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volume 4, issue 3, article 50, 2003.

Received 09 December, 2002;

accepted 15 April, 2003.

Communicated by:L.-E. Persson

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Journal of Inequalities in Pure and Applied Mathematics

CONVOLUTION INEQUALITIES AND APPLICATIONS

SABUROU SAITOH, VU KIM TUAN AND MASAHIRO YAMAMOTO

Department of Mathematics, Faculty of Engineering, Gunma University, Kiryu 376-8515 Japan.

E-Mail: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.

E-Mail:vu@mcs.sci.kuniv.edu.kw

Graduate School of Mathematical Sciences, The University of Tokyo,

3-8-1 Komaba Meguro Tokyo 153-8914 Japan.

E-Mail:myama@ms.u-tokyo.ac.jp

c

2000Victoria University ISSN (electronic): 1443-5756 138-02

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Convolution Inequalities and Applications

Saburou Saitoh, Vu Kim Tuanand Masahiro Yamamoto

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J. Ineq. Pure and Appl. Math. 4(3) Art. 50, 2003

http://jipam.vu.edu.au

Abstract

We introduce various convolution inequalities obtained recently and at the same time, we give new type of reverse convolution inequalities and their important applications to inverse source problems. We consider the inverse problem of determiningf(t),0< t < T, in the heat source of the heat equation∂_tu(x, t) =

∆u(x, t) +f(t)ϕ(x),x∈Rⁿ,t >0from the observationu(x0, t),0< t < T, at a remote pointx₀away from the support ofϕ. Under an a priori assumption that f changes the signs at mostN-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 in- equality, 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.

The authors wish to express our deep thanks for the referee for his or her careful reading of the manuscript. In particular, he or she found misprints in Theorem3.1 which also appeared in Theorem 3.1 in [17].

1. Introduction

For the Fourier convolution

(f ∗g)(x) = Z ∞

−∞

f(x−ξ)g(ξ)dξ, the Young’s inequality

(1.1) kf∗gkr ≤ kfkpkgkq, 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 off, g∈L2(R), the inequality does not hold. In a series of papers [12]–[15] (see also [5]) we obtained the following weightedL_p(p >1)norm inequality for convolution:

Proposition 1.1. ([15]) For two non-vanishing functionsρ_j ∈L₁(R)(j = 1,2), theL_p(p >1)weighted convolution inequality

(1.2)

((F₁ρ₁)∗(F₂ρ₂)) (ρ₁∗ρ₂)¹^p⁻¹ p

≤ kF₁k_L

p(R,|ρ₁|)kF₂k_L

p(R,|ρ₂|)

holds forFj ∈Lp(R,|ρj|)(j = 1,2). Equality holds here if and only if

(1.3) Fj(x) =Cje^αx,

where αis a constant such that e^αx ∈ Lp(R,|ρj|)(j = 1,2)(otherwise, C1 or C₂ = 0). Here

kFk_L_p(R,|ρ|) = Z ∞

−∞

|F(x)|^p|ρ(x)|dx ¹_p

.

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Unlike Young’s inequality, inequality (1.2) holds also in the casep= 2.

Note that the proof of inequality (1.2) in Proposition1.1is direct and fairly elementary. The proof will be done in three lines. 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 [18] for various convolutions. However, to determine the case that equality in (1.2) holds needs very delicate arguments. See [5] for the details.

In many cases of interest, the convolution is given in the form (1.4) ρ₂(x)≡1, F₂(x) = G(x),

whereG(x−ξ)is some Green’s function. Then the inequality (1.2) takes the form

(1.5) k(F ρ)∗Gk_p ≤ kρk¹⁻

1 p

p kGk_pkFk_L

p(R,|ρ|), whereρ, F, andGare such that the right hand side of (1.5) is finite.

Inequality (1.5) enables us to estimate the output function (1.6)

Z ∞

−∞

F(ξ)ρ(ξ)G(x−ξ)dξ

in terms of the input functionF in the related differential equation. For various applications, see [15]. We are also interested in the reverse type inequality of (1.5), namely, we wish to estimate the input functionF by means of the output (1.6). This kind of estimate is important in inverse problems. One estimate is obtained by using the following famous reverse Hölder inequality

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Proposition 1.2. ([19], see also [10, pp. 125-126]). For two positive functions f andgsatisfying

(1.7) 0< m≤ f

g ≤M <∞ on the setX, and forp, q >1, p⁻¹+q⁻¹ = 1,

(1.8)

Z

X

f dµ ¹_pZ

X

gdµ ¹_q

≤A_p,qm M

Z

X

f¹^pg¹^qdµ,

if the right hand side integral converges. Here

Ap,q(t) =p⁻¹^pq⁻¹^q t⁻^pq¹ (1−t)

1−t^p¹¹_p

1−t¹^q¹_q.

Then, by using Proposition1.2we obtain, as in the proof of Proposition1.1, the following:

Proposition 1.3. ([16]). LetF₁ andF₂be positive functions satisfying

0< m

1 p

1 ≤F1(x)≤M

1 p

1 <∞, (1.9)

0< m

1 p

2 ≤F₂(x)≤M

1 p

2 <∞, p > 1, x∈R.

Then for any positive continuous functionsρ₁ andρ₂, we have the reverseL_p–

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weighted convolution inequality (1.10)

A_p,q

m₁m₂ M₁M₂

−1

kF₁k_L

p(R,ρ1)kF₂k_L

p(R,ρ2)

≤

((F1ρ1)∗(F2ρ2)) (ρ1∗ρ2)¹^p⁻¹ p. Inequality (1.10) and others 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) replacingρ₂ by1, andF₂(x−ξ)byG(x−ξ), and taking integration with respect toxfromctodwe arrive at the following inequality (1.11) n

A_p,qm M

o−pZ ∞

−∞

ρ(ξ)dξ p−1

× Z ∞

−∞

F^p(ξ)ρ(ξ)dξ Z d−ξ

c−ξ

G^p(x)dx

≤ Z d

c

Z ∞

−∞

F(ξ)ρ(ξ)G(x−ξ)dξ p

dx

if the positive continuous functionsρ,F, andGsatisfy

(1.12) 0< m¹^p ≤F(ξ)G(x−ξ)≤M¹^p, x∈[c, d], ξ ∈R.

Inequality (1.11) is especially important when G(x−ξ) is a Green function.

We gave various concrete applications 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

Xa^p_k¹_pX b^q_k¹_q

−λX a_kb_k

for λ > 0, for p, q > 1 satisfying 1/p+ 1/q = 1 and for positive numbers {ak}ⁿ_k=1 and {bk}ⁿ_k=1, in detail. In their different approach, they showed that the constant A_p,q(t)in Proposition1.2is best possible in a sense. Note that the proof of Proposition1.2 is quite involved. In connection with Proposition 1.2 we note that the following version whose proof is surprisingly simple

Theorem 2.1. ([17]). In Proposition 1.2, replacing f and g by f^p and g^q, respectively, we obtain the reverse Hölder type inequality

(2.1)

Z

X

f^pdµ _p¹ Z

X

g^qdµ ¹_q

≤m M

−_pq¹ Z

X

f gdµ.

Proof. Sincef^p/g^q ≤M,g ≥M⁻¹^qf^p^q. Therefore f g ≥M⁻¹^qf¹⁺^p^q =M⁻¹^qf^p and so,

(2.2)

Z f^pdµ

¹_p

≤M^pq¹ Z

f gdµ ¹_p

.

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On the other hand, sincem≤f^p/g^q,f ≥m^1/pg^q/p. Hence Z

f gdµ ≥ Z

m¹^pg¹⁺^q^pdµ=m¹^p Z

g^qdµ,

and so,

Z f gdµ

¹_q

≥m^pq¹ Z

g^qdµ ¹_q

. Combining with (2.2), we have the desired inequality

Z f^pdµ

¹_pZ g^qdµ

¹_q

≤M^pq¹ Z

f gdµ ¹_p

m⁻¹^pq Z

f gdµ ¹_q

=m M

⁻¹_pq Z

f gdµ.

Remark 2.1. In a private communication, Professor Lars-Erik Persson pointed out the following very interesting result that

(LEP) A_p,q(t)< t⁻¹^pq

which implies that Theorem 2.1 can be derived from Proposition1.2, directly.

Its proof is noted as follows:

By using an obvious estimate and Hölder’s inequality we find that 1−t =

Z 1

t

1dt

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= Z 1

t

u^1/p²u^1/q²u^1/p−1/p²u^1/q−1/q²du u

<

Z 1

t

u^1/p²u^1/q²du u

≤ Z 1

t

u^1/pdu u

¹_p Z 1

t

u^1/qdu u

¹_q

= [p(1−t^1/p]^1/p[q(1−t^1/q]^1/q, which implies (LEP).

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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 reverse convolution inequality of new type.

Theorem 3.1. Letp≥1, δ >0,0≤α < T, andf, g∈L∞(0, T +δ)satisfy (3.1) 0≤f, g ≤M <∞, 0< t < T +δ.

Then

(3.2) kfk_L_p_(α,T₎kgk_L_p_(0,δ) ≤M^2p−2^p

Z T+δ

α

Z t

α

f(s)g(t−s)ds

dt

1 p

. In particular, for

(f∗g)(t) = Z t

0

f(t−s)g(s)ds, 0< t < T +δ

and forα= 0, we have

kfk_L_p_(0,T)kgk_L_p_(0,δ) ≤M^2p−2^p kf ∗gk

1 p

L1(0,T+δ). Proof. Since0≤f, g ≤M for0≤t≤T, we have

Z t

α

f(s)^pg(t−s)^pds= Z t

α

f(s)^p−1g(t−s)^p−1f(s)g(t−s)ds (3.3)

≤M^2p−2 Z t

α

f(s)g(t−s)ds.

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Hence Z T+δ

α

Z t

α

f(s)^pg(t−s)^pds

dt ≤M^2p−2 Z T+δ

α

Z t

α

f(s)g(t−s)ds

dt.

On the other hand, we have Z T+δ

α

Z t

α

f(s)^pg(t−s)^pds

dt = Z T+δ

α

Z T+δ

s

g(t−s)^pdt

f(s)^pds

= Z T+δ

α

Z T+δ−s

0

g(η)^pdη

f(s)^pds

≥ Z T

α

Z T+δ−s

0

g(η)^pdη

f(s)^pds

≥ Z T

α

Z δ

0

g(η)^pdη

f(s)^pds

=kfk^p_L

p(α,T)kgk^p_L

p(0,δ). Thus the proof of Theorem3.1is complete.

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4. Applications to Inverse Source Heat Problems

We consider the heat equation with a heat source:

(4.1) ∂_tu(x, t) = ∆u(x, t) +f(t)ϕ(x), x∈Rⁿ, t >0,

(4.2) u(x,0) = 0, x∈Rⁿ.

We assume thatϕis a given function and satisfiesϕ≥ 0, 6≡0 in Rⁿ,ϕhas compact support,

(4.3)

( ϕ∈C^∞(Rⁿ), if n≥4 and ϕ∈L2(Rⁿ), if n≤3.

Our problem is to derive a conditional stability in the determination off(t), 0< t < T, from the observation

(4.4) u(x₀, t), 0< t < T, wherex₀ 6∈supp ϕ.

We are interested only in the case ofx₀ 6∈suppϕ, because in the case where x₀ is in the interior ofsuppϕ, the problem can be reduced to a Volterra integral equation of the second kind by differentiation in t formula (4.8) stated below.

Moreoverx₀ 6∈suppϕmeans that our observation (4.4) is done far from the set where the actual process is occurring, and the design of the observation point is easy.

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Let

(4.5) K(x, t) = 1

(2√

πt)ⁿe⁻^|x|

2

4t , x∈Rⁿ, t >0.

Then the solutionuto (4.1) and (4.2) is represented by (4.6) u(x, t) =

Z t

0

Z

Rⁿ

K(x−y, t−s)f(s)ϕ(y)dyds, x∈Rⁿ, t >0,

(see e.g. Friedman [6]). Therefore, setting (4.7) µ_x₀(t) =

Z

Rⁿ

K(x₀−y, t)ϕ(y)dy, t >0,

we have

(4.8) u(x₀, t)≡h_x₀(t) = Z t

0

µ_x₀(t−s)f(s)ds, 0< t < T,

which is a Volterra integral equation of the first kind with respect tof. Since limt↓0

d^kµ_x₀

dt^k (t) = (∆^kϕ)(x0) = 0, k∈N∪ {0}

by x₀ 6∈ suppϕ (e.g. [6]), the equation (4.8) cannot be reduced to a Volterra equation of the second kind by differentiating int. Hence, even though, for any m ∈ N, we take theC^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.

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In Cannon and Esteva [3], an estimate of logarithmic type is proved: let n = 1 andϕ = ϕ(x)be the characteristic function of an interval (a, b) ⊂ R. Set

(4.9) V_M = (

f ∈C²[0,∞);f(0) = 0,

df dt

C[0,∞)

,

d²f dt²

C[0,∞)

≤M )

.

Letx₀ 6∈ (a, b). Then, forT >0, there exists a constantC =C(M, a, b, x₀)>

0such that

(4.10) |f(t)| ≤ C

|logku(x₀,·)k_L₂(0,∞)|², 0≤t ≤T,

for all f ∈ V_M. The stability rate is logarithmic and worse than any rate of Hölder type: ku(x₀,·)k^α_L

2(0,∞) for anyα > 0. For (4.9), the conditionf ∈ V_M prescribes a priori information and (4.9) is called conditional stability within the admissible set V_M. The rate of conditional stability heavily depends on the choice of admissible sets and an observation pointx₀. 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.

For fixedM >0andN ∈Nlet

(4.11) U ={f ∈C[0, T];kfk_C[0,T_]≤M,

f changes the signs at mostN-times}.

We takeUas an admissible set of unknownsf. Then, withinU, we can show an improved conditional stability of Hölder type:

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Theorem 4.1. Letϕsatisfy (4.3), andx₀ 6∈suppϕ. We set

(4.12) p >

( 4

4−n, n ≤3,

1, n ≥4.

Then, for an arbitrarily givenδ >0, there exists a constantC=C(x₀, ϕ, T, p, δ,U)

>0such that

(4.13) kfk_L_p_(0,T₎ ≤Cku(x₀,·)k

1 pN

L1(0,T+δ)

for anyf ∈ U.

We will see that limδ→0C = ∞ and, in order to estimate f over the time interval(0, T), we have to observeu(x₀,·)over a longer time interval(0, T+δ).

For a quite long proof of Theorem 4.1 and for the following remarks, see [17].

Remark 4.1. In the case ofn≥ 4, we can relax the regularity ofϕtoH^α(Rⁿ) with someα >0. In the case ofn ≤3, if we assume thatϕ ∈C^∞(Rⁿ)in (4.3), then in Theorem4.1we can take anyp >1.

Remark 4.2. As a subset ofU, we can take, for example,

P_N ={f;f is a polynomial whose order is at mostN andkfk_C[0,T_]≤M}.

The condition f ∈ U is quite restrictive at the expense of the practically rea- sonable estimate of Hölder type.

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Remark 4.3. The a priori boundedness kfk_C[0,T_] ≤ M is necessary for the stability. See [17] for a counter example.

Remark 4.4. For our stability, the finiteness of changes of signs is essential. In fact, we take

(4.14) f_n(t) = cosnt, 0≤t≤T, n∈N.

Then f_n oscillates very frequently and we cannot take any finite partition of (0, T)where the condition on signs in (4.11) holds true. We note that we can takeM = 1, that is,kf_nk_C[0,T_]≤1forn∈N. We denote the solution to (4.1) – (4.2) forf =f_nbyu_n(x, t). Then, we see that any stability cannot hold forf_n, n ∈N. See [17] for the proof.

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[1] R.A. ADAMS, Sobolev Spaces, Academic Press, New York (1975).

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[11] A. PAZY, Semigroups of Linear Operators and Applications to Partial Dif- ferential Equations, Springer-Verlag, Berlin (1983).

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[19] Xiao-Hua, L., On the inverse of Hölder inequality, Math. Practice and Theory, 1 (1990), 84–88.