International Journal of Analysis and Applications
ISSN 2291-8639
Volume 10, Number 2 (2016), 85-89
http://www.etamaths.com

BEURLING’S THEOREM AND Lp −Lq MORGAN’S THEOREM FOR THE
GENERALIZED BESSEL-STRUVE TRANSFORM

A. ABOUELAZ, A. ACHAK, R. DAHER, N. SAFOUANE∗

Abstract. The generalized Bessel-Struve transform satisfies some uncertainty principles similar to
the Euclidean Fourier transform. A generalization of Beurling’s theorem and Lp − Lq Morgan’s
theorem obtained for the generalized Bessel-Struve transform.

1. Introduction and Preliminaries

There are many theorems known which state that a function and its classical Fourier transform
on R cannot both be sharply localized. That is, it is impossible for a nonzero function and its
Fourier transform to be simultaneously small. Here a concept of the smallness had taken different
interpretations in different contexts. Morgan [5] and Beurling [3] for example interpreted the smallness
as sharp pointwise estimates or integrable decay of functions. In particular Beurling’s theorem, which
was found by Beurling and his proof was published much later by Hörmander [4], says that

Theorem 1. If f ∈ L2(R) satisfies that∫
R

∫
R
|f(x)||f̂(y)|e|x||y|dxdy < ∞,

then f = 0 a.e.

Morgan [5] has established a famous theorem stating that for γ > 2 and η = γ
γ−1, if (aγ)

1
γ (bη)

1
η >

(sin(π
2

(η− 1))
1
η , ea|x|

γ

f ∈ L∞(R) and eb|x|
η

F(f) ∈ L∞(R). then f is null almost everywhere. S. Ben
Farah and K. Mokni [2] have generalized Morgan’s theorem to an Lp −Lq−version where 1 ≤ p,q ≤
+∞.
The outline of the content of this paper is as follows. In section 2 we give an analogue of Beurling’s
theorem for FB,Sα,n . Section 3 is devoted to Lp −Lq-Morgan’s theorem for FB,Sα,n .
Let us now be more precise and describe our results. To do so, we need to introduce some notations.
Throughout this paper, the letter C indicates a positive constant not necessarily the same in each
occurrence. We denote by

•

(1) aα =
2Γ (α + 1)
√
πΓ
(
α + 1

2

)
where α > −1

2
.

• Mn the map defined by Mn(f(x)) = x2nf(x).
• Lpα(R) the class of measurable functions f on R for which ‖f‖p,α < ∞, where

‖f‖p,α =
(∫

R
|f(x)|p|x|2α+1dx

)1
p

, ifp < ∞,

and ‖f‖∞,α = ‖f‖∞ = ess supx≥0|f(x)|.
• Lpα,n(R) the class of measurable functions f on R for which

‖f‖p,α,n = ‖M−1n f‖p,α+2n < ∞.

2010 Mathematics Subject Classification. 42A38, 44A35, 34B30.
Key words and phrases. generalized Bessel-Struve transform; uncertainty principles.

c©2016 Authors retain the copyrights of
their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

85



86 ABOUELAZ, ACHAK, DAHER AND SAFOUANE

• K0 the space of functions f infinitely differentiable on R∗ with bounded support verifying for
all n ∈ N,

lim
y→0−

ynf(n)(y) and lim
y→0+

ynf(n)(y)

exist.
•

d

dx2
=

1

2x

d

dx
, where

d

dx
is the first derivative operator.

In this section we recall some facts about harmonic analysis related to the generalized Bessel-Struve
operator Fα,nB,S. We cite here, as briefly as possible, only some properties. For more details we refer to
[1].
For λ ∈ C and x ∈ R, put

Ψλ,α,n(x) = aα+2nx
2n

∫ 1
0

(1 − t2)α+2n−
1
2 eλxtdt.

Ψλ,α,n satisfies

∀ξ ∈ R, ∀ζ ∈ R,∀x ∈ R |Ψλ,α,n(i(ξ + iζ)x)| ≤ x2ne|x||ζ|.(2)

∀n ∈ N, ∀λ ∈ R, ∀x ∈ R, |
dn

dxn
(x−2nΨiλ,α,n(x))| ≤ |λ|n.(3)

Definition 1. The Generalized Bessel-Struve transform is defined on L1α,n(R) by

∀λ ∈ R, Fα,nB,S(f)(λ) =
∫
R
f(x)Ψ−iλ,α,n(x)|x|2α+1dx.

Definition 2. For f ∈ L1α,n(R) with bounded support, the integral transform Wα,n, given by

Wα,n(f(x)) = aα+2n

∫ +∞
|x|

(y2 −x2)α+2n−
1
2 y1−2nf(sgn(x)y)dy, x ∈ R\{0}

is called the generalized Weyl integral transform associated with Bessel-Struve operator.

Proposition 1. Wα,n is a bounded operator from L1α,n(R) to L
1(R), where L1(R) is the space of

lebesgue-integrable functions.

Remark 1. From Proposition 1 we can find a constant C such that∫
R
|Wα,n(f)(x)|dx ≤ C‖f‖α,n,1

Proposition 2. If f ∈ L1α,n(R) then

(4) Fα,nB,S = F ◦Wα,n,

where F is the classical Fourier transform defined on L1(R) by

F(g)(λ) =
∫
R
g(x)e−iλxdx.

Definition 3. Let α = k + 1
2
where k ∈ N. We define the operator Vα,n on K0 as follows

Vα,nf(x) = (−1)k+1
22k+4n+1(k + 2n)!

(2k + 4n + 1)!
x2n(

d

dx2
)k+2n+1(f(x)), x ∈ R∗.

Theorem 2. Let f ∈ K0, Vα,n and Wα,n are related by the following relation

Vα,n(Wα,n(f)) = f.



BEURLING’S THEOREM AND Lp −Lq MORGAN’S THEOREM 87

2. Beurling’s theorem for the Generalized Bessel-Struve transform

In this section we will prove Beurling’s theorem for the Generalized Bessel-Struve transform.

Theorem 3. Let k ∈ N, α = k + 1
2
and f ∈ L2α,n(R) satisfy

(5)
∫
R

∫
R
|f(x)||Fα,nB,S(f)(y)|e

|x||y||x|2(α+n)+1dxdy < ∞,

then f = 0 almost everywhere.

Proof. We start with the following lemma.

Lemma 1. We suppose that f ∈ L2α,n(R) satisfies (5), then f ∈ L1α,n(R).

Proof. We may assume that f 6= 0 in L2α,n(R). (5) and the Fubini theorem for almost every y ∈ R

|Fα,nB,S(f)(y)|
∫
R
|f(x)|e|x||y||x|2(α+n)+1dx < ∞,

since Fα,nB,S(f) 6= 0, there exist y0 ∈ R,y0 6= 0 such that F
α,n
B,S(f)(y0) 6= 0,

therefore ∫
R
|f(x)|e|x||y0||x|2(α+n)+1dx < ∞

∫
R

|f(x)|
x2n

e|x||y0||x|2(α+2n)+1dx < ∞∫
R
|M−1n f(x)|e

|x||y0||x|2(α+2n)+1dx < ∞,

since e|x||y0| > 1 for large |x| it follows
∫
R |M

−1
n f(x)||x|2(α+2n)+1dx < ∞.

This lemma and proposition 1 imply that Wα,n(f) is well-defined a.e on R.
By Remark 1 we can find a positif constant C such that∫

R
|Wα,n(f)(x)|dx ≤ C‖f‖α,n,1

≤ C‖M−1n f‖α+2n,1

≤ C
∫
R
|M−1n f(x)||x|

2(α+2n)+1dx

≤ C
∫
R
|
f(x)

x2n
||x|2(α+2n)+1dx

≤ C
∫
R
|f(x)||x|2(α+n)+1dx.

Thus∫
R

∫
R
|Wα,n(f(x))||F

α,n
B,S(f)(y)|e

|x||y|dxdy ≤ C
∫
R

∫
R
|f(x)||Fα,nB,S(f)(y)|e

|x||y||x|2(α+n)+1dxdy

< ∞.

It follows from Proposition 2 that∫
R

∫
R
|Wα,n(f(x))||F ◦Wα,n(f)(y)|e|x||y|dxdy < ∞.

According to Theorem 1, we can deduce that

Wα,n(f) = 0,

applying Lemma 1 we obtain
f = Vα,n ◦Wα,n(f) = 0.



88 ABOUELAZ, ACHAK, DAHER AND SAFOUANE

Corollaire 1. (Gelfand-Shilov) If f ∈ L2α,n(R) such that∫
R
|f(x)|e

|x|p
p |x|2(α+n)+1dx < ∞,

∫
R
|Fα,nB,S(f)(y)|e

|y|q
q dy < ∞

then f = 0.

Proof. Let M and M∗ be functions satisfying

xy ≤ M(x) + M∗(y).(6)

If ∫
R
|f(x)|eM(x)|x|2(α+n)+1dx < ∞,

∫
R
|FαB,S(f)(y)|e

M∗(y)dy < ∞

then

∫
R

∫
R
|f(x)||Fα,nB,S(f)(y)|e

|x||y||x|2(α+n)+1dxdy ≤
∫
R

∫
R
|f(x)||Fα,nB,S(f)(y)|e

M(x)+M∗(y)|x|2(α+n)+1dxdy

=

∫
R
|f(x)|eM(x)|x|2(α+n)+1dx

∫
R
|Fα,nB,S(f)(y)|e

M∗(y)dy < ∞.

Consequently, Beurling’s Theorem implies that f(y) = 0. In particular, if M(x) = |x|
p

p
and M∗(y) =

|y|q
q
, where p,q are conjugate exponents p−1 + q−1 = 1, then the pair (M, M∗) satisfies the condition

(6). Thus, we obtain an analogue of the Gelfand-Shilov uncertainty principle for the Bessel-Struve
transform.

3. Lp −Lq morgan’s theorem for the generalized bessel-struve transform

In this section, we prove Lp −Lq Morgan’s theorem for the Generalized Bessel-Struve transform.

Lemma 2. We assume that ρ ∈]1, 2[, q ∈ [1,∞], σ > 0 and B > σ sin(π
2

(ρ − 1)). If g is an entire
function on C verifying

|g(x + iy)| ≤ Ceσ|y|
ρ

∈ Lpα+2n(R)

eB|x|
ρ

g|R ∈ L
q
α+2n(R)

for all x, y ∈ R, then g = 0.

Proof. See [2].

Lemma 3. Let p ∈ [1,∞] , γ > 2 and f a measurable function on R verifying

(7) ∀a > 0, ea|x|
γ

f ∈ Lpα,n(R).

Then the function defined on C by

(8) Fα,nB,S(f)(z) =
∫
R
f(x)Ψ−iz,α,n(x)|x|2α+1dx

is well defined and entire on C. Moreover, we have

(9) ∀ξ,ζ ∈ R, |Fα,nB,S(f)(ξ + iζ)| ≤
∫
R
|f(x)|e|x||ζ||x|2α+1dx.

Proof. Relation (7) assert that Fα,nB,S(f) is well defined. Applying again relation (7), the analytic
theorem on (8) and the fact that Ψ−iλ,α,n(x) verifies (9), we deduce that z →F

α,n
B,S(f)(z) is an entire

on C. The relation (10) is obtained from relation (2).

Theorem 4. Let p,q ∈ [1,∞], a > 0, b > 0, γ > 2 and η = γ
γ−1. Suppose that f a measurable

function on R such that

ea|x|
γ

f ∈ Lpα,n(R) and e
b|x|ηFα,nB,S(f) ∈ L

q
α+2n(R).(10)

If (aγ)
1
γ (bη)

1
η > (sin(π

2
(η − 1))

1
η , then f is null almost everywhere.



BEURLING’S THEOREM AND Lp −Lq MORGAN’S THEOREM 89

Proof. We notice that ea|x|
γ

f ∈ Lpα,n ⇔ ea|x|
γ

M−1n f ∈ L
p
α+2n.

First case: 1 < p < ∞. Applying Hölder inequality, we get

|Fα,nB,S(f)(ξ + iζ)| ≤ ‖f‖α,n,p
(∫

R
e−ap

′|x|γe|x||ζ|p
′
|x|2α+1dx

) 1
p′

where p′ verifies 1
p′

+ 1
p

= 1.

Now, we take C ∈](bη)−η sin(π
2

(η − 1))
1
η ), (aγ)

1
γ [. Using a convexity’s inequality, we obtain

|x||ζ| ≤
Cγ

γ
|x|γ +

1

ηCη
|ζ|η(11)

and the following relation holds∫
R
e−ap

′|x|γe|x||ζ|p
′
|x|2α+1dx ≤ e

p′|ζ|η
ηCη

∫
R
e−p

′(a−C
γ

γ
|x|γ)|x|2α+1dx.

So we get
∀ξ,ζ ∈ R, |Fα,nB,S(f)(ξ + iζ)| ≤ const e

1
ηCη
|ζ|η.

Second case: p = 1 or p = +∞. From relations (2) and (11), we get

|FB,Sα,n (f)(ξ + iζ)| ≤ e
1

ηCη
|ξ|η
∫
R
ea|x|

γ

|f(x)|e−(a−
Cγ

γ
)|x|γ|x|2α+1dx.

Therefore
|FB,Sα,n (f)(ξ + iζ)| ≤ const e

1
ηCη
|ξ|η.

Hence

(12) ∀p ∈ [1,∞], ∀ξ,ζ ∈ R, |Fα,nB,S(f)(ξ + iζ)| ≤ const e
1

ηCη
|ζ|η.

By virtue of relations (10), (12) and Lemma 2, we obtain that Fα,nB,Sf = 0. The injectivity of the
generalized Bessel-Struve transform implies that f = 0 almost everywhere.

References

[1] A. Abouelaz, A. Achak, R. Daher and N. Safouane, Harmonic analysis associated with the generalized Bessel-Struve
operator on the real line, Int. Refereed J. Eng. Sci., 4 (2015), 72-84.

[2] S. Ben Farah, K. Mokni, Uncertainty Principle and the Lp − Lq-Version of Morgan’s Theorem on Semi-simple lie
Groups, Integrals Transform Spec. Funct. 16 (2005), 281-289.

[3] A. Beurling, The collect works of Arne Beurling, Birkhäuser. Boston, 1989, 1-2.
[4] L. Hörmander, A uniqueness theorem of Beurling for Fourier transform pairs, Ark. För Math., 2(1991), 237-240.
[5] G.W. Morgan, A note on Fourier transforms, J. London Math. Soc., 9 (1934), 188-192.

Department of Mathematics, Faculty of Sciences Aïn Chock, University of Hassan II, Casablanca
20100, Morocco

∗Corresponding author: safouanenajat@gmail.com