On the Beurling Convolution Algebra III

Mem. Fae. Educ., Kagawa Univ. II, 57 (2007), 17-27

On the Beurling Convolution Algebra III

K. ANZAI, K. HORIE and S. KOIZUMI

Kagawa University, Takamatsu National College of Technology and Keio University

Abstract

We treat the Beurling convolution algebra. Especially, Theorem IV and Theorem Vfil in [ 4]

are extended to the case of n-dimensional euclidean space Rn.

1. Introduction

A. Beurling [4] considered the theorems concerning a class of functions of

R1,

each member of which is the Fourier transform of an integrable function. The purpose of this paper is to extend his results to the class of functions on Rn.

Let us start to set notations, definitions and theorems, which we shall ask for, according to A. Beurling [ 4]. Let G be a locally compact Abelian group with the invariant measure dx. We consider a normed family Q of strictly positive functions

w

(x) on G which are measurable with respect to dx, and furthermore, together with the norm N (

w ) ,

satisfy the following conditions

( I·) For each w E Q, N( w), takes a finite value,

o< f

^wdx

<

^N(w).

( II ) If A is a positive number and w E Q, then A w E Q and N( 11.w)

=

11.N(w).

( ill ) If w _{1 ,} w _{2 E} Q, then sum w ₁

+

w ₂ as well as the convolution w ₁

*

w ₂ are

K. ANZAI, K. HORIE and S. KOIZUMI

also in Q and

N(w1

+

w2)

<

^N(w1)

⁺

^N(wi).

( N) Q is complete under the norm Nin the sense that for any sequence

1 w ⁿ ~ ~ C Q such that

L~

^{N ( w n)}

<

^{00 ,} it is satisfied that w

= L~

^{w n is in Q and}

We associate with each, the Banach space L~-1 (G) of measurable functions on G and having the norms

I

12 112

II

^F

II

^L~,

= ( twdx t : dx)

From these spaces, we obtain set of functions A ²

=

A ² ( G, Q ) by setting

and define

Now A. Beurling [ 4] proved the following Theorem I , II , ill,

N,

and V.

Theorem I. In the norm (1.3), A2(G) is the Banach algebra under addition and convolution, and we have

(1.1)

where Fi and F2 belong to A²(G).

A.Beurling [ 4] considered the theorems concerning a class of functions of RI, each member of which is the Fourier transform of an integrable function.

Let us remark that A ² (RI) c L ¹ (RI) by the inequality

J

_R 1

IF I dx < (f

_R

^{,wdx f}

_R

^,fildx)½

_W

On the Beurling Convolution Algebra III

and so the Fourier transform of F E A ² (RI) is well defined Let us define the normed ring

with norm

Let us also define the notation as follows

Let us introduce the following integral

This integral A (f) will be an important tool for the analysis of

A

^2•

Theorem II. A function f belongs to the normed ring

A

² (RI, Q) if and only if the following conditions are satisfied

(a) Jis continuous, ( b) lim f ( t)

=

0,

ltl---+⁰⁰

( C) A (f)

<

^{00 •}

Under these conditions, f is the Fourier transform of an FE A ² (R^1, Q), and the following inequalities hold

provided f

*

^{0 .}

Theorem

m.

Let g be a continuous function and a ~ontraction of the series

L~=

1 fv where each fv belongs to

A

² =

A

² (R^1, Q ) . Then we have·

where k (

<

5) is a constant depending only on the space.

K. ANZAI, K. RORIE and S. KOIZUMI

If, in a sequence of continuous functions { gm

l ,

each function is a contraction of senes

L:= ^{/v ,}

then the assumption

implies

Where M (g) denotes the supremum norm of g.

Next let us consider the family Q1 defined as the subset of Q consisting of functions :with the property

w

(0) =limw (x)

<

^oo.

x--+O

The norm in Q ₁ will be defined as

N(w)= w (O)+ JRnwdx.

We can define the Banach algebra d ₁²

= d/

^{(R\ Q} 1) •

We shall denote by B² = B² (Rn) and by B/ = B/ (Rn) the spaces of measurable functions on Rn which are L² over compact sets and have the norms

l

J

l<1>l

2

dx

l½' II

<I>

II

₈² = sup _lxl_:::;_r - - -

r >o Vn(r)

where Vn (r) denotes the volume of the sphere

I

x

I

~ r in Rn.

Theorem IV. B² = B² (Rn) is the conjugate space of the algebra A² =A² (R\ Q).

Similarly, B/ = B/ (Rn) is the conjugate space of the algebra

d/

⁼

d/

^{(Rn, Q} 1) .

Theorem V. The space .s#₁² = d ₁² (Rn, Q ₁₎ is the intersection of A² = A² (R^{1, Q} 1) and

L² (R^{1) ,} and the norms in these spaces satisfy the inequalities

(1.2)

II

Flld~>

IIFIIA

² (1.3)

IIFltr,i1~> IIFllt

(1.4)

IIFll.ss1:< IIFIIA

²

+ IIFllt

.

On the Beurling Convolution Algebra III

2. The class of functions on

Rn

The purpose of this paper is to extend results of Beurling to the class of functions on n -dimensional euclidean space Rn.

Let us remark that A ² (Rn) c L ¹ (Rn) by the inequality

and so the Fourier transform of F E A ² (Rn) is well defined.

Let us define the normed ring

A

²

={

^{f:f=F, FEA2 }}

with norm

Let us also define the notation as follows

where l1 ~f is the vector difference along a , that is

and

( ^{A (f)}

⁼ J

^{Rn 7] (a. f)}

^I

^{a da 13n/2 •}

In the preceding paper [1] , we proved the following theorem.

Theorem 1. A functionfbelongs to the normed ring

A

² (Rn) if and only if the following conditions are satisfied

(a) fis continuous, (c) limf (t)

=

0,

lrl-⁰⁰

(c) A (f)

<

^oo.

K. ANZAI, K. HORIE and S. . KOIZlJMI

Under these conditions, f is the Fourier transform of FE A ² (Rn), and the following inequalities hold

provided f

*

0, where en and dn are positive constants which are independent off

It is convenient to give at this instance some accounts of their basic properties, for the sake of completeness and the reasons of indispensable for further studies. These can be done by running the same lines as A.Beurling [ 4] . ·

( 1) The normed ring

A:2

(Rn) is complete and separable.

Since A.Beurling [4] proved that in Theorem I that the

1I2

(Rn) is the Banach algebra under addition and convolution. According to these results, the

A

² (Rn) is complete normed ring under pointwise addition and multiplication. Furthermore it is clear that

A

² (Rn) is separable and so

A

² (Rn) does too.

(2) The normed ring

A

²(Rn) satisfies the principle uniform contraction with certain constant.

We shall generalize Theorem III to functions on n-dimensional euclidean space Rn and this is the theorem in the preceding paper [1] without proof.

Theorem 2. Let g be a continuous function and a contraction of the series

L~

⁼ 1 fv , where each fv belongs to

A

² (Rn). Then we have

where constants en and dn are those of Theorem 1.

If, in a sequence of continuous functions j gm}, each function is a contraction of senes

L~=

1 fv , then the assumption

lim M(gm)

=

0

nz-HX>

implies

lim

II

gm

llx2 =

0

m-+oo

On the Beurling Convolution Algebra III

This theorem is the essential tool for further applications and so we shall give its proof.

Proof. Let us suppose that g be a contraction of the series

L:

^{= 1} fv · , fv E

A

² (Rn) . That is, for any t and a , the following properties

(2.1) and

(2.2) satisfied.

Then we have by the inequality (2.1) , lim g (t)

=

0

I tJ--->co

and by the inequality (2.2) ,

IJ (a,g)

< L:=

1 1J(a, fv) and so

A (g)

< L:=1

^{A (fv) ·}

Since we suppose that g (t) is continuous, we can conclude that g E

A:

² (Rn) by the Theoreml and we have

Next let us suppose that lim M(gm)

=

0, then we have

m---+oo

while

for fixed a and t. Hence by the Lebesgue theorem of dominated convergence, lim IJ (a, gm)

=

0 .

m---+oo

Similarly, we have

K. ANZAI, K. HORIE and S. KOIZUMI

11

_I

(a,gm)

< "N

^{11 (a,fv) 2 ( n)}

a 13n/2 - L.v=1

I

^a 13n/2 E L R

and we may apply the s~me theorem to the integral A (gm) obtaining Iim A (gm)

=

O and then by Theorem 1 we have lim

llgm

IIJ2

=

0.

m~oo m~oo

Thus we proved the theorem.

(3) The

A

² (Rn) satisfy the following property (2.3) ^sup

lg(t) I =

1,

llgllA,::; 1

II

g IIA2

(YtEK).

According to A.Beurling [4] , this proved as follows. Let us set G(x)

= /oX

^w0 (x) for any t0 E Rn with normalizedw₀ E Qand its Fourier transform

Then we have

and

llgllJ, = llall!,=Y!~f

w (x)dx

f ll~II

2

t1x

IGI

²

< f

^Wo (x) dx

f w-;

^dx

⁼ (f

^{w O dx) 2}

=

1.

On the other hand, we have

lg(to) I

<M(g)<

Jlcldx< IIGIIA2 llgll.A2·

Therefore we have

I

g (tJ

I =

I[ g

Ilk=

1 which shows the property (2.3) at t

= to .

We observe that at once (2.3) imply (2.4)

But this proved immediately by the definition of

A:

² (K)

We shall generalize Theorem V to functions on n-dimensional euclidean space Rn and this is the theorem in the preceding paper [1] without proof.

On the Beurling Convolution Algebra III

Theorem 3. The space

d/ ⁼

^.s;/

1 2

_{(Rn, Q}

1)

is the intersection of A

2 ₌

_A

2

_{(Rn, Q) and}

L ² (Rn), and the norms in these spaces satisfy the inequalities (2.5)

IIF IL?/~> IIF

IIA²

(2.6)

IIF lld:> IIF llt

(2.7)

IIFlld~< IIFIIA

²

+ IIFllt

Proof. The relation (2.5) is a consequence of the fact that Q 1 is subset of Q . As to the proof of (2.6) we recall that

2 l_

IJFIIA,

=

L~ ( (J

^RnW

(IX I

)dx)J R"

~(1:i')

^{dx )'}

I

. (( f I )J IF(x)

12

)2

~ l~t

^{w (0)}

⁺

^{RnW ( Ix )dx Rn W}

^(lxl)

^dx

By Theorem N, for

CD

^E

Bi

(Rn)=

(d/

(Rn))*,

II <I>IIB1=~~p(f

¹

·\~::(~)l

²

dx) ^{s (f,,,,} I <I>(x) l

²

dx)L II <I>I\L,

Hence, on taking

CD = F,

IIF t! = f

R"FCDdx

~ IIF lid: II

CD

IIB;< IIF

lid;

II

CD

t2,

and (2.6) follows.-For FE L² (Rn)

nA

² (Rn) We set a= ~FI\L,, b=

(f

^{Rn 1~ 12dx}

t

where w ^{E Q} is normalized and bis a number close to

IIF IIA2.

Clearly

w

1

(

Ix I)

== ab w (

Ix I)

a+bw(lxl)

belongs to Q 1 since w ^{1 (}

Ix I)

is nonincreasing function of

Ix I .

We have

N(w)

=

W1 (0)

+

tnwl

(lxl)dx

abw (0)

J

^{abw (}

lxl)

< - - - - + - - - - d x < a + b - a+bw(0) ^Rn a+bw(lxl)

K. ANZAI, K. HORIE and S. KOIZUMI

Thus we obtain that

1

IF

(x)

I

1 ^{2 · 2}

(

2 )

< (

^a

+

b)

bf

^{Rn W ( Ix}

I )

^dx

+ ; f

^{Rn IF (x)}

I

^dx

= (

a

+

b) which proves

That the strict inequality actually holds for F* 0 is easily established.

By inequality (2.5) and (2.6) ,

d ₁² (Rn) CA²(Rn)

nL

²(K).

By inequality (2.7),

Thus Theorem 3 is proved.

References

[1] K.Anzai, K.Horie and S. Koizumi, On the A. Beurling convolution algebra, Tokyo J.

Math. 19 (1996), 85-98.

[2] K.Anzai, K.Horie and S. Koizumi, On the Beurling convolution algepra II, Tokyo J.

Math. 26 (2003), 275-300.

[3] A.Beurling, On the spectral synthesis of bounded functions, Acta Math. 81 ( 1949) 225-238.

[ 4] A.Beurling, Construction and analysis of some convolution algebras, Ann. Inst. Fourier, 14 (1964), 1-32.

[5] A.Beurling, Co'ilectedWorks of Arne Beurling, Vol.2, Birkhauser (1989).

On the Beurling Convolution Algebra III

Present Addresses

KazuoANZAI

Department of Mathematics, Kagawa University, Saiwai-cho, Takamatsu, 760-8522, Japan

Kenji HORIE

Department of Mathematics, Takamatsu National College of Technology, Chokushi-cho, Takamatsu, 761-8058, Japan

Sumiyt;tki KQIZUMI

Department of Mathematics, Faculty of Sciences and Technology, Keio University, Hiyoshi, Kohoku-ku, Yokohama, 223-8522, Japan