CUBO A Mathematical Journal
Vol.16, No¯ 01, (95–104). March 2014

Lp local uncertainty inequality for the Sturm-Liouville
transform

Fethi Soltani
1

Department of Mathematics, Faculty of Science,

Jazan University,

P.O.Box 114, Jazan, Kingdom of Saudi Arabia.

fethisoltani10@yahoo.com

ABSTRACT

In this paper, we give analogues of local uncertainty inequality for the Sturm-Liouville

transform on [0, ∞[. A generalization of Donoho-Stark’s uncertainty principle is ob-

tained for this transform.

RESUMEN

En este art́ıculo entregamos resultados análogos de una desigualdad de incertidumbre

local de la transformada Sturm-Liouville en [0, ∞[. Una generalización del principio de

incertidumbre de Donoho-Stark se obtiene de esta transformación.

Keywords and Phrases: Sturm-Liouville transform; local uncertainty principle; Donoho-Stark’s

uncertainty principle.

2010 AMS Mathematics Subject Classification: 42B10; 44A20; 46G12.

1Author partially supported by DGRST project 04/UR/15-02 and CMCU program 10G 1503



96 Fethi Soltani CUBO
16, 1 (2014)

1 Introduction

We consider the second-order differential operator defined on ]0, ∞[ by

∆u := u′′ +
A′

A
u′ + ρ2u,

where A is a nonnegative function satisfying certain conditions and ρ is a nonnegative real number.

This operator plays an important role in analysis. For example, many special functions (orthogonal

polynomials) are eigenfunctions of an operator of ∆ type. The radial part of the Beltrami-Laplacian

in a symmetric space is also of ∆ type. Many aspects of such operators have been studied [2, 10,

17, 18, 19]. In particular, the two references [2, 17] investigate standard constructions of harmonic

analysis, such as translation operators, convolution product, and Fourier transform, in connection

with ∆.

Many uncertainty principles have already been proved for the Sturm-Liouville operarator ∆,

namely by Rösler and Voit [14] who established an uncertainty principle for Hankel transforms.

Bouattour and Trimèche [1] proved a Beurling’s theorem for the Sturm-Liouville transform. Daher

et al. [3, 4, 5] give some related versions of the uncertainty principle for the Sturm-Liouville

transform (Hardy’s theorem and Miyachi’s theorem). Ma [9] proved a Heisenberg uncertainty

principle for the Sturm-Liouville transform.

Building on the ideas of Faris [7] and Price [12, 13], we show a local uncertainty principle for

the Sturm-Liouville transform F. More precisely, we will show the following result. If 1 < p ≤ 2,

q = p/(p − 1) and 0 < a < (2α + 2)/q, there is a constant K(a) such that for every f ∈ Lp(µ) and

every measurable subset E ⊂ [0, ∞[ such that 0 < ν(E) < ∞,

(∫

E

|F(f)(λ)|qdν(λ)

)1/q

≤ K(a)
(

ν(E)
)

a
2α+2

‖xaf‖Lp(µ), (1.1)

where µ is the measure given by dµ(x) := A(x)dx, and ν is the Plancherel measure associated to

F. (For more details see the next section.)

This inequality generalizes the local uncertainty principle for the Hankel transform given by Ghob-

ber et al. [8] and Omri [11].

We shall use the local uncertainty principle (1.1); and building on the techniques of Donoho

and Stark [6], we show a continuous-time principles for the Lp theory, when 1 < p ≤ 2.

This paper is organized as follows. In Section 2 we list some basic properties of the Sturm-

Liouville transform F (Plancherel theorem, inversion formula,...). In Section 3 we show a local

uncertainty principle for the Sturm-Liouville F. The Section 4 is devoted to Donoho-Stark’s

uncertainty principle for the Sturm-Liouville transform F in the Lp theory, when 1 < p ≤ 2.



CUBO
16, 1 (2014)

Lp local uncertainty inequality for the Sturm-Liouville transform 97

2 The Sturm-Liouville transform F

We consider the second-order differential operator ∆ defined on ]0, ∞[ by

∆u := u′′ +
A′

A
u′ + ρ2u,

where ρ is a nonnegative real number and

A(x) := x2α+1B(x), α > −1/2,

for B a positive, even, infinitely differentiable function on R such that B(0) = 1. Moreover we

assume that A and B satisfy the following conditions:

(i) A is increasing and lim
x→∞

A(x) = ∞.

(ii)
A′

A
is decreasing and lim

x→∞

A′(x)

A(x)
= 2ρ.

(iii) There exists a constant δ > 0 such that

A′(x)

A(x)
= 2ρ + D(x) exp(−δx) if ρ > 0,

A′(x)

A(x)
=

2α + 1

x
+ D(x) exp(−δx) if ρ = 0,

where D is an infinitely differentiable function on ]0, ∞[, bounded and with bounded derivatives

on all intervals [x0, ∞[, for x0 > 0. This operator was studied in [2, 10, 17], and the following

results have been established:

(I) For all λ ∈ C, the equation

{
∆u = −λ2u

u(0) = 1, u′(0) = 0

admits a unique solution, denoted by ϕλ, with the following properties:

• for x ≥ 0, the function λ → ϕλ(x) is analytic on C;

• for λ ∈ C, the function x → ϕλ(x) is even and infinitely differentiable on R;

• for all λ, x ∈ R,

|ϕλ(x)| ≤ 1. (2.1)

(II) For nonzero λ ∈ C, the equation ∆u = −λ2u has a solution Φλ satisfying

Φλ(x) =
1

√

A(x)
exp(iλx)V(x, λ),

with limx→∞ V(x, λ) = 1. Consequently there exists a function (spectral function)

λ 7→ c(λ),



98 Fethi Soltani CUBO
16, 1 (2014)

such that

ϕλ = c(λ)Φλ + c(−λ)Φ−λ for nonzero λ ∈ C.

Moreover there exist positive constants k1, k2 and k such that

k1|λ|
2α+1 ≤ |c(λ)|−2 ≤ k2|λ|

2α+1

for all λ such that Imλ ≤ 0 and |λ| ≥ k.

Notation 2.1. We denote by

• µ the measure defined on [0, ∞[ by dµ(x) := A(x)dx; and by Lp(µ), 1 ≤ p ≤ ∞, the space

of measurable functions f on [0, ∞[, such that

‖f‖Lp(µ) :=

(∫
∞

0

|f(x)|pdµ(x)

)1/p

< ∞, 1 ≤ p < ∞,

‖f‖L∞(µ) := ess sup
x∈[0,∞[

|f(x)| < ∞;

• ν the measure defined on [0, ∞[ by dν(λ) :=
dλ

2π|c(λ)|2
; and by Lp(ν), 1 ≤ p ≤ ∞, the space

of measurable functions f on [0, ∞[, such that ‖f‖Lp(ν) < ∞.

The Fourier transform associated with the operator ∆ is defined on L1(µ) by

F(f)(λ) :=

∫
∞

0

ϕλ(x)f(x)dµ(x) for λ ∈ R.

Some of the properties of the Fourier transform F are collected bellow (see [2, 10, 17, 18]).

Theorem 2.2. (i) L1 − L∞-boundedness. For all f ∈ L1(µ), F(f) ∈ L∞(ν) and

‖F(f)‖L∞(ν) ≤ ‖f‖L1(µ). (2.2)

(ii) Inversion theorem. Let f ∈ L1(µ), such that F(f) ∈ L1(ν). Then

f(x) =

∫
∞

0

ϕλ(x)F(f)(λ)dν(λ), a.e. x ∈ [0, ∞[. (2.3)

(iii) Plancherel theorem. The Fourier transform F extends uniquely to an isometric isomor-

phism of L2(µ) onto L2(ν). In particular,

‖f‖L2(µ) = ‖F(f)‖L2(ν). (2.4)

Using relations (2.2) and (2.4) with Marcinkiewicz’s interpolation theorem [15, 16], we deduce

that for every 1 ≤ p ≤ 2, and for every f ∈ Lp(µ), the function F(f) belongs to the space Lq(ν),

q = p/(p − 1), and

‖F(f)‖Lq(ν) ≤ ‖f‖Lp(µ). (2.5)



CUBO
16, 1 (2014)

Lp local uncertainty inequality for the Sturm-Liouville transform 99

3 Lp local uncertainty inequality

This section is devoted to establish a local uncertainty principle for the Sturm-Liouville transform

F, more precisely, we will show the following theorem.

Theorem 3.1. If 1 < p ≤ 2, q = p/(p − 1) and 0 < a < (2α + 2)/q, then for all f ∈ Lp(µ) and

all measurable subset E ⊂ [0, ∞[ such that 0 < ν(E) < ∞,

(∫

E

|F(f)(λ)|qdν(λ)

)1/q

≤ K(a)
(

ν(E)
)

a
2α+2

‖xaf‖Lp(µ),

K(a) =
(

qa
)−

qa
2α+2

(

2α + 2 − qa
)

(q−1)a
2α+2

[

1 +
qa

2α + 2 − qa

(

sup
x∈[0,r0]

B(x)
)1/q

]

,

where

r0 =
(

qa
)

q
2α+2

(

2α + 2 − qa
)

1−q
2α+2

(

ν(E)
)− 1

2α+2

.

Proof. For r > 0, denote by χE, χ[0,r[ and χ[r,∞[ the characteristic functions.

Let f ∈ Lp(µ), 1 < p ≤ 2 and let q = p/(p − 1). By Minkowski’s inequality, for all r > 0,

‖F(f)χE‖Lq(ν) ≤ ‖F(fχ[0,r[)χE‖Lq(ν) + ‖F(fχ[r,∞[)χE‖Lq(ν)

≤
(

ν(E)
)1/q

‖F(fχ[0,r[)‖L∞(ν) + ‖F(fχ[r,∞[)‖Lq(ν);

hence it follows from (2.2) and (2.5) that

‖F(f)χE‖Lq(ν) ≤
(

ν(E)
)1/q

‖fχ[0,r[‖L1(µ) + ‖fχ[r,∞[‖Lp(µ). (3.1)

On the other hand, by Hölder’s inequality,

‖fχ[0,r[‖L1(µ) ≤ ‖x
−aχ[0,r[‖Lq(µ)‖x

af‖Lp(µ).

By hypothesis a < (2α + 2)/q,

‖x−aχ[0,r[‖Lq(µ) ≤
r−a+(2α+2)/q

(2α + 2 − qa)1/q

(

sup
x∈[0,r]

B(x)
)1/q

,

and therefore,

‖fχ[0,r[‖L1(µ) ≤
r−a+(2α+2)/q

(2α + 2 − qa)1/q

(

sup
x∈[0,r]

B(x)
)1/q

‖xaf‖Lp(µ). (3.2)

Moreover,

‖fχ[r,∞[‖Lp(µ) ≤ ‖x
−aχ[r,∞[‖L∞(µ)‖x

af‖Lp(µ) ≤ r
−a‖xaf‖Lp(µ). (3.3)

Combining the relations (3.1), (3.2) and (3.3), we deduce that

‖F(f)χE‖Lq(ν) ≤

[

r−a +
(

ν(E)
)1/q r−a+(2α+2)/q

(2α + 2 − qa)1/q

(

sup
x∈[0,r]

B(x)
)1/q

]

‖xaf‖Lp(µ).



100 Fethi Soltani CUBO
16, 1 (2014)

We choose r = r0 =
(

qa
)

q
2α+2

(

2α + 2 − qa
)

1−q
2α+2

(

ν(E)
)− 1

2α+2

, we obtain the desired inequality.

2

Remark 3.2. (i) The Local uncertainty principle for the Sturm-Liouville transform F generalizes

the local uncertainty principle for the Hankel transform (see [8, 11]).

(ii) If 1 < p ≤ 2 and 0 < a < (2α + 2)/q, where q = p/(p − 1), then for every f ∈ Lp(µ),

sup
E⊂[0,∞[, 0<ν(E)<∞

[

(

ν(E)
)− a

2α+2

‖F(f)χE‖Lq(ν)

]

≤ K(a)‖xaf‖Lp(µ).

The left hand side is known to be an equivalent norm of F(f) in the Lorentz-space Lpa,q(ν), where

pa =
q(2α + 2)

2α + 2 − qa
.

4 Lp Donoho-Stark uncertainty principle

Let T and E be measurable subsets of [0, ∞[. We introduce the time-limiting operator PT by

PT f := fχT , (4.1)

and, we introduce the partial sum operator SE by

F(SEf) = F(f)χE. (4.2)

Lemma 4.1. If ν(E) < ∞ and f ∈ Lp(µ), 1 ≤ p ≤ 2,

SEf(x) =

∫

E

ϕλ(x)F(f)(λ)dν(λ).

Proof. Let f ∈ Lp(µ), 1 ≤ p ≤ 2 and let q = p/(p − 1). Then by (2.1), Hölder’s inequality and

(2.5),

‖F(f)χE‖L1(ν) =

∫

E

|F(f)(λ)|dν(λ)

≤
(

ν(E)
)1/p

‖F(f)‖Lq(ν)

≤
(

ν(E)
)1/p

‖f‖Lp(µ),

and

‖F(f)χE‖L2(ν) =

(∫

E

|F(f)(λ)|2dν(λ)

)1/2

≤
(

ν(E)
)

q−2
2q

‖F(f)‖Lq(ν)

≤
(

ν(E)
)

q−2
2q

‖f‖Lp(µ).



CUBO
16, 1 (2014)

Lp local uncertainty inequality for the Sturm-Liouville transform 101

Thus F(f)χE ∈ L
1(µ) ∩ L2(µ) and by (4.2),

SEf = F
−1
(

F(f)χE

)

.

This combined with (2.3) gives the result. 2

Let T and E be measurable subsets of [0, ∞[. We say that a function f ∈ Lp(µ), 1 ≤ p ≤ 2, is

ε-concentrated to T in Lp(µ)-norm, if there is a measurable function g(t) vanishing outside T such

that ‖f − g‖Lp(µ) ≤ ε‖f‖Lp(µ). Similarly, we say that F(f) is ε-concentrated to E in L
q(ν)-norm,

q = p/(p−1), if there is a function h(λ) vanishing outside E with ‖F(f)−h‖Lq(ν) ≤ ε‖F(f)‖Lq(ν).

If f is εT -concentrated to T in L
p(µ)-norm (g being the vanishing function) then by (4.1),

‖f − PT f‖Lp(µ) =

(∫

[0,∞[\T

|f(t)|pdµ(t)

)1/p

≤ ‖f − g‖Lp(µ) ≤ εT ‖f‖Lp(µ) (4.3)

and therefore f is εT -concentrated to T in L
p(µ)-norm if and only if

‖f − PT f‖Lp(µ) ≤ εT ‖f‖Lp(µ).

From (4.2) it follows as for PT that F(f) is εE-concentrated to E in L
q(ν)-norm, q = p/(p−1),

if and only if

‖F(f) − F(SEf)‖Lq(ν) ≤ εE‖F(f)‖Lq(µ). (4.4)

Let Bp(E), 1 ≤ p ≤ 2, be the set of functions f ∈ L
p(µ) that are bandlimited to E (i.e.

f ∈ Bp(E) implies SEf = f).

The spaces Bp(E) satisfy the following property.

Lemma 4.2. Let T and E be measurable subsets of [0, ∞[ such that 0 < ν(E) < ∞. For f ∈ Bp(E),

1 < p ≤ 2 and 0 < a < (2α + 2)(1 − 1
p
),

‖PTf‖Lp(µ) ≤ K(a)
(

µ(T)
)1/p(

ν(E)
)

1
p
+ a

2α+2

‖xaf‖Lp(µ),

where K(a) is the constant given by Theorem 3.1.

Proof. If µ(T) = ∞, the inequality is clear. Assume that µ(T) < ∞. For f ∈ Bp(E), 1 < p ≤ 2,

from Lemma 4.1,

f(t) =

∫

E

ϕλ(t)F(f)(λ)dν(λ),

and by (2.1), Hölder’s inequality and Theorem 3.1,

|f(t)| ≤
(

ν(E)
)1/p

(∫

E

|F(f)(λ)|qdν(λ)

)1/q

≤ K(a)
(

ν(E)
)

1
p
+ a

2α+2

‖xaf‖Lp(µ),



102 Fethi Soltani CUBO
16, 1 (2014)

where q = p/(p − 1). Hence,

‖PT f‖Lp(µ) =

(∫

T

|f(t)|pdµ(t)

)1/p

≤ K(a)
(

µ(T)
)1/p(

ν(E)
)

1
p
+ a

2α+2

‖xaf‖Lp(µ),

which yields the result. 2

It is useful to have uncertainty principle for the Lp(µ)-norm.

Theorem 4.3. Let T and E be measurable subsets of [0, ∞[ such that 0 < ν(E) < ∞; and let

f ∈ Bp(E), 1 < p ≤ 2 and 0 < a < (2α + 2)(1 −
1
p
). If f is εT -concentrated to T, then

‖f‖Lp(µ) ≤
K(a)

1 − εT

(

µ(T)
)1/p(

ν(E)
)

1
p
+ a

2α+2

‖xaf‖Lp(µ).

Proof. Let f ∈ Bp(E), 1 < p ≤ 2. Since f is εT -concentrated to T in L
p(µ)-norm, then by (4.3)

and Lemma 4.2,

‖f‖Lp(µ) ≤ εT ‖f‖Lp(µ) + ‖PT f‖Lp(µ)

≤ εT ‖f‖Lp(µ) + K(a)
(

µ(T)
)1/p(

ν(E)
)

1
p
+ a

2α+2

‖xaf‖Lp(µ).

Thus,

(1 − εT )‖f‖Lp(µ) ≤ K(a)
(

µ(T)
)1/p(

ν(E)
)

1
p
+ a

2α+2

‖xaf‖Lp(µ),

which gives the result. 2

Another uncertainty principle for the Lp(µ) theory is obtained.

Theorem 4.4. Let E be measurable subset of [0, ∞[ such that 0 < ν(E) < ∞; and let f ∈ Lp(µ),

1 < p ≤ 2 and 0 < a < (2α + 2)(1 − 1
p
). If F(f) is εE-concentrated to E in L

q(ν)-norm,

q = p/(p − 1), then

‖F(f)‖Lq(ν) ≤
K(a)

1 − εE

(

ν(E)
)

a
2α+2

‖xaf‖Lp(µ).

Proof. Let f ∈ Lp(µ), 1 < p ≤ 2. Since F(f) is εE-concentrated to E in L
q(ν)-norm, q = p/(p−1),

then by (4.4) and Theorem 3.1,

‖F(f)‖Lq(ν) ≤ εE‖F(f)‖Lq(ν) +

(∫

E

|F(f)(λ)|qdν(λ)

)1/q

≤ εE‖F(f)‖Lq(ν) + K(a)
(

ν(E)
)

a
2α+2

‖xaf‖Lp(µ).

Thus,

(1 − εE)‖F(f)‖Lq(ν) ≤ K(a)
(

ν(E)
)

a
2α+2

‖xaf‖Lp(µ),

which proves the result. 2

Received: March 2013. Accepted: September 2013.



CUBO
16, 1 (2014)

Lp local uncertainty inequality for the Sturm-Liouville transform 103

References

[1] L. Bouattour and K. Trimèche, Beurling-Hörmander’s theorem for the Chébli-Trimèche trans-

form, Glob. J. Pure Appl. Math. 1(3) (2005) 342–357

[2] H. Chébli, Théorème de Paley-Wiener associé à un opérateur différentiel singulier sur (0, ∞),

J. Math. Pures Appl. 58(1) (1979) 1–19.

[3] R. Daher and T. Kawazoe, Generalized of Hardy’s theorem for Jacobi transform, Hiroshima J.

Math. 36(3) (2006) 331–337.

[4] R. Daher and T. Kawazoe, An uncertainty principle on Sturm-Liouville hypergroups, Proc.

Japan Acad. 83 Ser. A (2007) 167–169.

[5] R. Daher, T. Kawazoe and H. Mejjaoli, A generalization of Miyachi’s theorem, J. Math. Soc.

Japan 61(2) (2009) 551–558.

[6] D.L. Donoho and P.B. Stark, Uncertainty principles and signal recovery, SIAM J. Appl. Math.

49(3) (1989) 906–931.

[7] W.G. Faris, Inequalities and uncertainty inequalities, J. Math. Phys. 19 (1978) 461–466.

[8] S. Ghobber and P. Jaming, Strong annihilating pairs for the Fourier-Bessel transform, J. Math.

Anal. Appl. 377 (2011) 501–515.

[9] R. Ma, Heisenberg uncertainty principle on Chébli-Trimèche hypergroups, Pacific J. Math.

235(2) (2008) 289–296.

[10] M.M. Nessibi, L.T. Rachdi and K. Trimèche, The local central limit theorem on the product

of the Chébli-Trimèche hypergroup and the Euclidean hypergroup Rn, J. Math. Sci. (Calcutta)

9(2) (1998) 109–123.

[11] S. Omri, Local uncertainty principle for the Hankel transform, Int. Trans. Spec. Funct. 21(9)

(2010) 703–712.

[12] J.F. Price, Inequalities and local uncertainty principles, J. Math. Phys. 24 (1983) 1711–1714.

[13] J.F. Price, Sharp local uncertainty principles, Studia Math. 85 (1987) 37–45.

[14] M. Rösler and M. Voit, An uncertainty principle for Hankel transforms, Proc. Amer. Math.

Soc. 127(1) (1999) 183–194.

[15] E.M. Stein, Interpolation of linear operators, Trans. Amer. Math. Soc. 83 (1956) 482–492.

[16] E.M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton

Univ. Press., Princeton, N.J, (1971).



104 Fethi Soltani CUBO
16, 1 (2014)

[17] K. Trimèche, Transformation intégrale de Weyl et théorème de Paley-Wiener associés à un

opérateur différentiel singulier sur (0, ∞), J. Math. Pures Appl. 60(1) (1981) 51–98.

[18] Z. Xu, Harmonic analysis on Chébli-Trimèche hypergroups, Ph.D. thesis, Murdoch University,

Perth, Western Australia, 1994.

[19] H. Zeuner, The central limit theorem for Chébli-Trimèche hypergroups, J. Theoret. Probab.

2(1) (1989) 51–63.