On Convolution Operators in the Spaces of Almost Periodic Functions and Lp Spaces

On Convolution Operators in the Spaces of Almost Periodic Functions and

^Lp

Spaces

Giordano Bruno

Dipartimento MeMoMat

Universita di Roma \La Sapienza\' Italia e-mail:

bigi@dmmm.uniroma1.it

Alexander Pankov

Department of Mathematics

Vinnitsa State Pedagogical University, Ukraine e-mail:

pankov@apmth.rts.vinnica.ua

November 3, 1998

Abstract.

We consider operators of convolution generated by ^L1 func- tions in ^Lp and various spaces of almost periodic functions. It turns out to be that if such an operator is invertible in one of these spaces, then it is in- vertible in all the spaces we consider. Further, we prove that any convolution has identical norms in many natural couples of functional spaces.

Key words:

convolution operators, almost periodic functions

AMS subject classication:

47G10

1 Introduction

We consider convolution operators acting in the spaces ^Lp and the spaces of Bohr, Stepanov, and Besicovich almost periodic (a.p.) functions. We prove that in all the spaces we consider the invertibility for convolutions take place, or do not take place, simultanuously.

Next, we study norms of convolutions in natural couples of functional spaces: ^Lp and ^Bp, ^BSp and ^Sp, ^Cb and ^CAP. We prove that convolution operators have identical norms in each of two members of any such couple.

Our study is motivated by the results of M. Shubin 1] and 2]. In those papers the results on norms and invertibility (spectra) were proved for a wide class of a.p. pseudo dierential operators, but only in the^L2-^B2 setting. The case ^p6= 2, as well as the case of Stepanov spaces, are not considered there.

We attempt here to enlarge the range of spaces in which such \coincideness"

results take place. However, since ^Lp-theory and, all the more, ^Bp- and ^Sp- theories of pseudo dierential operators are not well-developed, we restrict ourself to the case of convolutions only. Even in this case a rich picture appears. We remark also that ^Lp-^Bp setting was studied in 3] and 4] for nonlinear dierential operators. However, in these works the value of ^p de- pends on the structure of operators under consideration and plays the same r^ole as ^p= 2 in the linear theory.

In the present paper we deal with convolutions dened on the real line only. Nevertheless, all the results and techniques may be extended to the case of convolutions on ^Rn without any diculties.

2 Preliminaries

We use the standard notations ^Lp (resp. ^{Lpl oc}), 1 ^{p 1}, for the (resp.

local) Lebesgue spaces of complex valued measurable functions on the real line ^R. Let ^{' 2 L1} and ^{2 C}. It is well-known 6] that the operator ^A dened by the formula

Au =^u+^'u (1)

wherestands for the convolution operation, acts continuously in each space

L

p, 1^p1. All such operators form an algebra which is denoted by ^A. Now we introduce other spaces in which operator (1) will be considered.

Let us denote by ^Cb the closed subspace of ^L1 formed by all bounded con- 1

tinuous functions. Futhermore, for any^p1 let us denote by ^BSp the space of all functions ^{f 2Lpl oc} for which

kfk

S

p = sup

t2R Z

t+1

t

jf(^x)^jpdx

1=p

<1: (2)

This is the space of Stepanov bounded functions (with exponent ^p). The closure in ^BSp of the set of all trigonometric polynomials

X

finite a

ke^{ikx ak2C k 2R}

is denoted by^Sp. It consists of all Stepanov a.p. functions (with the exponent

p). Similarly, the closure of all trigonometric polynomials in the space ^Cb consists of all Bohr a.p. functions. The last space is denoted by ^CAP.

For any ^{f 2CAP} it is well-dened the mean value

M ffg = lim

T!1

21^T

Z

T

;T

f(^t)^dt:

As a consequence, for any ^{f 2CAP} one can dene the norm

kfk

B

p =^Mfjfjpg1=p:

We dene the space ^Bp of Besicovich a.p. functions (with the exponent ^p) as the completion of^CAP with respect to the norm ^kkBp.

For more details on a.p. functions we refer to 4], 5], 7] and 8]. We remark only that all the spaces just introduced are Banach spaces and the following continuous and dense embeddings

C

b

BS p

p1

CAP S p

S q

CAP B p

B q

pq

take place.

Now we show that the algebra ^A acts naturally in all spaces we intro- duced.

Lemma 1

^{Let A2A. Then}

2

1)

^A acts continuously in ^Cb

2)

^A acts continuously in ^CAP.

Proof. The rst part of the statement is well-known. Let us prove the second one. Since ^CAP is a closed subspace in ^Cb, one need only to show that ^Au2CAP provided ^u2CAP. By the Bochner criterion,^u2CAP i the family ^fu(+^y)^gy2R is precompact in^Cb. Now let ^u2CAP. Since^A is translation invariant, we have

(^Au)(+^y) =^Au(+^y)] ^y2R:

By continuity of^A in^Cb, we see that the family^f(^Au)(+^y)^gy2R is precom- pact. ²

Lemma 2

Any operator ^{A2 A} acts continuously in ^BSp, 1 ^p<1, and leaves the subspace ^Sp invariant.

Proof. Without loss of generality we can assume that = 0. Let ^{u 2}

BS

p, ^{p >} 1, and ^p0 the dual exponent, 1^=p0 + 1^=p = 1. Using the Holder inequality, we have the following inequalities

Z

+1

Z

R

'(^t)^u(^x;t)^dt

p

dx Z

+1

Z

R

j'(^t)^1=p0j'(^t)^j1=pju(^x;t)^jdt

dx

Z

+1

Z

R

j'(^t)^dt

1=p 0

Z

R

j'(^t)^jju(^x;t)^jpdt

dx

C Z

+1

Z

R

j'(^t)^jju(^x;t)^jpdt

dx=

=^C

Z

R

j'(^t)^j

Z

+1

ju(^x;t)^jpdx

dtCk'k

L 1 kuk

p

S p

where ^{C >}0 does not depend on ^u. Thus,

kAuk

S

p Ckuk

S p

and the operator ^A acts continuously in ^BSp, ^p>1.

The case ^p= 1 may be considered in the similar way it is even simpler.

The proof of the second part of the lemma may be carried out exactly as in Lemma 1, using the version of Bochner criterion for Stepanov a.p. functions.

2.

3

Lemma 3

^{Let A2Aand p}1. Then there exists a constant^C =^Cp(^A)^>

0 which depends only on ^A and ^p such that

kAuk

B p

Ckuk

B p

for all ^u2CAP.

Proof. It goes along the same lines as in the proof of the rst statement of Lemma 2. ²

Lemma 3 permits us to extend, by continuity, any operator ^A2A to an operator acting in ^Bp. Such the extension is still denoted by ^A.

Remark 1

It is easy to see that the norms of all the operators we consider are estimated by ^Ck'kL1.

Remark 2

It is well-known that, given ^{A 2 A}, the adjoint operator ^A in the sense of the scale ^Lp also belongs to ^A. Moreover, a simple calculation show us that the extension of ^A to ^Bp is in fact the adjoint operator of ^A in the sense of the scale ^Bp.

3 Invertibility of Convolutions

To any operator ^{A 2 A} of the form (1) one can associate its symbol ^a() dened by the formula

a() =+ ^^'() (3) where ^^' is the Fourier transform of ^'. It is well-known (see, e.g., 9]) that

a() is a continuous function on ^R and lim

!1

a() = ^:

Therefore, ^a() may be regarded as a continuous function on ^R, one-point compactication of the real line. The set ^A is a commutative algebra with the natural involution ^{A 7! A}. The map ^{A 7! a}() is a homomophism

A;!C(^R) of algebras with involution. Here the algebra^C(^R) of continuous functions on ^R is endowed with the natural involution, complex conjugation.

Theorem 1

^{Let A2A}. The following statements are equivalent:

4

(ⁱ) ^a() is nowhere vanishing on ^R

(ⁱⁱ⁰) ^A is invertible in ^Lp0 for some ^{p0 2}1¹) (ⁱⁱ) ^A is invertible in ^Lp for all ^p21¹) (ⁱⁱⁱ⁰) ^A is invertible in ^Bp0 for some ^{p0 2}1¹) (ⁱⁱⁱ) ^A is invertible for all ^p21¹)

(^iv) ^A is invertible in ^Cb (^v) ^A is invertible in ^CAP

(^vi0) ^A is invertible in ^BSp0 for some ^{p0 2}1¹) (^vi) ^A is invertible in ^BSp for all ^p21¹) (^vii0) ^A is invertible in ^Sp0 for some ^{p0 2}1¹) (^vii) ^A is invertible in ^Sp for all ^p21¹).

Proof. Assume that ^a() ⁶= 0 for all ^{2 R}. By the classical Wiener theorem the function ^b() = 1^=a() is of the form

b() = 1

+ ^^'() where ^'2L1. Hence, the operator

Bu= 1

u+^'u

acts in all the spaces we consider and is the inverse to ^A. Thus, (ⁱ) implies all other statements listed above. Moreover, it is easy to see that (ⁱⁱ⁰) with

p

0 = 1 implies (ⁱ).

Now let us suppose that (ⁱⁱ⁰) is fullled. Without loss of generality, we may assume ^{p0 >}1. Consider the operators

Tu= ^u and (^J(^u))(^x) =^u(^;x)^:

5

It is obvious that ^T2 = ^J2 = ^I, ^J is linear and ^T is antilinear. A direct calculation shows that

A=^JTATJ: (4)

Since ^A is invertible in ^Lp00, ^p00 being the dual exponent, (4) implies that

A is invertible in ^Lp00. By the Riesz{Thorin theorem, ^A is invertible in ^L2. Since

(^Auc)() =^a()^^u() we have proved (ⁱ).

The rest of the proof is simple. For example, assume (^v) to hold. Let

e

(^x) = exp(^ix) ^2R:

Then ^Ae = ^a()^e and ^kekCAP = 1. If there exists a bounded inverse operator ^A;1, then we easily have

1^a()^{kA;1k 2R:}

By continuity of^a(), the same inequality holds for all^2R. Hence,^a()⁶= 0 for all ^2R and (^v)⁾ (ⁱ).

The same argument proves that each of the statements (ⁱⁱⁱ⁰), (^iv), (^vi0), (^vii0) implies (ⁱ) and the proof is complete . ²

Remark 3

The implications (ⁱ)^,(ⁱⁱ⁰)^,(ⁱⁱ) are well-known (see e.g. 10].

We have included here the proof for the sake of completeness.

4 Norms of Convolutions

Now we want to study connections between norms of convolution operators acting in various function spaces. As usual, ^L(^E) stands for the space of all bounded linear operators in a Banach space ^E.

We start with the spaces ^BSp and ^Sp. We need the following additional continuity property of convolutions.

Lemma 4

^{Let A 2 A. If uk} is bounded in ^BSp and ^{uk ! u} in ^{Lpl oc}, then

Au

k

!Au in ^{Lpl oc}.

6

Proof. Without loss of generality, we may assume that = 0 and ^u= 0.

Fix ^{x0 >}0. By assumption,

Z

x

0

;x0 ju

k(^x;t)^jpdxC (5) where ^{C >} 0 is independent on ^k and ^t. As in the proof of Lemma 2, we have

Z

jxjx

0 jAu

k(^x)^{jpdx C}

Z

jxjx

0 Z

R

j'(^t)^jjuk(^x;t)^jpdt

dx:

Then, for any ^{t0 >}0,

Z

jxjx0 jAu

k(^x)^{jpdx C}

Z

jtjt0

+

Z

jtjt0

j'(^t)^j

Z

jxjx0 ju

k(^x;t)^jpdx

dt=

= ^I1 +^I2: Let ^">0. Due to (5),

I

1 C

Z

jtjt

0

j'(^t)^jdt and ^{I1 "} if ^t0 is choosen large enough. Now

I

2

C

Z

jtjt

0

j'(^t)^jdt

Z

jyjx

0 +t

0 ju

k(^y)^jpdy

Ck'k

L 1

Z

jyjx0+t0 ju

k(^y)^jpdy:

Since ^{uk !} 0 in ^{Lpl oc}, we see that ^{I2 "} if ^k is large enough. The proof is complete. ²

Remark 4

It is easy to verify that if ^uk is bounded in ^BSp and ^{uk !u} in

L p

l oc, then ^u2BSp and

liminf^{kukkSp kukSp:}

Theorem 2

^{Let A2A}. Then for any ^p21¹)

kAk

L(BS p

) =^kAjSpkL(Sp): (6) 7

Proof. Let ^aand ^b stand for the left- and right-hand sides of (6), respec- tively. Since ^Sp is a closed subspace of ^BSp, we have ^{b a}, and we need only to prove that

ab: (7)

By denition,

a= sup^fkAukSp :^u2BSpkukSp 1^g

b = sup^fkAvkSp :^{v 2SpkvkSp} 1^g:

Let^u2BSp. Given ^{T >}0, let^uT be the 2^T-periodic extension of^uj;TT]to

R. It is easy to verify that

kuk

S

p = lim

T!1 ku

T k

S p

kuk

S p

kuk

S p

1^:

Also it is obvious that ^{uT ! u} in ^{Lpl oc}, as ^{T ! 1}. Since ^A is a bounded operator, we see that ^AuT is uniformly bounded in ^BSp and 2^T-periodic (hence, a.p.). By Lemma 4, ^{AuT !Au} in ^{Lpl oc}. By Remark 4,

liminf

T!1 kAu

T k

S p

kAuk

S p:

This implies (7), and we conclude. ²

Remark 5

A similar approach was used in the proof of Proposition 2.2 in 11].

In the same way, using uniform convergence on compact sets instead of

L p

l oc-convergence, one can prove the following

Theorem 3

^{Let A2A. Then}

kAk

L(C

b

)=^{kAjCAPkL(CAP)}=^kAjC0kL(C0)

where ^C0 =^fu2Cb : lim^x!1u(^x) = 0^g is the closed subspace of ^Cb. Now we consider the spaces ^Lp and ^Bp.

Theorem 4

^{Let A2A. Then}

kAk

L(L p

)=^kAkL(Bp): (8) 8

Proof. First we give a separate proof for the case ^p= 1, since it claries the duality between this case and Theorem 3. Then we consider the general case. It is well-known (see e.g 6] or 12]) that

kuk

L

1 = sup

j(^uv)^j

kvk

L 1

:^{v 2 L1v 6}= 0

:

By the classical Lusin Theorem (6], 12]) one can replace here ^L1 by ^Cb. Therefore,

kAk

L(L 1

) = sup

j(^Auv)^j

kuk

L 1kvk

C

b

:^{u2L1v 2Cbu6}= 0^{v 6}= 0

: (9) Now we recall that ^B1 and ^CAP are naturally isomorphic to ^L1(^RB) and

C(^RB) respectively, where ^RB is the so-called Bohr compactication 4], 5]

and ^L1(^RB) is regarded with respect to the Haar measure on^RB. Hence, we have, as above,

kAk

L(B 1

)= sup

j(^Auv)^Bj

kuk

B 1kvk

CAP

:^{u2B1v 2CAPu6}= 0^{v 6}= 0^:

(10) Since

(^Auv) = (^uAv) (^Auv)^B = (^uAv)^B we conclude that

kAk

L(L 1

) =^kAkL(Cb)

kAk

L(B 1

) =^kAkL(CAB) and Theorem 3 implies the required.

Now we consider the general case assuming for denitness that ^{p >} 1.

For the sake of brevity, let us denote by ^aand^bthe left- and right-hand sides of (8), respectively. We have

a= sup

kAuk

L p

kuk

L p

:^u2Csupp^{uRu 6}= 0

b = sup

kAvk

B p

kvk

B p

:^{v 2CAPv 6}= 0

:

9

Recall that "^{X Y}" means that^X is a compact subset of ^Y.

Since compactly supported continuous functions are dense in ^L1, we can choose a sequence ^'n of such functions such that^{'n !'}in^L1. By Remark 1, ^{kAnkBp ! kAkBp} and ^{kAnkLp ! kAkLp}, where ^An stands for the oper- ator generated by ^'n. Therefore, we can assume that the kernel ^' itself is continuous and compactly supported.

Now let ^u2C and supp ^u;T0T0]. For ^{T T0} we denote by ^uT the 2^T-periodic extention of ^uj;T0T0] to^R, and set

u

T = (2^T)^1=puT: It is easy to verify that

ku

T k

B

p =^kukLp: (11)

Since ^'is compactly supported, we see that

A(^uT) = (^Au)^T provided ^T is large enough. Due to (11) we have

kAu

T k

B

p =^kAukLp (12)

for such ^T.

Now using (11) and (12), we have

kAuk

L

p =^{kAuTkBp bkuTkBp bkukLp} for ^T being large enouph. This implies that ^ab.

Now let us prove that ^{b a}. For any ^{T >}0, let ^T be the characteristic function of the interval ^;TT]. For ^{v 2CAP} it is easy to see that

lim

T!1

21^{Tk TvkpLp} =^kvkpBp: (13) We want to show that

lim

T!1

21^TkA( ^Tv)^kpLp =^kAvkpBp: (14) In view of (13), to do this it suces to prove that

lim

T!1

21^{Tk TAv;A}( ^Tv)^kpLp = 0^: (15)

10

First of all, we observe that the operator

B

T = ^{TA;A T} is an integral operator with the kernel

'

T(^xt) = ^T(^x)^'(^x;t)^;'(^x;t) ^T(^t)^:

Moreover, ^BT acts in ^L1 and is uniformly (with respect to ^T) bounded in this space. In particular, we see that for ²(01)

lim

T!1

21^T

Z

T;T jxjT+T jB

T

v(^x)^jpdx= 0 (16) since the domain of integration has Lebesgue measure of order^T.

Now we have

Z

jxjT+T jB

T

v(^x)^jpdx

Z

jxjT+T dx

Z

jtjT

j'(^x;t)^jjv(^t)^jdt

p

:

Here ^{jx;tj T}. Since the support of ^' is compact, we see that in the right-hand side ^'(^x;t) = 0 if^T is large enough. Hence,

21^T

Z

jxjT+T jB

T

v(^x)^jpdx= 0^: (17) Similarly,

21^T

Z

jxjT;T jB

T

v(^x)^jpdx= 0^: (18) Puting together (16){(18), we get (15) and, hence, (14).

Now

kAvk p

B

p = lim

T!1

21^TkA( ^Tv)^{kpLp ap} lim

T!1

21^{Tk TvkpLp} =

= ^apkvkpBp: Therefore, ^{b a}. Hence, ^a=^b. ²

Acknowledgements

. The authors thank A. Avantaggiati for helpful dis- cussions. The paper was started during the visit of the second author (A.P.)

11

to the Dipartimento MeMoMat, Univ. "La Sapienza" in May, 1998, and completed during the visit of A.P. to the Institut fur Mathematik, Humboldt Universitat (September | December, 1998) under support from Deutsche Forschungsgemeinschaft. A.P. is very grateful to the members of MeMoMat and of the Institut fur Mathematik, Humboldt Universitat, for their kind hospitality. The work of A.P. is also partially supported by the Ukrainian Fundamental Research Foundation, grant 1.4/44.

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