≤ M , g ≥ M − q f q , therefore
Reverse Convolution
Inequalities and Applications to
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Saburou Saitoh, Vũ Kim Tuấn and
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1
p
1
f g ≥ M − q f 1+ q = M − q f p
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and so,
Z
(2.2)
Close
p1
Z
p1
1
f dµ
≤ M pq
f gdµ .
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On the other hand, since m ≤
Z
fp
,
gq
Z
f gdµ ≥
and so,
1
1
p
m g
1+ pq
1q
Z
f gdµ
q
f ≥ m p g p , hence
≥m
dµ = m
1
pq
Z
1
p
Z
g q dµ,
1q
g dµ .
q
Combining with (2.2), we have the desired inequality
Z
p1
1q
p1 Z
1q
Z
Z
−1
1
q
g dµ
≤ M pq
f gdµ m pq
f gdµ
f dµ
Z
m −1
pq
f gdµ.
=
M
p
Reverse Convolution
Inequalities and Applications to
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3.
New Reverse Convolution Inequalities
In reverse convolution inequality (1.10), similar type inequalities for m1 =
m2 = 0 are also important as we see from our example in Section 4. For
these, we obtain a new reverse convolution inequality.
Theorem 3.1. Let p ≥ 1, δ > 0, 0 ≤ α < T , and f, g ∈ L∞ (0, T ) satisfy
0 ≤ f, g ≤ M < ∞,
(3.1)
0 < t < T.
Then
(3.2)
kf kLp (α,T ) kgkLp (0,δ) ≤ M
2p−2
p
Z
T +δ
Z
α
t
p1
f (s)g(t − s)ds dt .
α
Reverse Convolution
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Saburou Saitoh, Vũ Kim Tuấn and
Masahiro Yamamoto
In particular, for
Z
(f ∗ g)(t) =
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t
f (t − s)g(s)ds,
0<t<T
0
and for α = 0, we have
kf kLp (0,T ) kgkLp (0,δ) ≤ M
2p−2
p
1
kf ∗ gkLp 1 (0,T +δ) .
Proof. Since 0 ≤ f, g ≤ M for 0 ≤ t ≤ T , we have
Z t
Z t
p
p
(3.3)
f (s) g(t − s) ds =
f (s)p−1 g(t − s)p−1 f (s)g(t − s)ds
α
α
Z t
2p−2
≤M
f (s)g(t − s)ds.
α
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Hence
Z T +δ Z
α
t
Z
2p−2
f (s) g(t − s) ds dt ≤ M
p
T +δ
p
α
Z
t
f (s)g(t − s)ds dt.
α
α
On the other hand, we have
Z T +δ Z t
Z T +δ Z T +δ
p
p
p
f (s) g(t − s) ds dt =
g(t − s) dt f (s)p ds
α
α
α
s
Z T +δ Z T +δ−s
p
=
g(η) dη f (s)p ds
α
0
Z T Z T +δ−s
p
≥
g(η) dη f (s)p ds
α
0
Z T Z δ
p
≥
g(η) dη f (s)p ds
α
0
= kf kpLp (α,T ) kgkpLp (0,δ) .
Thus the proof of Theorem 3.1 is complete.
Reverse Convolution
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4.
Applications to Inverse Source Heat Problems
and Results
We consider the heat equation with a heat source:
(4.1)
(4.2)
∂t u(x, t) = ∆u(x, t) + f (t)ϕ(x),
u(x, 0) = 0,
x ∈ Rn , t > 0
x ∈ Rn .
We assume that ϕ is a given function and satisfies

ϕ ≥ 0, 6≡ 0
in Rn ,





ϕ has compact support, ϕ ∈ C ∞ (Rn ), if n ≥ 4 and
(4.3)





ϕ ∈ L2 (Rn ),
if n ≤ 3.
Our problem is to derive a conditional stability in the determination of f (t),
0 < t < T , from the observation
Reverse Convolution
Inequalities and Applications to
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(4.4)
u(x0 , t),
0 < t < T,
where x0 6∈ supp ϕ.
We are interested only in the case of x0 6∈ supp ϕ, because in the case where
x0 is in the interior of supp ϕ, the problem can be reduced to a Volterra integral
equation of the second kind by differentiation in t formula (4.8) stated below.
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Moreover x0 6∈ supp ϕ means that our observation (4.4) is done far from the set
where the actual process is occuring, and the design of the observation point is
easy.
Let
(4.5)
K(x, t) =
|x|2
1
− 4t
√
e
,
(2 πt)n
x ∈ Rn , t > 0.
Then the solution u to (4.1) and (4.2) is represented by
Z tZ
(4.6)
u(x, t) =
K(x − y, t − s)f (s)ϕ(y)dyds,
0
x ∈ Rn , t > 0
Rn
(e.g., Friedman [6]). Therefore, setting
Z
(4.7)
µx0 (t) =
K(x0 − y, t)ϕ(y)dy,
Reverse Convolution
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t > 0,
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Rn
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we have
Z
(4.8)
u(x0 , t) ≡ hx0 (t) =
t
µx0 (t − s)f (s)ds,
0 < t < T,
0
which is a Volterra integral equation of the first kind with respect to f . Since
dk µx0
lim
(t) = (∆k ϕ)(x0 ) = 0,
t↓0 dtk
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k ∈ N ∪ {0}
by x0 6∈ supp ϕ (e.g., [6]), the equation (4.8) cannot be reduced to a Volterra
equation of the second kind by differentiating in t. Hence, even though, for any
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m ∈ N, we take the C m -norms for data h, the equation (4.8) is ill-posed, and
we cannot expect a better stability such as of Hölder type under suitable a priori
boundedness.
In Cannon and Esteva [3], an estimate of logarithmic type is proved: let
n = 1 and ϕ = ϕ(x) be the characteristic function of an interval (a, b) ⊂ R.
Set
(
)
2 d
f
df
(4.9) VM = f ∈ C 2 [0, ∞); f (0) = 0, ,
≤M .
dt dt2 C[0,∞)
C[0,∞)
Let x0 6∈ (a, b). Then, for T > 0, there exists a constant C = C(M, a, b, x0 ) >
0 such that
(4.10)
|f (t)| ≤
C
,
| log ku(x0 , ·)kL2 (0,∞) |2
Reverse Convolution
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0 ≤ t ≤ T,
for all f ∈ VM . The stability rate is logarithmic and worse than any rate of
Hölder type: ku(x0 , ·)kαL2 (0,∞) for any α > 0. For (4.10), the condition f ∈ VM
prescribes a priori information and (4.10) is called conditional stability within
the admissible set VM . The rate of conditional stability heavily depends on
the choice of admissible sets and an observation point x0 . As for other inverse
problems for the heat equation, we can refer to Cannon [2], Cannon and Esteva
[4], Isakov [7] and the references therein.
We arbitrarily fix M > 0 and N ∈ N. Let
(4.11) U = {f ∈ C[0, T ]; kf kC[0,T ] ≤ M,
f changes the signs at most N -times}.
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We take U as an admissible set of unknowns f . Then, within U, we can show
an improved conditional stability of Hölder type:
Theorem 4.1. Let ϕ satisfy (4.3), and x0 6∈ supp ϕ. We set
 4

, n ≤ 3,

4−n
(4.12)
p>


1,
n ≥ 4.
Then, for an arbitrarily given δ > 0, there exists a constant C = C(x0 , ϕ, T, p, δ, U)
> 0 such that
(4.13)
1
N
kf kLp (0,T ) ≤ Cku(x0 , ·)kLp 1 (0,T +δ)
for any f ∈ U.
We will see that limδ→0 C = ∞ and, in order to estimate f over the time
interval (0, T ), we have to observe u(x0 , ·) over a longer time interval (0, T +δ).
Remark 4.1. In the case of n ≥ 4, we can relax the regularity of ϕ to H α (Rn )
with some α > 0, but we will not go into the details. In the case of n ≤ 3, if we
assume that ϕ ∈ C ∞ (Rn ) in (4.3), then in Theorem 4.1 we can take any p > 1.
Remark 4.2. As a subset of U, we can take, for example,
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PN = {f ; f is a polynomial whose order is at most N and kf kC[0,T ] ≤ M }.
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The condition f ∈ U is quite restrictive at the expense of the practically reasonable estimate of Hölder type (4.4).
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Remark 4.3. The a priori boundedness kf kC[0,T ] ≤ M is necessary for the
stability.
Example 4.1. Let n = 1, p > 2 and


0
|x| < r, |x| > R,


√
(4.14)
ϕ(x) =
π



, r < |x| < R.
R−r
We set x0 = 0. Then, by (4.7), we have
Z
y2
1
e− 4t ϕ(y)dy,
(4.15)
µ0 (t) = √
2 πt r<|y|<R
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so that
(4.16)
R2
r2
1
1
√ e− 4t ≤ µ0 (t) ≤ √ e− 4t ,
t
t
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t > 0.
We choose fn as
(4.17)
1
1
fn (t) = √ e− nt ,
t
t > 0, n ∈ N.
Then fn does not change the signs in (0, T ) and limn→∞ max0≤t≤T |fn (t)| = ∞.
The corresponding solution un (x, t) of (4.1) – (4.2) with fn is estimated as
follows:
Z t
Z t
r2
1
1
1
√
|un (0, t)| = µ0 (t − s)fn (s)ds ≤
e− 4(t−s) √ e− ns ds,
s
t−s
0
0
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and so
Z
0
T
T
t
r2
1 −1
1
− 4(t−s)
√ e ns ds dt
√
e
|un (0, t)|dt ≤
s
t−s
0
0
Z T Z T
r2
1
1
1
− 4(t−s)
√
=
e
dt √ e− ns ds
s
t−s
0
s
Z T Z T
2
1
1 −r
≤
√ e 4η dη √ ds
η
s
0
0
Z
T
√
1 − r2
=2 T
√ e 4η dη.
η
0
Z
Z
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Next
Z
T
p
fn (t) dt = n
p
−1
2
0
Z
nT
p
p
e− η η − 2 dη.
0
Therefore, for any γ ∈ (0, 1), we have
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R
p1
p1
p
1
1
p
nT
fn (t)p dt
n 2 − p 0 e− η η − 2 dη
R
γ ≥ √ R
γ −→ ∞ as n −→ ∞
2
T
T 1 −r
4η dη
√
u
(0,
t)dt
2
T
e
n
0
η
0
R
T
0
by p > 2. Hence the stability of the type (4.4) is impossible for p > 2.
Remark 4.4. For our stability, the finiteness of changes of signs is essential. In
fact, we take
(4.18)
fn (t) = cos nt,
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0 ≤ t ≤ T, n ∈ N.
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Then fn oscillates very frequently and we cannot take any finite partition of
(0, T ) where the condition on signs in (4.1) holds true. We note that we can
take M = 1, that is, kfn kC[0,T ] ≤ 1 for n ∈ N. We denote the solution to (4.1)
– (4.2) for f = fn by un (x, t). Then
Z t
un (x0 , t) =
µx0 (t − s)fn (s)ds
0
Z t
=
µx0 (s)fn (t − s)ds
0
Z t
Z t
= cos nt
µx0 (s) cos nsds − sin nt
µx0 (s) sin nsds.
0
0
By µx0 ∈ L1 (0, T ), the Riemann-Lebesgue lemma yields limn→∞ un (x0 , t) = 0
for all t ∈ [0, T + δ]. Moreover we readily see that
Z T +δ
|un (x0 , t)| ≤
µx0 (s)ds < ∞,
n ∈ N, 0 ≤ t ≤ T + δ.
Reverse Convolution
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Masahiro Yamamoto
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0
Therefore, by the Lebesgue convergence theorem, we can conclude that
lim kun (x0 , ·)kL1 (0,T +δ) = 0.
n→∞
For n = 1, we can choose p = 2 in Theorem 4.1. We have
Z T
T
sin 2nT
fn (t)2 dt = +
,
2
4n
0
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so that limn→∞ kfn kL2 (0,T ) 6= 0. Thus any stability cannot hold for fn , n ∈ N.
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5.
Proof of Theorem 4.1
Suppose that f changes the signs at 0 < t1 < t2 < ... < tI = T, I ≤ N .
Without loss of generality, we may assume that f ≥ 0 on (0, t1 ). Since f ∈ U,
we see that f satisfies (3.1). Meanwhile since µx0 (t) is positive and bounded,
for some constant B > 0, Bµx0 (t) satisfies (3.1). We apply Theorem 3.1 on
1
(0, t1 ), setting α = 0 and g(t) = Bµx0 (t). Setting C1 = B 1− p kµx0 kLp (0,δ)
(C1 > 0), we obtain
(5.1)
kf kLp (0,t1 ) ≤ C1−1 M
2p−2
p
1
ku(x0 , ·)kLp 1 (0,t1 +δ) .
Next we will prove
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Saburou Saitoh, Vũ Kim Tuấn and
Masahiro Yamamoto
1
(5.2)
|u(x0 , t1 )| ≤ C2 ku(x0 , ·)kLp 1 (0,t1 +δ) ,
where the constant C2 > 0 depends on ϕ, T, δ, p. Henceforth the constants
Cj > 0, j ≥ 2, are independent of the choice of 0 < t1 < t2 < · · · < tN < T .
Proof of (5.2). Let L2 (Rn ) be the usual L2 -space with the norm k · k and let −A
be the operator defined by
(5.3)
(−Au)(x) = ∆u(x),
x ∈ Rn ,
D(A) = H 2 (Rn ).
Then −A generates an analytic semigroup e−tA , t > 0 and, by the definition of
H 2` (Rn ) and the interpolation inequality (e.g., Lions and Magenes [9]), we see
that
(DA` ) = H 2` (Rn ),
kukH 2` (Rn ) ≤ C3 (`)kA` uk,
u ∈ D(A` ).
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Moreover
kA` e−tA k ≤ C3 (`)t−`
(5.4)
and
Z
(5.5)
u(t) = u(·, t) =
t
e−(t−s)A f (s)ϕ(·)ds,
t>0
0
(e.g., Pazy [11]). By the Sobolev inequality: H 2` (Rn ) ⊂ L∞ (Rn ) if 4` > n
(e.g., Adams [1]), we have D(A` ) ⊂ L∞ (Rn ) and
(5.6)
kukL∞ (Rn ) ≤ C4 (`)kA` uk,
Reverse Convolution
Inequalities and Applications to
Inverse Heat Source Problems
u ∈ D(A` ).
Case: n ≤ 3.
We can take
n
+ ε0 < 1 with a sufficiently small ε0 > 0.
4
4
Let q > 1 satisfy p1 + 1q = 1. Since p > 4−n
, we have q < n4 , therefore we can
choose ε0 > 0 sufficiently small such that
(5.7)
Saburou Saitoh, Vũ Kim Tuấn and
Masahiro Yamamoto
`=
q` < 1.
(5.8)
≤ C5
−q`
(t1 − s)
0
1q Z
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t1
p
|f (s)| ds
ds
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0
t1
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Hence, by (5.4), (5.5) and the Hölder inequality, we obtain
Z t1
`
(5.9)
kA u(t1 )k ≤
|f (s)|kA` e−(t1 −s)A ϕkds
0
Z t1
≤ C5
(t1 − s)−` |f (s)|ds
Z
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p1
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.
0
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Consequently, by (5.8), we have
`
kA u(t1 )k ≤ C5
t1−q`
1
1 − q`
! 1q
kf kLp (0,t1 ) ≤ C5
T
1 − q`
1q
kf kLp (0,t1 ) .
Therefore (5.1) and (5.6) yield (5.2).
Reverse Convolution
Inequalities and Applications to
Inverse Heat Source Problems
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Masahiro Yamamoto
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Case: n ≥ 4.
By ϕ ∈ C0∞ (Rn ), we have ϕ ∈ D(A` ) for ` ∈ N. Therefore the estimate of
kA` u(t)k is simpler than in (5.9):
Z t1
`
−(t1 −s)A `
kA u(t1 )k = f (s)e
A ϕds
0
Z t1
≤
|f (s)|ke−(t1 −s)A kkA` ϕkds
≤
0
C60 kf kL1 (0,t1 )
≤ C6 kf kLp (0,t1 ) .
Reverse Convolution
Inequalities and Applications to
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Thus (5.1) and (5.6) complete the proof of (5.2).
Saburou Saitoh, Vũ Kim Tuấn and
Masahiro Yamamoto
Next we will estimate kf kLp (t1 ,t2 ) . By (4.8), we have
Z t
(5.10) −u(x0 , t) = −u(x0 , t1 ) +
µx0 (t − s)(−f (s))ds,
t1 ≤ t ≤ t2 .
t1
Taking α = t1 , T = t2 in Theorem 3.1, we obtain
Z t2 +δ
p1
2p−2
−1
kf kLp (t1 ,t2 ) ≤ M p C1
(5.11)
| − u(x0 , t) + u(x0 , t1 )|dt
t1
≤M
2p−2
p
1
C1−1 (ku(x0 , ·)kL1 (t1 ,t2 +δ) + T |u(x0 , t1 )|) p .
(5.12)
kf kLp (t1 ,t2 ) ≤
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Therefore we apply (5.2), and
C1−1 M
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2p−2
p
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1
p
ku(x0 , ·)kL1 (t1 ,t2 +δ)
+ C1−1 M
2p−2
p
1
1
p
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1
p2
T p C2 ku(x0 , ·)kL1 (0,t1 +δ)
J. Ineq. Pure and Appl. Math. 3(5) Art. 80, 2002
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1
1
− 12
2
p
ku(x0 , ·)kL1 (tp1 ,t2 +δ) ku(x0 , ·)kLp 1 (t1 ,t2 +δ)
1
1
1
p
p2
p
+ T C2 ku(x0 , ·)kL1 (0,t1 +δ)
C1−1 M
2p−2
p
≤ C1−1 M
2p−2
p
≤
((T M 0 )
p−1
p2
1
1
1
2
+ T p C2p )ku(x0 , ·)kLp 1 (0,t2 +δ) .
Here, since u(x0 , t) is bounded, we take a positive M 0 such that |u(x0 , t)| ≤ M 0 .
By (5.1) and (5.12), we can estimate kf kLp (0,t2 ) . Continuing this argument until
tI = T , we can complete the proof of Theorem 4.1.
Reverse Convolution
Inequalities and Applications to
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Saburou Saitoh, Vũ Kim Tuấn and
Masahiro Yamamoto
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References
[1] R.A. ADAMS, Sobolev Spaces, Academic Press, New York, 1975.
[2] J.R. CANNON, The One-dimensional Heat Equation, Addison-Wesley,
Reading, Massachusetts, 1984.
[3] J.R. CANNON AND S.P. ESTEVA, An inverse problem for the heat equation, Inverse Problems, 2 (1986), 395–403.
[4] J.R. CANNON AND S.P. ESTEVA, Uniqueness and stability of 3D heat
sources, Inverse Problems, 7 (1991) 57–62.
Reverse Convolution
Inequalities and Applications to
Inverse Heat Source Problems
[5] M. CWICKEL AND R. KERMAN, On a convolution inequality of Saitoh,
Proc. Amer. Math. Soc., 124 (1996), 773–777.
Saburou Saitoh, Vũ Kim Tuấn and
Masahiro Yamamoto
[6] A. FRIEDMAN, Partial Differential Equations of Parabolic Type, Krieger,
Malabar, Florida, 1983.
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[7] V. ISAKOV, Inverse Problems for Partial Differential Equations, SpringerVerlag, Berlin, 1998.
[8] S. IZUMINO AND M. TOMINAGA, Estimations in Hölder type inequalities, Mathematical Inequalities & Applications, 4 (2001), 163–187.
[9] J.-L. LIONS AND E. MAGENES, Non-homogeneous Boundary Value
Problems and Applications, Springer-Verlag, Berlin, 1972.
[10] D.S. MITRINOVIĆ, J.E. PEČARIĆ AND A.M. FINK, Classic and New
Inequalities in Analysis, Kluwer Academic Publishers, The Netherlands,
1993.
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[11] A. PAZY, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, Berlin, 1983.
[12] S. SAITOH, A fundamental inequality in the convolution of L2 functions
on the half line, Proc. Amer. Math. Soc., 91 (1984), 285–286.
[13] S. SAITOH, Inequalities in the most simple Sobolev space and convolutions of L2 functions with weights, Proc. Amer. Math. Soc., 118 (1993),
515–520.
[14] S. SAITOH, Various operators in Hilbert space introduced by transforms,
International. J. Appl. Math., 1 (1999), 111–126.
[15] S. SAITOH, Weighted Lp -norm inequalities in convolutions, Survey on
Classical Inequalities, Kluwer Academic Publishers, The Netherlands,
2000, 225–234.
[16] S. SAITOH, V. K. TUAN AND M. YAMAMOTO, Reverse weighted
Lp -norm inequalities in convolutions and stability in inverse problems,
J. Inequal. Pure and Appl. Math., 1 (2000), Article 7. [ONLINE:
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[17] H.M. SRIVASTAVA AND R.G. BUSCHMAN, Theory and Applications
of Convolution Integral Equations, Kluwer Academic Publishers, The
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[18] L. XIAO-HUA, On the inverse of Hölder inequality, Math. Practice and
Theory, 1 (1990), 84–88.
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