A result of Strichartz on a generalization of Wiener’s characterization of continuous

A result of Strichartz on a generalization of Wiener’s characterization of continuous

measures revisited

B´ erenger Akon KPATA

Laboratoire de Math´ ematiques et Informatique, UFR des Sciences Fondamentales et Appliqu´ ees, Universit´ e Nangui Abrogoua,

02 BP 801 Abidjan 02, Cˆ ote d’Ivoire kpata akon@yahoo.fr

Received: 19.8.2014; accepted: 18.5.2015.

Abstract. We give the reasons for which the continuous part of a measure that belongs to the Wiener amalgam space M

²

does not contribute to the Bohr mean of its Fourier transform.

Keywords: Fourier transform, Radon measure, Wiener amalgam space MSC 2010 classification: primary 42B10, secondary 42B35

1 Introduction

In this paper, for 1 < p < ∞, we denote by p ⁰ the real number satisfying

1

p + _p ¹

0

= 1. A Radon measure µ on R ^d belongs to the Wiener amalgam space M ^p , (1 ≤ p < ∞) if ₁ kµk _p < ∞ with

r kµk _p =



 X

k∈Z

^d

|µ|(I _k ^r ) ^p





1 p

, r > 0

where I _k ^r = Q d

i=1 [k _i r, (k _i + 1)r) for k = (k ₁ , . . . , k _d ) ∈ Z ^d and |µ| denotes the total variation of µ.

The Fourier transform on amalgam spaces has been studied by various authors including F. Holland ([8], [9]), J. Stewart [11], J. P. Bertrandias and C. Dupuis [2], J. J. F. Fournier ([6], [7]) and I. Fofana [5].

In [8], the Fourier transform f 7→ b f defined on the usual Lebesgue space L ¹ by

f (x) = (2π) b ⁻

^d2

Z

R

^d

f (y) e ^−ixy dy, x ∈ R ^d

http://siba-ese.unisalento.it/ c 2015 Universit` a del Salento

has been extended by F. Holland to the spaces (L ^q , l ^p ) defined for 1 ≤ q, p ≤ ∞ as follows:

(L ^q , l ^p ) = n

f ∈ L ⁰ | ₁ kf k _{q, p} < ∞ o

where L ⁰ stands for the space of (equivalence classes modulo the equality Lebesgue almost everywhere of) all complex-valued functions defined on R ^d and for r > 0,

r kf k _{q, p} =



 

 



 

 





 X

k∈Z

^d

 f χ I

_k^r

q

 p





1 p

if 1 ≤ p < ∞, sup

x∈R

^d

f χ _J

_x^r

q if p = ∞,

where J _x ^r = Q d

i (x i − ^r ₂ , x i + ^r ₂ ) for x = (x 1 , . . . , x d ) ∈ R ^d , χ I

^r_k

denotes the characteristic function of I _k ^r and k · k q is the usual Lebesgue norm. In the same paper, he extended to the spaces M ^p (1 < p ≤ 2) the Fourier transform µ 7→ µ b defined on the space M ¹ of finite Radon measures on R ^d by

µ(x) = (2π) b ⁻

^d2

Z

R

^d

e ^−ixy dµ(y), x ∈ R ^d .

In fact, he proved that if µ belongs to M ^p (1 < p ≤ 2), then there exists a unique element µ ∈ (L b ^p

⁰

, l ^∞ ) such that for any sequence (r n ) _n≥1 of positive real numbers increasing to ∞, the sequence ( \ µbJ ₀ ^r

ⁿ

) _n≥1 converges in (L ^p

⁰

, l ^∞ ) to b µ, where µbJ ₀ ^r

ⁿ

is the measure defined by (µbJ ₀ ^r

ⁿ

) (N ) = µ (J ₀ ^r

ⁿ

∩ N ) for any Borel subset N of R ^d . In addition,

Z

R

^d

g (x) µ (x) dx = b Z

R

^d

b g(x) dµ (x) , g ∈ L ^p , l ¹
. In [5], I. Fofana has proved the following Hausdorff-Young inequality :

r ⁻

^p0d

_r k µk b _p

0

, ∞ ≤ C

1

r

kµk _p , r > 0 (1.1) where the real constant C does not depend on µ and r.

The relation between the properties of a Radon measure and the asymptotic behavior of its Fourier transform is an important topic in Harmonic Analysis.

A well-known theorem of Wiener [14] states that if µ ∈ M ¹ , then

r→∞ lim r ^−d Z

J

₀^r

| µ(y)| b ² dy = X

a∈D

|µ ({a})| ² , (1.2)

where D = { a ∈ R ^d | µ({a}) 6= 0 } is a countable set.

The left-hand side of (1.2) is the so-called Bohr mean of µ. Wiener’s result b has found many applications in various areas of mathematics, including ergodic theory and optimal control theory. Since then, Wiener-type characterizations of continuous measures are extensively investigated (see for example [1], [3], [4], [12] and [13, Section 12.5]). Strichartz has established that equality (1.2) continues to hold if the measure µ belongs to M ² (see Theorem 4.4 in [12]).

The aim of this note is to give, by a new proof, the reasons for which the continuous part of a measure belonging to M ² does not contribute to formula (1.2). To this end , we prove in Section 2 the following result: if µ is a continuous measure that belongs to M ^p , then

r→0 lim r kµk _p = 0.

In Section 3, we show that if µ is a discrete measure that belongs to M ^p , then

r→0 lim r kµk _p = X

a∈D

|µ ({a})| ^p

!

¹

p

.

Since any Radon measure on R ^d has a decomposition into discrete and continu- ous parts, we combine these above results with inequality (1.1) to establish, in Section 4, the generalization of Wiener’s theorem obtained by Strichartz.

2 M ^p -estimate of a continuous measure

Let us recall the definition of a continuous Radon measure.

Definition 1. A Radon measure µ on R ^d is continuous if for any x ∈ R ^d we have µ({x}) = 0.

Notice that for any Radon measure µ on R ^d and any x ∈ R ^d , we have

|µ({x})| = |µ|({x}). Therefore, a Radon measure on R ^d is continuous if and only if its total variation is continuous.

The following proposition will be useful in the proof of the main result of this section.

Proposition 1. [10]. Let 1 ≤ p < ∞. If µ ∈ M ^p and 0 < r < s < ∞, then (i) _s kµk _p ≤ Int _r ^s
+ 2

^p0d

2

^dp

_r kµk _p , where Int ^s _r

denotes the greatest integer not exceeding _r ^s ;

(ii) r kµk _p ≤ 3

^p0d

2

^dp

s kµk _p .

We can now prove the following result.

Proposition 2. Suppose that 1 ≤ p < ∞ and µ is a continuous measure that belongs to M ^p . Then we have

(i) lim r→0 sup _k∈Z

^d

|µ|(I _k ^r ) = 0, (ii) p > 1 =⇒ lim r→0 r kµk _p = 0.

Proof. For each non-negative integer n, set s n = sup

k∈Z

^d

|µ|(I _k ²

⁻ⁿ

).

(i) Notice that (s _n ) _n is a decreasing sequence of positive real numbers. Suppose that lim n→∞ s n = s > 0. For each non-negative integer n, let

K n = n

k ∈ Z ^d | |µ|(I _k ²

⁻ⁿ

) > s 2

o . Let us notice that for each positive integer n,

0 < card K _n ≤  2

s ²

⁻ⁿ

kµk _p

 p

≤  2

s ¹ kµk _p

 p

< ∞ and

k ∈ K n+1 =⇒ ∃ l ∈ K n : I _k ²

⁻⁽ⁿ⁺¹⁾

⊂ I _l ²

⁻ⁿ

,

where card K _n denotes the cardinality of K _n . So there exists a sequence (k _n ) _n of elements of Z ^d such that for each non-negative integer n we have

k n ∈ K _n and I _k ²

_n+1⁻⁽ⁿ⁺¹⁾

⊂ I _k ²

_n⁻ⁿ

. It follows that ∩ n I _k ²

⁻ⁿ

n

is a one point set and |µ|(∩ n I _k ²

⁻ⁿ

n

) > 0. This is in contra- diction with the fact that µ is a continuous measure. So

n→∞ lim s _n = 0.

Now let us consider a real number ε > 0. There exists an integer n ₀ such that s n

0

< ₂ ^ε

d

. Let us notice that for each element (r, k) of (0, 2 ⁻ⁿ

⁰

) × Z ^d , the set I _k ^r intersects at most 2 ^d elements of { I _l ²

⁻ⁿ⁰

| l ∈ Z ^d }. So

|µ|(I _k ^r ) ≤ 2 ^d s n

0

, (r, k) ∈ (0, 2 ⁻ⁿ

⁰

) × Z ^d . It follows that

sup

k∈Z

^d

|µ|(I _k ^r ) < ε, r ∈ (0, 2 ⁻ⁿ

⁰

).

Therefore,

r→0 lim sup

k∈Z

^d

|µ|(I _k ^r ) = 0.

(ii) Suppose p > 1. Let us consider a real number t > 0. There exists a finite subset L of Z ^d such that

X

k∈Z

^d

\L

|µ|(I _k ¹ ) ^p < t.

Let us set

E = [

k∈L

I _k ¹

and for each non-negative integer n, L _n = n

k ∈ Z ^d | I _k ²

⁻ⁿ

⊂ E o

, α _n = X

k∈L

n

|µ|(I _k ²

⁻ⁿ

) ^p and t _n = X

k∈Z

^d

\L

n

|µ|(I _k ²

⁻ⁿ

) ^p .

Then, for each non-negative integer n,

2

⁻ⁿ

kµk _p = (α _n + t _n )

^1p

, α _n ≤ s ^p−1 _n |µ|(E) and t _n ≤ t.

Let us notice that

|µ|(E) ≤ card L ₁ kµk _p < ∞

(where card L denotes the cardinality of L) and ( ₂

⁻ⁿ

kµk _p ) _n is a decreasing sequence. It follows that

n→∞ lim 2

⁻ⁿ

kµk _p ≤ t

^p1

.

Since the above inequality holds for an arbitrary real number t > 0, we deduce that

n→∞ lim 2

⁻ⁿ

kµk _p = 0.

Further, for each non-negative integer n, it follows from Proposition 1 (ii) that

0 ≤ r kµk _p ≤ 3

^p0d

2

^dp

₂

⁻ⁿ

kµk _p , r ∈ (0, 2 ⁻ⁿ ).

So

r→0 lim r kµk _p = 0.

QED

3 M ^p -estimate of a discrete measure

This section is devoted to the proof of the following result.

Proposition 3. Suppose that 1 ≤ p < ∞ and µ is a discrete measure that belongs to M ^p . Then

r→0 lim r kµk _p = X

a∈D

|µ({a})| ^p

!

¹_p

,

where D = { a ∈ R ^d | µ({a}) 6= 0 }.

Proof. Let us consider a real number ε > 0. There exists a finite subset K of Z ^d such that



 X

k∈Z

^d

\K

|µ| I _k ¹
p





1 p

< ε 2

^dp

.

Set

E = [

k∈K

I _k ¹ , D n =



a ∈ D | |µ({a})| ≥ 1 n



and µ n (B) = µ(B ∩ D n )

for any positive integer n and any Borel subset B of R ^d . Then, for each positive integer n and each element r of (0, 1), we have

r kµk _p ≤



 X

k∈Z

^d

|µ| (I _k ^r ∩ E) ^p





1 p

+



 X

k∈Z

^d

|µ| (I _k ^r \ E) ^p





1 p

≤



 X

k∈Z

^d

|µ| (I _k ^r ∩ E) ^p





1 p

+



2 ^d X

k∈Z

^d

\K

|µ| I _k ¹
p





1 p

≤



 X

k∈Z

^d

|µ| (I _k ^r ∩ E) ^p





1 p

+ ε

≤



 X

k∈Z

^d

|µ _n | (I _k ^r ∩ E) ^p





1 p

+ |µ| (E ∩ (D \ D _n )) + ε

≤



 X

k∈Z

^d

|µ _n | (I _k ^r ) ^p





1 p

+ |µ| (E ∩ (D \ D n )) + ε.

Since for each positive integer n the set D _n is finite, then there exists an element r _n of (0, 1) such that for all a, b ∈ D _n ,

a 6= b =⇒ ∃ i ∈ { 1, . . . , d } : |a i − b _i | > r _n and for each element r of (0, r n )



 X

k∈Z

^d

|µ _n | (I _k ^r ) ^p





1 p

= X

a∈D

n

|µ({a})| ^p

!

_p¹

≤ X

a∈D

|µ({a})| ^p

!

¹_p

.

Therefore, for each element r of (0, r n ) we have

r kµk _p ≤ X

a∈D

|µ({a})| ^p

!

¹_p

+ |µ| (E ∩ (D \ D n )) + ε.

Since |µ|(E) < ∞ and (D \ D n ) n is a decreasing sequence that converges to the empty set, we get

lim sup

r→0 r kµk _p ≤ X

a∈D

|µ({a})| ^p

!

¹_p

+ ε.

As this inequality holds for an arbitrary real number ε > 0, we deduce that

lim sup

r→0 r kµk _p ≤ X

a∈D

|µ({a})| ^p

!

¹_p

.

Further, for all real numbers r > 0, we have

X

a∈D

|µ({a})| ^p

!

¹_p

=



 X

k∈Z

^d



 X

a∈D∩I

_k^r

|µ({a})| ^p









1 p

≤ _r kµk _p .

So

r→0 lim r kµk _p = X

a∈D

|µ({a})| ^p

!

¹_p

.

QED

4 Generalization of Wiener’s theorem

Throughout this section, for 1 < p < ∞ and µ ∈ M ^p , we set D = { a ∈ R ^d | µ({a}) 6= 0 }.

Proposition 4. Let 1 < p < ∞. If µ ∈ M ^p then

r→0 lim r kµk _p = X

a∈D

|µ({a})| ^p

!

¹_p

.

Proof. Let us write µ = µ _c + µ _δ , where µ _c is a continuous measure and µ _δ is a discrete measure. For each real number r > 0, we have

r kµ _δ k _p ≤ _r kµk _p ≤ _r kµ _c k _p + _r kµ _δ k _p .

The desired result follows from Propositions 2 and 3.

^QED

As an immediate consequence of (1.1) and Proposition 4, we have the fol- lowing result.

Corollary 1. Let 1 < p ≤ 2. There exists a real constant C > 0 such that for any µ ∈ M ^p we have

lim sup

r→∞

r ⁻

d

p0

r k µk b p

⁰

, ∞ ≤ C X

a∈D

|µ({a})| ^p

!

¹_p

.

In the case p = 2, we obtain the following generalization of Wiener’s theorem.

Proposition 5. If µ ∈ M ² then

r→∞ lim r ⁻

^d2

Z

J

₀^r

| µ(y)| b ² dy

!

¹₂

= X

a∈D

|µ({a})| ²

!

¹₂

.

Proof. a) Let us write µ = µ _c + µ _δ , where µ _c is a continuous measure and µ _δ is a discrete measure. It follows from Propositions 2 and 3 that

r→∞ lim

¹r

kµ _c k ₂ = 0 and lim

r→∞

¹r

kµ _δ k ₂ = X

a∈D

|µ({a})| ²

!

¹₂

< ∞.

Note that for each element (r, x) of (0, ∞) × R ^d , we have Z

J

_x^r

| µ(y)| b ² dy = Z

J

_x^r

| c µ _δ (y)| ² dy + 2Re Z

J

_x^r

c µ _δ (y) µ b c (y) dy + Z

J

_x^r

| µ b c (y)| ² dy,

where Re Z

J

_x^r

c µ _δ (y) µ b _c (y) dy denotes the real part of Z

J

_x^r

c µ _δ (y) µ b _c (y) dy.

By the H¨ older’s inequality and (1.1), we have 

    Z

J

_x^r

c µ _δ (y) µ b _c (y) dy     

≤ Z

J

_x^r

| c µ _δ (y)| ² dy

!

¹₂

Z

J

_x^r

| µ b _c (y)| ² dy

!

¹₂

≤ _r k c µ _δ k _{2, +∞ r} k µ b c k _{2, +∞}

≤ C ²

1

r

k µ _δ k ₂

1

r

k µ _c k ₂ r ^d and

Z

J

_x^r

| µ b c (y)| ² dy ≤ r k µ b c k ² _{2, +∞} ≤ C ²

1 r

kµ _c k ² ₂ r ^d .

It follows that

lim sup

r→∞

r ⁻

^d2

r k µk b 2, ∞ = lim sup

r→∞

r ⁻

^d2

r k c µ δ k _{2, ∞} and

lim inf

r→∞ r ⁻

^d2

r k µk b 2, ∞ = lim inf

r→∞ r ⁻

^d2

r k c µ _δ k _{2, ∞} .

Therefore, it suffices to consider the case where µ is a discrete measure to prove the desired result.

b) Suppose now that µ is a discrete measure.

Let us consider an element ε of



0, ¹ ₃ P

a∈D |µ({a})| ²

¹₂

 . For each positive integer n and each Borel subset B of R ^d , set

D _n =



a ∈ D | |µ({a})| ≥ 1 n



and µ _n (B) = µ(B ∩ D _n ).

There exists a positive integer N such that



 X

a∈D\D

N

|µ({a})| ²





1 2

< ε

3(C + 1) , (4.1)

where C is a real constant as in Corollary 1. So

lim sup

r→+∞

r ⁻

^d2

r k µ − b µ c N k _{2, ∞} ≤ C



 X

a∈D\D

N

|µ({a})| ²





1 2

< ε 3 .

Thus, there exists a real number R ₁ > 0 such that for any r ≥ R ₁ and any x ∈ R ^d ,

     

r ⁻

^d2

Z

J

_x^r

| b µ(y)| ² dy

!

¹₂

− r ⁻

^d2

Z

J

_x^r

| µ c _N (y)| ² dy

!

¹₂

     

< ε

3 . (4.2)

Since µ _N is a finite measure, it follows from (1.2) that

r→∞ lim r ⁻

^d2

Z

J

₀^r

| µ c N (y)| ² dy

!

¹₂

=



 X

a∈D

N

|µ({a})| ²





1 2

.

Thus, there exists a real number R ₂ > 0 such that for any r ≥ R ₂ , 

     

r ⁻

^d2

Z

J

₀^r

| µ c N (y)| ² dy

!

¹₂

−



 X

a∈D

N

|µ({a})| ²





1 2

      

< ε

3 . (4.3)

Let R = max{ R ₁ , R ₂ }. From inequalities (4.1), (4.2) and (4.3), we obtain 

    

r ⁻

^d2

Z

J

₀^r

| µ(y)| b ² dy

!

¹

2

− X

a∈D

|µ({a})| ²

!

¹

2

     

< ε, r ≥ R.

Hence

r→+∞ lim r ⁻

^d2

Z

J

₀^r

| µ(y)| b ² dy

!

¹₂

= X

a∈D

|µ({a})| ²

!

¹₂

.

QED

As an immediate consequence of Proposition 5 and Definition 1, we have the following criterion.

Corollary 2. Let µ ∈ M ² . Then µ is continuous if and only if

r→∞ lim r ^−d Z

J

₀^r

| µ(y)| b ² dy = 0.

Acknowledgements. The author would like to thank Professor I. Fofana for his help and guidance in the preparation of this paper.

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