JJ II

(1)

volume 1, issue 1, article 7, 2000.

Received and acepted 14 December, 1999.

Communicated by:J.E. Pe˘cari´c

Abstract Contents

JJ II

J I

Home Page Go Back

Close Quit

Journal of Inequalities in Pure and Applied Mathematics

REVERSE WEIGHTEDL_P–NORM INEQUALITIES IN CONVOLUTIONS

SABUROU SAITOH, V ˜U KIM TU ´ÂN AND MASAHIRO YAMAMOTO

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

2000c School of Communications and Informatics,Victoria University of Technology ISSN (electronic): 1443-5756

018-99

(2)

Reverse WeightedLp–Norm Inequalities in Convolutions

Saburou Saitoh,V ˜u Kim Tu ´ânand Masahiro Yamamoto

Title Page Contents

JJ II

J I

Go Back Close

Quit Page2of16

J. Ineq. Pure and Appl. Math. 1(1) Art. 7, 2000

http://jipam.vu.edu.au

Abstract

Various weighted L_p(p > 1)–norm inequalities in convolutions were derived by using Hölder’s inequality. Therefore, by using reverse Hölder inequalities one can obtain reverse weightedLp–norm inequalities. These inequalities are important in studying stability of some inverse problems.

2000 Mathematics Subject Classification:44A35, 26D20

Key words: Convolution, weightedLpinequality, reverse Hölder inequality, inverse problems, Green’s function, integral transform, stability.

The authors wish to express their sincere thanks to Professor Josip Pe˘cari´c 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.

1. Introduction

For the Fourier convolution

(f ∗g)(x) = Z ∞

−∞

f(x−ξ)g(ξ)dξ, the Young’s inequality

(1.1)

kf∗gk_r ≤ kfk_pkgk_q, f ∈L_p(R), g ∈L_q(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.

Proposition 1.1. ([7]). For two nonvanishing functionsρ_j ∈ L₁(R) (j = 1,2) the followingL_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 forF_j ∈L_p(R,|ρ_j|) (j = 1,2). Equality holds if and only if

(1.3) Fj(x) =Cje^αx,

whereαis a constant such thate^αx ∈L_p(R,|ρ_j|) (j = 1,2).

Here

kFk_L_p_(R;ρ) = Z ∞

−∞

|F(x)|^pρ(x)dx ¹_p

.

(4)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page4of16

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

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.

The 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 this paper we are interested in the reverse type inequality for (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. 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 andgsatisfying

(1.7) 0< m≤ f

g ≤M <∞

(5)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page5of16

on the setX, and forp, q >0, 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

A_p,q(t) = p⁻¹^pq⁻¹^qt⁻^pq¹(1−t)

1−t¹^p−_p¹

1−t¹^q−¹_q

.

(6)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page6of16

2. A general reverse weighted L

_p

convolution in- equality

Our main result is the following

Theorem 2.1. LetF₁ andF₂be positive functions satisfying (2.1)

0< m

1 p

1 ≤F₁(x)≤M

1 p

1 <∞, 0< m

1 p

2 ≤F₂(x)≤M

1 p

2 <∞, p >1, x∈R. Then for any positive functions ρ₁ and ρ₂ we have the reverse L_p–weighted convolution inequality

(2.2)

((F₁ρ₁)∗(F₂ρ₂)) (ρ₁∗ρ₂)¹^p⁻¹ p ≥

A_p,q

m1m2

M₁M₂ −1

kF₁k_L

p(R,ρ1)kF₂k_L

p(R,ρ2). 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) = F₁^p(ξ)F₂^p(x−ξ)ρ₁(ξ)ρ₂(x−ξ), g(ξ) =ρ₁(ξ)ρ₂(x−ξ).

Then condition (2.1) implies m₁m₂ ≤ f(ξ)

g(ξ) ≤M₁M₂, ξ ∈R.

(7)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page7of16

Hence, one can apply the reverse Hölder inequality (1.8) forf andg to get A_p,q

m1m2

M₁M₂ Z ∞

−∞

F₁(ξ)ρ₁(ξ)F₂(x−ξ)ρ₂(x−ξ)dξ

≥ Z ∞

−∞

F₁^p(ξ)F₂^p(x−ξ)ρ₁(ξ)ρ₂(x−ξ)dξ

¹_pZ ∞

−∞

ρ₁(ξ)ρ₂(x−ξ)dξ ¹⁻_p¹

.

Hence, (2.3)

Z ∞

−∞

F1(ξ)ρ1(ξ)F2(x−ξ)ρ2(x−ξ)dξ

pZ ∞

−∞

ρ1(ξ)ρ2(x−ξ)dξ 1−p

≥

A_p,q

m₁m₂ M₁M₂

−pZ ∞

−∞

F₁^p(ξ)F₂^p(x−ξ)ρ₁(ξ)ρ₂(x−ξ)dξ.

Taking integration of both sides of (2.3) with respect to xfrom−∞ to∞we obtain the inequality

(2.4) Z ∞

−∞

Z ∞

−∞

F₁(ξ)ρ₁(ξ)F₂(x−ξ)ρ₂(x−ξ)dξ

pZ ∞

−∞

ρ₁(ξ)ρ₂(x−ξ)dξ ^1−p

dx

≥

A_p,q

m₁m₂ M₁M₂

^−pZ ∞

−∞

F₁^p(ξ)ρ₁(ξ)dξ Z ∞

−∞

F₂^p(x)ρ₂(x)dx.

Raising both sides of the inequality (2.4) to power ¹_p yields the inequality (2.2).

(8)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page8of16

Inequality (1.8) reverses the sign if 0 < p < 1. Hence, inequality (2.2) reverses the sign if0< p <1.

In formula (2.3) replacingρ₂ by1, andF₂(x−ξ)by G(x−ξ), and taking integration with respect toxfromctodwe arrive at the following inequality (2.5)

Z d c

Z ∞

−∞

F(ξ)ρ(ξ)G(x−ξ)dξ p

dx

≥n Ap,q

m M

o−pZ ∞

−∞

ρ(ξ)dξ

p−1Z ∞

−∞

F^p(ξ)ρ(ξ)dξ Z d−ξ

c−ξ

G^p(x)dx,

valid if positive continuous functionsρ,F, andGsatisfy

(2.6) 0< m¹^p ≤F(ξ)G(x−ξ)≤M¹^p, 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.

(9)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page9of16

3. Examples

3.1. The first order differential equation

The solutiony(x)of the first order differential equation y⁰(x) +λ y(x) =F(x), y(0) = 0,

is represented in the form

y(x) = Z x

0

F(t)e^−λ(x−t)dt.

So we shall consider the integral transform f(x) =

Z x 0

F(t)ρ(t)e^−λ(x−t)dt, λ >0.

Take

G(x) =

e^−λx, x >0 0, x <0 . The condition (2.6) reads

(3.1) 0< m¹^p ≤F(t)e^−λ(x−t) ≤M¹^p. It will be satisfied for0≤t≤x≤d <∞, if we have (3.2) 0< m¹^pe^λd−λt ≤F(t)≤M^p¹, 0< d < 1

pλlog M m.

(10)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page10of16

Notice that

Z d−ξ c−ξ

G^p(x)dx=

( _e−λpc−e^−λpd

λp e^λpξ, ξ < c,

1−e^λpξ−λpd

λp , c < ξ < d.

Thus the inequality (2.5) yields (3.3)

Z d c

f^p(x) Z x

0

ρ(t)dt 1−p

dx

≥n

A_p,qm M

o−p 1 λp

(e^−λpc−e^−λpd) Z c

0

F^p(ξ)ρ(ξ)e^λpξdξ

+ Z d

c

F^p(ξ)ρ(ξ)(1−e^−λpde^λpξ)dξ

. Here we assume thatρis a positive continuous function on[0, d], andF satisfies (3.2).

3.2. Picard transform

Note that ¹₂e^−|x−t|is the Green’s function for the boundary value problem y⁰⁰−y= 0, lim

x→±∞y(x) = 0.

So, we shall consider the Picard transform f(x) = 1

2 Z ∞

−∞

F(t)ρ(t)e^−|x−t|dt.

(11)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page11of16

TakeG(x) =e^−|x|. Since

e^−ae^|t|≤e^|x−t| ≤e^ae^|t|, |x| ≤a, we see that the condition (2.6)

(3.4) 0< m¹^p ≤F(t)e^−|x−t| ≤M¹^p, holds if

(3.5) 0< m¹^pe^ae^|t|≤F(t)≤M^p¹e^−ae^|t|, t∈R, 0< a < 1

2plogM m. We have

Z d−t c−t

G^p(x)dx= Z d−t

c−t

e^−p|x|dx=







e^pt p

e^−pc−e^−pd

, t < c,

e^−pt p

e^pd−e^pc

, t > d,

1

p 2−e^pc−pt−e^pt−pd

, c < t < d.

Thus, for−a≤c, d≤athe inequality (2.5) yields (3.6)

Z d c

f^p(x)dx≥ 1 2^pp

n

A_p,qm M

o^−pZ ∞

−∞

ρ(t)dt p−1

e^−pc−e^−pd Z c

−∞

F^p(t)ρ(t)e^ptdt+ e^pd−e^pc Z ∞

d

F^p(t)ρ(t)e^−ptdt

+ Z d

c

F^p(t)ρ(t) 2−e^pc−pt−e^pt−pd dt

, ifρis positive continuous, andF satisfies (3.5).

(12)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page12of16

3.3. Poisson integrals

Consider the Poisson integral

(3.7) u(x, y) = 1

π Z ∞

−∞

F(ξ)ρ(ξ) y

(x−ξ)²+y²dξ.

Take

G(x) = y x² +y². Let

ξ ∈[a, b], x∈[c, d].

Denote

α= max{|a−c|,|a−d|,|b−c|,|b−d|}.

We have

y

α²+y² ≤ y

(x−ξ)²+y² ≤ 1 y. Thus,

Z d−ξ c−ξ

G^p(x)dx= Z d−ξ

c−ξ

y x²+y²

p

dx ≥(d−c) y

α²+y² p

.

Hence, for a functionF satisfying α²+y²

y m¹^p ≤F(ξ)≤y M¹^p,

(13)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page13of16

and for a positive continuous functionρon[a, b]we obtain (3.8)

Z d c

u^p(x, y)dx≥ (d−c) π^p

y α²+y²

p

n

A_p,qm M

o^−p

Z b a

ρ(ξ)dξ

^p−1Z b a

F^p(ξ)ρ(ξ)dξ.

Consider now the conjugate Poisson integral

(3.9) v(x, y) = 1

π Z ∞

−∞

F(ξ)ρ(ξ) x−ξ

(x−ξ)²+y²dξ.

Take

G(x) = x x² +y². For

ξ ∈[a, b], x∈[c, d], (b < c), we have

c−b

(d−a)²+y² ≤ x−ξ

(x−ξ)²+y² ≤ d−a (c−b)²+y². Thus,

Z d−ξ c−ξ

G^p(x)dx= Z d−ξ

c−ξ

x x²+y²

p

dx≥(d−c)

c−b (d−a)²+y²

p

. Hence, for a functionF satisfying

(d−a)²+y²

c−b m¹^p ≤F(ξ)≤ (c−b)²+y² d−a M¹^p,

(14)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page14of16

and for a positive continuous functionρon[a, b]we obtain (3.10)

Z d c

v^p(x, y)dx≥ (d−c) π^p

c−b (d−a)²+y²

p

n

A_p,qm M

o−p

Z b a

ρ(ξ)dξ

^p−1Z b a

F^p(ξ)ρ(ξ)dξ.

3.4. Heat equation

We consider the Weierstrass transform (3.11) u(x, t) = 1

√4πt Z ∞

−∞

F(ξ)ρ(ξ) exp

−(x−ξ)² 4t

dξ,

which gives the formal solutionu(x, t)of the heat equation u_t = ∆u on R+×R,

subject to the initial condition

u(x,0) = F(x)ρ(x) on R.

Take

G(x) = e⁻^x

2 4t. Let

x∈[−a, a], ξ∈[−b, b], a+b≤ s

4t p logM

m.

(15)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page15of16

From

1≤exp

(x−ξ)² 4t

≤exp

(a+b)² 4t

, we have

0< m¹^p ≤F(ξ) exp

−(x−ξ)² 4t

≤M^p¹,

if

(3.12) m¹^p exp

(a+b)² 4t

≤F(ξ)≤M¹^p, ξ∈[−b, b].

It is easy to see that Z d−ξ

c−ξ

e⁻^px

2

4t dx=

rπt p

erf

√

p(d−ξ) 2√

t

−erf √

p(c−ξ) 2√

t

, where

erf (x) = 2

√π Z x

0

e^−t²dt

is the error function. Therefore, for−a≤c < d≤a,the inequality (2.5) yields (3.13)

Z d c

u(x, t)^pdx≥ 1 2^p(πt)^(p−1)/2√

p n

A_p,qm M

o−pZ b

−b

ρ(ξ)dξ ^p−1

Z b

−b

F^p(ξ)ρ(ξ)

erf √

p(d−ξ) 2√

t

−erf √

p(c−ξ) 2√

t

dξ, whereρis a positive continuous function on[−b, b], andF satisfies (3.12).

(16)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page16of16

References

[1] M. CWICKELANDR. 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. MITRINOVI ´C, J.E. PE ˘CARI ´CANDA.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), 515- 520.

[6] S. SAITOH, Various operators in Hilbert spaces introduced by transforms, Intern. J. Appl. Math., 1 (1999), 111-126.

[7] S. SAITOH, Weighted L_p-norm inequalities in convolutions, Handbook on Classical Inequalities, Kluwer Academic Publishers, Dordrecht (to ap- pear).

[8] H.M. SRIVASTAVAANDR.G. BUSCHMAN, Theory and Applications of Convolution Integral Equations, Kluwer Academic Publishers, Dordrecht, 1992.