Uncertainty principles for integral operators
Saifallah Ghobber, Philippe Jaming
To cite this version:
Saifallah Ghobber, Philippe Jaming. Uncertainty principles for integral operators. Studia
Mathematica, INSTYTUT MATEMATYCZNY * POLSKA AKADEMIA NAUK, 2014, 220,
pp.197–220. <hal-00704805>
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UNCERTAINTY PRINCIPLES FOR INTEGRAL OPERATORS
SAIFALLAH GHOBBER AND PHILIPPE JAMING
Abstract. The aim of this paper is to prove new uncertainty principles for an integral operator
T with a bounded kernel for which there is a Plancherel theorem. The first of these results is an
extension of Faris’s local uncertainty principle which states that if a nonzero function f ∈ L2 (Rd , µ)
is highly localized near a single point then T (f ) cannot be concentrated in a set of finite measure.
The second result extends the Benedicks-Amrein-Berthier uncertainty principle and states that a
nonzero function f ∈ L2 (Rd , µ) and its integral transform T (f ) cannot both have support of finite
measure. From these two results we deduce a global uncertainty principle of Heisenberg type for the
transformation T . We apply our results to obtain a new uncertainty principles for the Dunkl and
Clifford Fourier transforms.
1. Introduction
Uncertainty principles are mathematical results that give limitations on the simultaneous concentration of a function and its Fourier transform. They have implications in two main areas: quantum
physics and signal analysis. In quantum physics they tell us that a particle’s speed and position cannot
both be measured with infinite precision. In signal analysis they tell us that if we observe a signal
only for a finite period of time, we will lose information about the frequencies the signal consists of.
There are many ways to get the statement about concentration precise. The most famous of them
is the so called Heisenberg Uncertainty Principle [29] where concentration is measured by dispersion
and the Hardy Uncertainty Principle [26] where concentration is measured in terms of fast decay. A
little less known one consists in measuring concentration in terms of smallness of support. A considerable attention has been devoted recently to discovering new formulations and new contexts for
the uncertainty principle (see the surveys [4, 23] and the book [27] for other forms of the uncertainty
principle).
Our aim here is to consider uncertainty principles in which concentration is measured either by
(generalized) dispersion like in Heisenberg’s uncertainty principle or by the smallness of the support.
The transforms under consideration are integral operators T with polynomially bounded kernel K and
for which there is a Plancherel Theorem and include the usual Fourier transform, the Fourier-Bessel
(Hankel) transform, the Fourier-Dunkl transform and the Fourier-Clifford transform as particular
cases.
b be two convex cones in Rd (i.e. λx ∈ Ω if λ > 0 and
Let us now be more precise. Let Ω, Ω
x ∈ Ω) with non-empty interior. We endow them with Borel measures µ and µ
b. The Lebesgue
spaces Lp (Ω, µ), 1 ≤ p ≤ ∞, are then defined in the usual way. We assume that the measure µ is
absolutely continuous with respect to the Lebesgue measure and has a polar decomposition of the
form dµ(rζ) = r2a−1 dr Q(ζ) dσ(ζ) where dσ is the Lebesgue measure on the unit sphere Sd−1 of Rd
and Q ∈ L1 (Sd−1 , dσ), Q 6= 0. Then µ is homogeneous of degree 2a in the following sense: for every
continuous function f with compact support in Ω and every λ > 0,
Z
Z
x
2a
(1.1)
f
f (x) dµ(x).
dµ(x) = λ
λ
Ω
Ω
We define b
a accordingly for µ
b and assume that b
a = a.
Date: June 6, 2012.
1991 Mathematics Subject Classification. 42A68;42C20.
Key words and phrases. Uncertainty principles, annihilating pairs, Dunkl transform, Fourier-Clifford transform,
integral operators.
1
2
SAIFALLAH GHOBBER AND PHILIPPE JAMING
b −→ C be a kernel such that
Next, let K : Ω × Ω
(1) K is continuous;
b
(2) K is polynomially bounded: |K(x, ξ)| ≤ cT (1 + |x|)m (1 + |ξ|)m
;
(3) K is homogeneous: K(λx, ξ) = K(x, λξ).
One can then define the integral operator T on S(Ω) by
Z
b
(1.2)
T (f )(ξ) =
f (x)K(x, ξ) dµ(x), ξ ∈ Ω.
Ω
For ρ > 0, we define the measures dµρ (x) = (1 + |x|)ρ dµ(x) and db
µρ (ξ) = (1 + |ξ|)ρ db
µ(ξ). Then T
1
extends into a continuous operator from L (Ω, µm ) to
(
)
|f
(ξ)|
b
Cm
f continuous s.t.kf k∞,m
<∞ .
b (Ω) =
b := sup
m
b
b (1 + |ξ|)
ξ∈Ω
bλ , λ > 0:
Further, if we introduce the dilation operators Dλ , D
1 x
bλ f (x) = 1 f x ,
, D
Dλ f (x) = a f
λ
λ
λba
λ
then the homogeneity of K implies
(1.3)
b1T .
T Dλ = D
λ
Also, from the fact that µ, µ
b are absolutely continuous with respect to the Lebesgue measure, these
b µ
dilation operators are continuous from (0, ∞) × L2 (Ω, µρ ) –resp. (0, ∞) × L2 (Ω,
bρ )– to L2 (Ω, µρ )
2 b
–resp. L (Ω, µ
bρ ).
The integral operators under consideration will be assumed to satisfy some of the following proprieties that are common for Fourier-like transforms:
b µ
(1) T has an Inversion Formula: When both f ∈ L1 (Ω, µm ) and T (f ) ∈ L1 (Ω,
bm
b ) we have
f ∈ Cm (Ω) and
Z
f (x) = T −1 [T (f )](x) =
T (f )(ξ)K(x, ξ) db
µ(ξ), x ∈ Ω.
b
Ω
(2) T satisfies Plancherel’s Theorem: for every f ∈ S(Ω), kT (f )kL2 (Ω,b
b µ) = kf kL2 (Ω,µ) . In
2
b µ
particular, T extends uniquely to a unitary transform from L (Ω, µ) onto L2 (Ω,
b).1
This family of transforms include for instance the Fourier transform and the Fourier-Dunkl transform. We will also slightly relax the conditions to include the Fourier-Clifford transform. We will
here concentrate on uncertainty principles where concentration is measured in terms of dispersion or
in terms of smallness of support. Our first result will be the following local uncertainty principle that
we state here in the case m = m
b = 0 for simplicity:
Theorem A.
b be a measurable subset of finite measure 0 < µ
Assume m = m
b = 0. Let Σ ⊂ Ω
b(Σ) < ∞. Then
(1) if 0 < s < a, there is a constant C such that for all f ∈ L2 (Ω, µ),
s
h
i 2a
s kT (f )kL2 (Σ,bµ) ≤ C µ
b(Σ) |x| f L2 (Ω,µ) ;
(2) if s > a, there is a constant C such that for all f ∈ L2 (Ω, µ),
h
i 12 a a
1− s
|x|s f s 2
kT (f )kL2 (Σ,bµ) ≤ C µ
b(Σ) f L2 (Ω,µ)
.
L (Ω,µ)
1This condition implies b
a = a.
UNCERTAINTY PRINCIPLES FOR INTEGRAL OPERATORS
3
This
theorem implies that if f is highly localized in the neighborhood of 0, i.e. the dispersion
|x|s f 2
takes a small value, then T (f ) cannot be concentrated in a subset Σ of finite measure.
L (Ω,µ)
We can refer to [22, 39, 40, 41] for the history of these uncertainty inequalities.
Another uncertainty principle which is of particular interest is: a function f and its integral transform T (f ) cannot both have small support. In other words we are interested in the following adaptation of a well-known notion from Fourier analysis:
Definition.
b be two measurable subsets. Then
Let S ⊂ Ω, Σ ⊂ Ω
• (S, Σ) is a weak annihilating pair2 if, supp f ⊂ S and supp T (f ) ⊂ Σ implies f = 0.
• (S, Σ) is called a strong annihilating pair if there exists C = C(S, Σ) such that for every
f ∈ L2 (Ω, µ)
2
2
2
(1.4)
kf kL2 (Ω,µ) ≤ C kf kL2 (S c ,µ) + kT (f )kL2 (Σc ,bµ) ,
b The constant C(S, Σ) will be called the
where Ac is the complementary of the set A in Ω or Ω.
annihilation constant of (S, Σ).
Of course, every strong annihilating pair is also a weak one. To prove that a pair (S, Σ) is a
strong annihilating pair, it is enough to shows that there exists a constant D(S, Σ) such that for all
f ∈ L2 (Ω, µ) supported in S
(1.5)
kf k2L2 (Ω,µ) ≤ D(S, Σ)kT (f )k2L2 (Σc ,bµ) .
The qualitative (or weak) uncertainty principle has been considered in various places [2, 3, 13, 21,
30, 33, 35, 42]. Our main concern here is the quantitative (or strong) uncertainty principles of the
form (1.4). In his paper [16], de Jeu proved a quite general uncertainty principle for integral operators
with bounded transform. This result states that if S, Σ are sets with sufficiently small measure, then
(S, Σ) is a strong annihilating pair. One is thus lead to ask whether any pair of sets of finite measure
is strongly annihilating.
In the case of the Fourier transform, this was proved by Amrein-Berthier [1] (while the weak
counter-part was proved by Benedicks [3]). It is interesting to note that, when f ∈ L2 (Rd ) the
optimal estimate of C, which depends only on Lebesgue’s measures |S| and |Σ|, was obtained by F.
Nazarov [38] (d = 1), while in higher dimension the question is not fully settled unless either S or Σ is
convex (see [32] for the best result today). For the Fourier-Bessel/Hankel transform, this was done by
the authors in [25]. Our main result will be the following adaptation of the Benedicks-Amrein-Berthier
uncertainty principle:
Theorem B.
b be a pair of measurable subsets with 0 < µ2m (S), µ
Let S ⊂ Ω, Σ ⊂ Ω
b2m
b (Σ) < ∞. Then there exists
a constant C(S, Σ) such that for any function f ∈ L2 (Ω, µ),
kf k2L2 (Ω,µ) ≤ C(S, Σ) kf k2L2 (S c ,µ) + kT (f )k2L2 (Σc ,bµ) .
For the Fourier transform the proof of this theorem in stated in [1] where the translation and the
modulation operators plays a key role. Our theorem include essentially integral operators for which
the translation operator is not explicit (the Dunkl transform for example) or does not behave like the
ordinary translation (the Fourier-Bessel transform for example). To do so we will replace translation
by dilation and use the fact that the dilates of a C0 -function are linearly independent (see Lemma
3.4).
Finally, from either Theorem A or Theorem B we will deduce the following global uncertainty
inequality:
2see also the very similar notion of Heisenberg uniqueness pairs [28].
4
SAIFALLAH GHOBBER AND PHILIPPE JAMING
Theorem C.
For s, β > 0, there exists a constant Cs,β such that for all f ∈ L2 (Ω, µ),
2β
s s+β
|x| f 2
L (Ω,µ)
β
2s
|ξ| T (f ) s+β
2 b
L (Ω,b
µ)
≥ Cs,β kf k2L2 (Ω,µ) .
In particular when s = β = 1 we obtain a Heisenberg uncertainty principle type for the transformation T .
The structure of the paper is as follows: in the next section we will prove the local uncertainty
inequality for the transformation T . Section 3 is devoted to our Benedicks-Amrein-Berthier type
theorem and in Section 4 we apply our results for the Dunkl and the Clifford Fourier transforms.
Notation. Throughout
inner product in Rd , we
pthis paper we denote by h., .i the usual Euclidean
d
d
write for x ∈ R , |x| = hx, xi and if S is a measurable subset in R , we will write |S| for its Lebesgue
measure.
Finally, Sd−1 is the unit sphere on Rd endowed with the normalized surface measure dσ.
We will write c(T ) (resp. c(s, T )...) for a constant that depends on the parameters a, m, m̂ and cT
defined above (resp. to indicate the dependence on some other parameter s...). This constants may
change from line to line.
2. Local uncertainty principle
Local uncertainty inequalities for the Fourier transform were firstly obtained by Faris [22], and they
were subsequently sharpened and generalized by Price and Sitaram [39, 40]. Similar inequalities on
Lie groups of polynomial growth were established by Ciatti, Ricci and Sundari in [10] which is based
on [41] and further extended in [36]
First from the polar decomposition of our measure we remark that

Z
dµ(x)


< ∞, 0 < s < a;
C1 (s) :=
2s
ZΩ∩{|x|≤1} |x|
(2.6)
dµ(x)


< ∞,
s > a.
C2 (s) :=
2s
Ω (1 + |x|)
Theorem 2.1.
b be a measurable subset of finite measure 0 < µ
Let Σ ⊂ Ω
b2m
b (Σ) < ∞. Then
(1) if 0 < s < a, there is a constant c(s, T ) such that for all f ∈ L2 (Ω, µ),

s
h
i 2(a+m)
s 
c(s, T ) µ
|x| f 2
b2m
, if µ
b2m
b (Σ)
b (Σ) ≤ 1;
L (Ω,µ)
s
(2.7)
kT (f )kL2 (Σ,bµ) ≤
h
i 2a

|x|s f 2
c(s, T ) µ
b2m
,
if µ
b2m
b (Σ)
b (Σ) > 1;
L (Ω,µ)
(2) if a ≤ s ≤ a + m then, for every ε > 0 there is a constant c(s, T , ε) such that for all
f ∈ L2 (Ω, µ),

1
h
i 2(1+m/a)
−ε 1− a +ε a

c(s, T , ε) µ
f 2 s |x|s f s 2−ε , if µ
b2m
b2m
b (Σ)
b (Σ) ≤ 1;
L (Ω,µ)
L (Ω,µ)
kT (f )kL2 (Σ,bµ) ≤
h
i 21 −ε a a

c(s, T , ε) µ
f 1−2 s +ε |x|s f s 2−ε ,
b2m
if µ
b2m
b (Σ) > 1;
b (Σ)
L (Ω,µ)
L (Ω,µ)
(3) if s > m + a, there is a constant c(s, T ) such that for all f ∈ L2 (Ω, µ),

h
i 21 a a

c(s, T ) µ
f 1−2 s |x|s f s 2
b2m
(Σ)
, if m = 0;
b
L (Ω,µ)
L (Ω,µ)
(2.8)
kT (f )kL2 (Σ,bµ) ≤
h
i 12 
c(s, T ) µ
f 2
b2m
,
otherwise.
b (Σ)
L (Ω,µ2s )
UNCERTAINTY PRINCIPLES FOR INTEGRAL OPERATORS
5
Proof. As for the first part take r > 0 and let χr = χΩ∩{|x|≤r} and χ˜r = 1 − χr . We may then write
kT (f )kL2 (Σ,bµ) = kT (f )χΣ kL2 (Ω,b
b µ) ≤ kT (f χr )χΣ kL2 (Ω,b
b µ) + kT (f χ˜r )kL2 (Ω,b
b µ) ,
hence, it follows from Plancherel’s theorem that
1/2
kT (f )kL2 (Σ,bµ) ≤ µ
b2m
kT (f χr )k∞,m
b (Σ)
b + kf χ˜r kL2 (Ω,µ) .
Now we have
kT (f χr )k∞,m
b
≤
≤
On the other hand,
so that
s −s
cT kf χr kL1 (Ω,µm ) ≤ cT |x| (1 + |x|)m χr L2 (Ω,µ) |x| f L2 (Ω,µ)
p
s cT C1 (s) (1 + r)m ra−s |x| f L2 (Ω,µ) .
s s −s kf χ˜r kL2 (Ω,µ) ≤ |x| χ˜r L∞ (Ω,µ) |x| f L2 (Ω,µ) = r−s |x| f L2 (Ω,µ) ,
p
s 1/2 kT (f )kL2 (Σ,bµ) ≤ r−s + 2cT C1 (s) (1 + r)m ra−s µ
b2m
|x| f L2 (Ω,µ) .
b (Σ)
−1/2a
If µ
b2m
b2m
< 1 (thus (1 + r)m ≤ 2m ) to obtain that there is a constant
b (Σ) > 1 we take r = µ
b (Σ)
C depending only on s and T such that
s s/2a kT (f )kL2 (Σ,bµ) ≤ C µ
b2m
|x| f L2 (Ω,µ) .
b (Σ)
−1/2(a+m)
If µ
b2m
b2m
> 1 (thus (1 + r)m ≤ 2m rm ) to obtain that there is a
b (Σ) < 1 we take r = µ
b (Σ)
constant C depending only on s and T such that
s s/2(a+m) kT (f )kL2 (Σ,bµ) ≤ C µ
b2m
|x| f L2 (Ω,µ) .
b (Σ)
Next, take 0 < σ < a ≤ s ≤ a + m, apply the first part with σ replacing s and then apply the
classical inequality
1− σ
σ
s
k|x|σ f kL2 (Ω,µ) ≤ C(σ, s)kf kL2 (Ω,µ)
k|x|s f kLs 2 (Ω,µ) .
As for the last part we write
Moreover
1/2
1/2
kT (f )kL2 (Σ,bµ) ≤ µ
b2m
kT (f )k∞,m
b2m
kf kL1 (Ω,µm ) .
b (Σ)
b (Σ)
b ≤ cT µ
2
kf kL1 (Ω,µm )
=
Z
Ω
=
≤
Z
2
(1 + |x|) |f (x)| dµ(x)
m
2
(1 + |x|)−(s−m) (1 + |x|)s |f (x)| dµ(x)
Ω
Z
C2 (s − m) (1 + |x|)2s |f (x)|2 dµ(x).
Ω
Further, if m = 0, then this last inequality implies
2
kf k2L1 (Ω,µ) ≤ 22s C2 (s) kf k2L2 (Ω,µ) + |x|s f L2 (Ω,µ) .
Replacing f by Dλ f , λ > 0 in this inequality, gives
2
s 2
2
kf kL1 (Ω,µ) ≤ 22s C2 (s) λ−2a f L2 (Ω,µ) + λ2(s−a) |x| f L2 (Ω,µ) .
Minimizing the right hand side of that inequality over λ > 0, we obtain the desired result.
We now show that local uncertainty principle implies a global uncertainty principle type for T .
For sake of simplicity, we will assume that m = m
b = 0. The general case will be treated in the next
section.
6
SAIFALLAH GHOBBER AND PHILIPPE JAMING
Corollary 2.2.
Assume that m = m
b = 0. For s, β > 0, s 6= a there exists a constant C = C(s, β, T ) such that for all
f ∈ L2 (Ω, µ),
β
s
s s+β
β
s+β
|x| f 2
(2.9)
b µ) ≥ Ckf kL2 (Ω,µ) .
L (Ω,µ) |ξ| T (f ) L2 (Ω,b
b ∩ {x : |x| ≤ r} and Brc = Ω
b \ Br .
Proof. In this proof, we will denote by Br = Ω
Let 0 < s < a and β > 0. Then, using Plancherel’s theorem and Theorem 2.1 (1),
2
kf kL2 (Ω,µ)
=
≤
≤
2
2
2
kT (f )kL2 (Ω,b
b µ) = kT (f )kL2 (Br ,b
µ) + kT (f )kL2 (Brc ,b
µ)
2
2
s s
−2β β
a
|ξ| T (f ) L2 (Ω,b
c(s, T )b
µ(Br ) |x| f L2 (Ω,µ) + r
b µ)
2
2
s
c′ (s, T )r2s |x| f L2 (Ω,µ) + r−2β |ξ|β T (f )L2 (Ω,b
b µ) .
The desired result follows by minimizing the right hand side of that inequality over r > 0.
For s > a and β > 0 we deduce from Plancherel’s theorem and Theorem 2.1 (3) that
kf k2L2 (Ω,µ)
2
2
kT (f )k2L2 (Ω,b
b µ) = kT (f )kL2 (Br ,b
µ) + kT (f )kL2 (Brc ,b
µ)
2a
2a
2−
s
s
c(s, T )2 f L2 (Ω,µ)
µ
b(Br )|x| f Ls2 (Ω,µ) + kT (f )k2L2 (B c ,bµ) .
=
≤
(2.10)
r
But, using Plancherel’s theorem again,
2− 2a
2a
2
2a
2− 2a
s
s
s
kT (f )kL2 (Brc ,bµ) ≤ kT (f )kLs2 (B c ,bµ) kT (f )kL2 (Ω,b
b µ) = kT (f )kL2 (B c ,b
µ) kf kL2 (Ω,µ)
r
r
2− 2a
s
to obtain
so that, in (2.10), we may simplify by kf kL2 (Ω,µ)
2a
kf kLs2 (Ω,µ)
≤
≤
s 2a
2a
c(s, T )2 µ
b(Br )|x| f Ls2 (Ω,µ) + kT (f )kLs2 (B c ,bµ) .
r
s 2a
2a
2aβ ′
2a −
β
s
c (s, T ) r |x| f 2
+ r s |ξ| T (f ) s
L (Ω,µ)
b µ) .
L2 (Ω,b
The desired result follows by minimizing the right hand side of that inequality over r > 0.
Inequality (2.9) has been obtained by Cowling and Price [12] for the Fourier transform on Rd and
later generalized in [36] for any pair of positive self-adjoint operators on a Hilbert space. In particular
when s = β = 1 we obtain a version of Heisenberg’s uncertainty principle for the operator T . Moreover
if the function f ∈ L2 (Ω, µ) is supported in a subset S of finite measure one can easily obtain bounds
on T (f ) that limit the concentration of T (f ) in any small set and may provide lower bounds for the
concentration of T (f ) in sufficiently large sets. For instance we have this simple local uncertainty
inequality : if f is supported in a set S with finite measure µ2m (S) < ∞, then
kT (f )k2L2 (Σ,bµ)
(2.11)
≤
≤
2
2
2
b2m
µ
b2m
b (Σ)kT (f )k∞,m
b (Σ)kf kL1 (Ω,µm )
b ≤ cT µ
2
c2T µ2m (S)b
µ2m
b (Σ)kf kL2 (Ω,µ) ,
−2
which implies that the pair (S, Σ) is strongly annihilating provided that µ2m (S)b
µ2m
b (Σ) < cT . In the
next section we will prove this result for arbitrary subsets S and Σ of finite measure.
3. Pairs of sets of finite measure are strongly annihilating
b are sets of finite measure 0 < µ2m (S), µ
In this section we will show that, if S ⊂ Ω, Σ ⊂ Ω
b2m
b (Σ) <
∞, then the pair (S, Σ) is strongly annihilating for the operator T . In order to prove this, we will
need to introduce a pair of orthogonal projections on L2 (Ω, µ) defined by
ES f = χS f,
FΣ = T −1 EΣ T ,
b are measurable subsets.
where S ⊂ Ω and Σ ⊂ Ω
We will need the following well-known lemma (see e.g. [25, Lemma 4.1]):
UNCERTAINTY PRINCIPLES FOR INTEGRAL OPERATORS
7
Lemma 3.1.
If kES FΣ k := kES FΣ kL2 (Ω,µ)→L2 (Ω,µ) < 1, then
(3.12)
kf k2L2(Ω,µ) ≤ (1 − kES FΣ k)−2 kES c f k2L2 (Ω,µ) + kFΣc f k2L2 (Ω,µ) .
Unfortunately, showing that kES FΣ k < 1 is in general difficult. However, the Hilbert-Schmidt
norm kES FΣ kHS is much easier to compute. Let us illustrate this fact by showing that, if S and Σ are
subsets with sufficiently small measure then the pair (S, Σ) is strongly annihilating. We can deduce
this result easily from (2.11), but we will give here another proof that we will use later.
Lemma 3.2.
−2
2
If µ2m (S)b
µ2m
b (Σ) < cT , then for all function f ∈ L (Ω, µ),
−2 p
2
2
kf k2L2(Ω,µ) ≤ 1 − cT µ2m (S)b
µ2m
(Σ)
kf
k
+
kT
(f
)k
2
c
2
c
b
L (S ,µ)
L (Σ ,b
µ) .
b
Proof. We have, for f ∈ L2 (Ω, µ), |T (f )(η)| ≤ cT (1+|ξ|)m
kf k∞,m thus if µ
b2m
b (Σ) < ∞, χΣ (η)T (f )(η) ∈
1 b
L (Ω, µ
bm
b ). The Inversion Formula for T thus gives
Z
ES FΣ f (y) = χS (y)
χΣ (η)T (f )(η)K(y, η) db
µ(η)
b
Z
ZΩ
µ(η)
= χS (y)
χΣ (η)
f (x)K(x, η) dµ(x) K(y, η) db
b
Ω
Ω
Z
f (x)N (x, y) dµ(x),
=
Ω
where
N (x, y)
Z
χΣ (η)K(x, η)K(y, η) db
µ(η)
=
χS (y)
=
Z
χS (y) χΣ (η)K(y, η)K(x, η) db
µ(η)
=
χS
b
Ω
b
Ω
(y)T −1
[χΣ (·)K(y, ·)] (x).
Here we appealed repeatedly to Fubini’s theorem which is justified by the fact that µ
b2m
b (Σ) < ∞ and
b
K is bounded by cT (1 + |x|)m (1 + |ξ|)m
.
This shows that ES FΣ is an integral operator with kernel N . But, with Plancherel’s theorem,
Z
Z
−1
2
2
2
T [χΣ (·)K(y, ·)] (x) dµ(x) dµ(y)
kN kL2 (Ω,µ)⊗L2 (Ω,µ) =
|χS (y)|
Ω
Z Ω
Z
2
|χΣ (η)K(y, η)| db
µ(η) dµ(y)
|χS (y)|
=
Ω
b
Ω
≤ c2T µ2m (S)b
µ2m
b (Σ)
b
since |K(y, η)| ≤ cT (1 + |y|)m (1 + |η|)m
. It follows that the Hilbert-Schmidt norm of ES FΣ is bounded:
p
(3.13)
kES FΣ kHS = kN kL2 (Ω,µ)⊗L2 (Ω,µ) ≤ cT µ2m (S)b
µ2m
b (Σ).
Now using the fact that kES FΣ k ≤ kES FΣ kHS , we obtain
p
kES FΣ k ≤ cT µ2m (S)b
µ2m
b (Σ).
It follows from Lemma 3.1 that
−2 p
(3.14)
kf k2L2 (Ω,µ) ≤ 1 − cT µ2m (S)b
µ2m
kES c f k2L2 (Ω,µ) + kFΣc f k2L2 (Ω,µ) .
b (Σ)
Plancherel’s theorem then gives kFΣc f k2L2 (Ω,µ) = kT (f )k2L2 (Σc ,bµ) which allows to conclude.
8
SAIFALLAH GHOBBER AND PHILIPPE JAMING
Remark 3.3.
Let S, Σ be two sets with µ2m (S), µ
b2m
b (Σ) < ∞. Let ε1 , ε2 > 0. Assume that there is a function
f ∈ L2 (Ω, µ) with kf kL2 (Ω,µ) = 1 that is ε1 -concentrated on S, i.e. kES c f kL2 (Ω,µ) ≤ ε1 and ε2 bandlimited on Σ for the transformation T , i.e. kFΣc f kL2 (Ω,µ) ≤ ε2 .
−2
Then either µ2m (S)b
µ2m
b (Σ) ≥ cT or we may apply Inequality (3.14) and obtain
q
p
µ2m
ε21 + ε22 .
1 − cT µ2m (S)b
b (Σ) ≤
In both cases, we obtain
2
q
−2
2 + ε2
,
µ2m (S)b
µ2m
(Σ)
≥
c
1
−
ε
b
1
2
T
(3.15)
which is Donoho-Stark’s uncertainty inequality for the integral operator T . This inequality improves
slightly the result of de Jeu [16]. In the case of the Fourier transform, it dates back to Donoho and
Stark [17] in a slightly weaker form and to [31] to the form (3.15).
Before proving our main theorem, we will now prove the following lemma which results directly
from a similar result in [25] for functions in C0 (R+ ).
Lemma 3.4.
Let f be a function in L2 (Ω, µ) and assume that 0 < µ(supp f ) < ∞. Then the dilates {Dλ f }λ>0 are
linearly independent.
Proof. Let ζ ∈ Sd−1 ∩ Ω and consider
(
ta−1/2 f (tζ), for t > 0;
fζ (t) =
0,
for t < 0.
For ζ ∈ Sd−1 ∩ Ωc , we just define fζ = 0.
Then, there exists ζ such that fζ ∈ L2 (R) and 0 < |supp fζ | < ∞, in particular, fζ ∈ L1 (R). Indeed
the first property holds for almost every ζ since
Z
Z
2
|fζ (t)|2 dt Q(θ) dσ(θ) = kf kL2 (Ω,µ) < ∞.
Sd−1
R
As for the second one, notice that
|supp fζ | ≤ |[0, 1]| +
Z
r2a−1 dr.
supp fζ ∩[1,∞)
Integrating with respect to Q(ζ) dσ(ζ) we get
Z
|supp fζ |Q(ζ) dσ(ζ) ≤ kQkL1 (Sd−1 ∩Ω) + µ (supp f ∩ {|x| > 1}) < ∞.
Sd−1 ∩Ω
We thus proved that |supp fζ | < ∞ for almost every ζ. Finally, |supp fζ | > 0 on a set of ζ’s of positive
dσ measure, otherwise the support of f would have Lebesgue measure 0, thus µ-measure zero.
Now assume that we had a vanishing linear combination of dilates of f :
X
(3.16)
αi f (x/λi ) = 0.
f inite
Then, for t > 0 and the above ζ
a−1/2 a−1/2 X
X
t
t
1
λi
βi fζ (t/λi ) = 0
f
ζ = a−1/2
αi
t
λi
λi
t
f inite
f inite
a−1/2
where we have set βi = αi λi
. Thus
X
f inite
βi fζ (t/λi ) = 0.
UNCERTAINTY PRINCIPLES FOR INTEGRAL OPERATORS
9
Taking the Euclidean Fourier transform F , we obtain
X
βi λi F (fζ )(λi x) = 0.
f inite
1
But, as fζ ∈ L (R), it follows from Riemann-Lebesgue’s Lemma that F (fζ ) ∈ C0 (R). It remains to
invoke [25, Lemma 2.1] to see that the dilates of F (fζ ) are linearly independent so that the βi ’s thus
the αi ’s are 0.
We can now state our main theorem:
Theorem 3.5.
b be a pair of measurable subsets with 0 < µ2m (S), µ
Let S ⊂ Ω, Σ ⊂ Ω
b2m
b (Σ) < ∞. Then any function
f ∈ L2 (Ω, µ) vanishes as soon as f is supported in S and T (f ) is supported in Σ. In other words,
(S, Σ) is a weak annihilating pair.
Proof. We will write ES ∩ FΣ for the orthogonal projection onto the intersection of the ranges of ES
and FΣ and we denote by Im P the range of a linear operator P.
First we will need the following elementary fact on Hilbert-Schmidt operators:
(3.17)
2
2
dim(Im ES ∩ Im FΣ ) = kES ∩ FΣ kHS ≤ kES FΣ kHS .
Since µ2m (S), µ2m
b (Σ) < ∞, from Inequality (3.13) we deduce that
(3.18)
dim(Im ES ∩ Im FΣ ) < ∞.
Assume now that there exists f0 6= 0 such that S0 := supp f0 and Σ0 := supp T (f0 ) have both
finite measure 0 < µ2m (S0 ), µ
b2m
b (Σ0 ) < ∞, thus also µ(S0 ) < ∞ so that Lemma 3.4 applies.
b of finite measure 0 < µ2m (S1 ) < ∞
Next, let S1 (resp. Σ1 ) be a measurable subset of Ω (resp. Ω)
(resp. 0 < µ
b2m
b (Σ1 ) < ∞), such that S0 ⊂ S1 (resp. Σ0 ⊂ Σ1 ). Since for λ > 0,
2
µ2m (S1 ∪ λS0 ) = kχλS0 − χS1 kL2 (Ω,µ2m ) + hχλS0 , χS1 iL2 (Ω,µ2m ) ,
the function λ 7→ µ2m (S1 ∪ λS0 ) is continuous on R+ \{0}. The same holds for λ 7→ µ
b2m
b (Σ1 ∪ λΣ0 ).
From this, one easily deduces that, there exists an infinite sequence of distinct numbers (λj )∞
j=0 ⊂
∞
∞
[
[
1
Σ0 ,
R+ \{0} with λ0 = 1, such that, if we denote by S =
λj S0 and Σ =
λ
j=0
j=0 j
µ2m (S) < 2µ(S0 ),
µ
b2m
µ2m
b (Σ) < 2b
b (Σ0 ).
ab
We next define fi = Dλi f0 , so that supp fi = λi S0 ⊂ S. Since T (fi ) = λa−b
D λ1 T (f0 ), we have
i
i
supp T (fi ) = λ1i Σ0 ⊂ Σ.
As supp f0 has finite measure, it follows from Lemma 3.4 that (fi )∞
i=0 are linearly independent
vectors belonging to (Im ES ∩ Im FΣ ), which contradicts (3.18).
Remark 3.6.
The theorem can be extended to operators T that take their values in a finite dimensional Banach
algebra.
The proof given here follows roughly the scheme of Amrein-Berthier’s original one in [1]. It can
obviously be adapted so as to replace dilations by actions of more general groups on measure spaces.
The main difficulty would be to prove that this action leads to linearly independent functions as in
Lemma 3.4. As we have no specific application in mind, we refrain from stating a more general result.
A simple well known functional analysis argument allows us to obtain the following improvement
(see e.g. [4, Proposition 2.6]):
Corollary 3.7.
b be a pair of measurable subsets of finite measure, 0 < µ2m (S), µ
Let S ⊂ Ω, Σ ⊂ Ω
b2m
b (Σ) < ∞. Then
there exists a constant C(S, Σ) such that for all f ∈ L2 (Ω, µ),
kf k2L2 (Ω,µ) ≤ C(S, Σ) kf k2L2 (S c ,µ) + kT (f )k2L2 (Σc ,bµ) .
10
SAIFALLAH GHOBBER AND PHILIPPE JAMING
Proof. Assume there is no such constant D(S, Σ) such that for every function f ∈ L2 (Ω, µ) supported
in S,
kf k2L2(Ω,µ) ≤ D(S, Σ)kT (f )k2L2 (Σc ,bµ) .
Then, there exists a sequence fn ∈ L2 (Ω, µ) with kfn kL2 (Ω,µ) = 1 and with support in S such that
2
kEΣc T (fn )kL2 (Ω,b
b µ) converge to 0. Moreover, we may assume that fn is weakly convergent in L (Ω, µ)
with some limit f . As T (fn )(ξ) is the scalar product of fn and ES K(·, ξ), it follows that T (fn ) converge
b
to T (f ). Finally, as |T (fn )(ξ)|2 is bounded by c2T µ2m (S)(1+|ξ|)2m
, we may apply Lebesgue’s theorem,
2 b
thus EΣ T (fn ) converges to EΣ T (f ) in L (Ω, µ
b). But we have supp f ⊂ S and supp T (f ) ⊂ Σ so by
Theorem 3.5, f is 0, which contradicts the fact that f has norm 1.
Now we will show a global uncertainty inequality type for the transformation T . But this time we
will use Corollary 3.7 and the proof here is simpler than that using the local uncertainty principle and
not necessary with the same constant.
Corollary 3.8.
Let s, β > 0. Then there exists a constant C = C(s, β, a) such that for all f ∈ L2 (Ω, µ),
2β
2s
β
s+β
s s+β
2
|x| f 2
(3.19)
b µ) ≥ Ckf kL2 (Ω,µ) .
L (Ω,µ) |ξ| T (f ) L2 (Ω,b
b1 = Ω
b ∩ {ξ : |ξ| ≤ 1}. Let B c = Ω \ B1 and B
bc = Ω
b \B
b1 .
Proof. Let B1 = Ω ∩ {x : |x| ≤ 1} and B
1
1
c
From Corollary 3.7 there exists a constant C = C(B1 , B1 ) such that
kf k2L2 (Ω,µ) ≤ C kf k2L2(B1c ,µ) + kT (f )k2L2 (Bbc ,bµ) .
1
It follows then
kf k2L2(Ω,µ)
2
2
≤ C |x|s f L2 (B c ,µ) + |ξ|β T (f )L2 (Bbc ,bµ)
1
1
β
2
s 2
≤ C |x| f L2 (Ω,µ) + |ξ| T (f ) L2 (Ω,b
b µ) .
Replacing f by Dλ f in the last inequality we have by (1.3)
β
Dλ f 2 2
|x|s Dλ f 2 2
|ξ| D
b 1 T (f )2 2 b
(3.20)
≤
C
+
,
L (Ω,µ)
L (Ω,µ)
L (Ω,b
µ)
λ
which gives
2
2
2s s 2
−2β β
f 2
2 b
≤
C
λ
|x|
f
+
λ
|ξ|
T
(f
)
.
2
L (Ω,µ)
L (Ω,µ)
L (Ω,b
µ)
The desired result follows by minimizing the right hand side of that inequality over λ > 0.
Let us notice that Theorem 3.5 is valid in the L1 -version. Precisely we have the following proposition:
Proposition 3.9.
b be a pair of measurable subsets with 0 < µ2m (S), µ
Let S ⊂ Ω, Σ ⊂ Ω
b2m
b (Σ) < ∞. Suppose
f ∈ L1 (Ω, µm ) (in particular f ∈ L1 (Ω, µ)) verifies supp f ⊂ S and supp T (f ) ⊂ Σ, then f = 0.
b
b µ
Proof. If f ∈ L1 (Ω, µm ), then (1 + |ξ|)−m
T (f ) ∈ L∞ (Ω,
b). Then
−m
b
kT (f )kL1 (Ω,b
=
kχ
T
(f
)k
≤
µ
b
(Σ)
(1
+
|ξ|)
T
(f
)
1
Σ
2
m
b
b µm
b µm
L (Ω,b
c)
c)
b µ)
L∞ (Ω,b
< ∞.
−m
b µ
This implies that T (f ) ∈ L1 (Ω,
bm
f ∈ L∞ (Ω, µ). Finally,
b ), thus (1 + |x|)
Z
kf k2L2 (Ω,µ) =
(1 + |x|)−m |f (x)||f (x)|(1 + |x|)m dµ(x)
Ω
≤ (1 + |x|)−m f L∞ (Ω,µ) kf kL1 (Ω,µm ) < ∞,
hence f ∈ L2 (Ω, µ). By Theorem 3.5 we have f = 0.
The same argument as the one used in the proof of Corollary 3.7 gives the following result :
UNCERTAINTY PRINCIPLES FOR INTEGRAL OPERATORS
11
Proposition 3.10.
b be a pair of measurable subsets with 0 < µ2m (S), µ
Let S ⊂ Ω, Σ ⊂ Ω
b2m
b (Σ) < ∞. Then there exists
a constant D(S, Σ) such that for all function f ∈ L1 (Ω, µ) supported in S,
kf kL1 (Ω,µ) ≤ D(S, Σ)kT (f )kL1 (Σc ,bµ) .
4. Examples
4.1. The Fourier transform and the Fourier-Bessel transform.
Let dµ(x) = (2π)−d/2 dx the Lebesgue measure and T = F the Fourier transform. For f ∈
1
L (Rd , µ) ∩ L2 (Rd , µ), the Fourier transform is defined by
Z
F (f )(ξ) =
f (x)e−ihx,ξi dµ(x),
ξ ∈ Rd ;
Rd
2
d
and is then extended to all L (R , µ) in the usual way. In this case we take cT = 1, a = d/2 and
m = m
b = 0. Then (3.15) is Donoho-Stark’s theorem [17, 31], Corollary 3.7 is Amrein-Berthier’s
theorem [1] while the local and the global uncertainty principles for the Fourier transform date back
respectively to [39, 40] and [12]. Note that our proof of Theorem 3.5 is inspired by the one established
in [1] where we replace translation by dilation.
If f (x) = f0 (|x|) is a radial function on Rd , then
Z ∞
1
f0 (t)jd/2−1 (t|ξ|)td−1 dt = Fd/2−1 (f0 )(|ξ|),
F (f )(ξ) = d/2−1
2
Γ(d) 0
where Fd/2−1 is the Fourier-Bessel transform of index d/2 − 1. For α ≥ −1/2, jα is the Bessel function
given by
∞
x 2n
X
Jα (x)
(−1)n
jα (x) = 2α Γ(α + 1) α := Γ(α + 1)
,
x
n!Γ(n + α + 1) 2
n=0
where Γ is the gamma function.
1
x2α+1 dx, then for f ∈ L1 (R+ , µα ) ∩
We have |jα | ≤ 1 and if we denote dµα (x) = 2α Γ(α+1)
L2 (R+ , µα ), the Fourier-Bessel (or Hankel) transform is defined by
Z ∞
Fα (f )(ξ) =
f (x)jα (xξ) dµα (x), ξ ∈ R+ ;
0
and extends to an isometric isomorphism on L2 (R+ , µα ) with Fα−1 = Fα . Theorem A and Theorem
B has been stated in [25] for this transformation. Moreover we have the following two new results.
Theorem 4.1 (Donoho-Stark’s uncertainty principle for Fα ).
Let S, Σ be a pair of measurable subsets of R+ and α > −1/2. If f ∈ L2 (R+ , µα ) of unit L2 -norm is
ε1 -concentrated on S and ε2 -bandlimited on Σ for the the Fourier-Bessel transform, then
2
2
q
q
2
2
2
2
(4.21)
µα (S)µα (Σ) ≥ 1 − ε1 + ε2
and |S||Σ| ≥ cα 1 − ε1 + ε2 ,
where cα is a numerical constant that depends only on α.
This result improves the estimate in [37] (which has already improved [49]) showing that, if f of
unit L2 -norm is ε1 -concentrated on S and ε2 -bandlimited on Σ, then
|S||Σ| ≥ c′α (1 − ε1 − ε2 )2 .
Theorem 4.2 (Global uncertainty principle for Fα ).
For s, β > 0, there exists a constant Cs,β,α such that for all f ∈ L2 (R+ , µα ),
2β
2s
s s+β
β
s+β
2
x f 2 +
L (R ,µα ) ξ Fα (f ) L2 (R+ ,µα ) ≥ Cs,β,α kf kL2 (R+ ,µα ) .
The case when s = β = 1 has been established in [5, 45] with the optimal constant C1,1,α = α + 1.
12
SAIFALLAH GHOBBER AND PHILIPPE JAMING
4.2. The Fourier-Dunkl transform.
In this section we will deduce new uncertainty principles for the Dunkl transform. Uncertainty
principles for this transformation have been considered in various places, e.g. [43, 47] for a Heisenberg
type inequality or [24] for Hardy type uncertainty principles and recently [11, 34] for a generalization
and a variant of Cowling-Price’s theorem, Beurling’s theorem, Miyachi’s theorem and Donoho-Stark’s
uncertainty principle.
Let us fix some notation and present some necessary material on the Dunkl transform. Let G be
a finite reflection group on Rd , associated with a root system R and R+ the positive subsystem of R
(see [15, 19, 46]). We denote by k a nonnegative multiplicity function defined on R with the property
that k is G-invariant. We associate with k the index
X
γ := γ(k) =
k(ξ) ≥ 0
ξ∈R+
and the weight function wk defined by
wk (x) =
Y
|hξ, xi|2k(ξ) .
ξ∈R+
Further we introduce the Mehta-type constant ck by
−1
Z
2
− |x|
ck =
,
e 2 dµk (x)
Rd
3
where dµk (x) = wk (x) dx. Moreover
Z
wk (x) dσ(x) =
Sd−1
c−1
k
γ+d/2−1
2
Γ(γ
+ d/2)
= dk .
By using the homogeneity of wk it is shown in [46] that for a radial function f ∈ L1 (Rd , µk ) the
function fe defined on R+ by f (x) = fe(|x|), for all x ∈ Rd is integrable with respect to the measure
r2γ+d−1 dr. More precisely,
Z Z
Z
f (x)wk (x) dx =
wk (ry) dσ(y) fe(r)rd−1 dr
Sd−1
Rd
R+
Z
= dk
(4.22)
fe(r)r2γ+d−1 dr.
R+
Introduced by C. F. Dunkl in [18], the Dunkl operators Tj , 1 ≤ j ≤ d on Rd associated with the
reflection group G and the multiplicity function k are the first-order differential-difference operators
given by
X
∂f
f (x) − f (σξ (x))
Tj f (x) =
k(ξ)ξj
+
,
x ∈ Rd ;
∂xj
hξ, xi
ξ∈R+
where f is an infinitely differentiable function on Rd , ξj = hξ, ej i, (e1 , . . . , ed ) being the canonical
basis of Rd and σξ denotes the reflection with respect to the hyperplane orthogonal to ξ.
The Dunkl kernel Kk on Rd × Rd has been introduced by C. F. Dunkl in [19]. For y ∈ Rd the
function x 7→ Kk (x, y) can be viewed as the solution on Rd of the following initial problem
Tj u(x, y) = yj u(x, y); 1 ≤ j ≤ d,
d
u(0, y) = 1.
d
This kernel has a unique holomorphic extension to C × C . M. Rösler has proved in [44] the following
integral representation for the Dunkl kernel
Z
Kk (x, z) =
ehy,zi dµkx (y),
x ∈ Rd ,
z ∈ Cd ;
Rd
3we chose here to stick to the notation that is usual in Dunkl analysis rather than that of the previous section in
which µk is simply denoted by µ.
UNCERTAINTY PRINCIPLES FOR INTEGRAL OPERATORS
13
where µkx is a probability measure on Rd with support in the closed ball B|x| . We have (see [44]) for
all λ ∈ C, z, z ′ ∈ Cd and x, y ∈ Rd
Kk (z, z ′ ) = Kk (z ′ , z), Kk (λz, z ′ ) = Kk (z, λz ′ ), Kk (−iy, x) = Kk (iy, x), |Kk (−iy, x)| ≤ 1.
The Dunkl transform Fk of a function f ∈ L1 (Rd , µk ) ∩ L2 (Rd , µk ) which was introduced by C. F.
Dunkl (see [15, 20]), is given by
Z
Kk (−iξ, x)f (x) dµk (x),
ξ ∈ Rd ;
Fk (f )(ξ) := ck
Rd
and extends uniquely to an isometric isomorphism on L2 (Rd , µk ) with Fk−1 (f )(ξ) = Fk (f )(−ξ).
The Dunkl transform Fk provides a natural generalization of the Fourier transform F , to which it
reduces in the case k = 0, and if f (x) = fe(|x|) is a radial function on Rd , then
Fk (f )(ξ) = Fγ+d/2−1 (fe)(|ξ|),
where Fγ+d/2−1 is the Fourier-Bessel transform of index γ + d/2 − 1.
Now if we take cT = ck , a = γ + d/2 and m = m
b = 0, then from Section 2 and 3 we obtain a new
uncertainty principles for the Dunkl transform Fk .
Theorem 4.3 (Donoho-Stark’s uncertainty principle for Fk ).
Let S, Σ be a pair of measurable subsets of Rd . If f ∈ L2 (Rd , µk ) of unit L2 -norm is ε1 -concentrated
on S and ε2 -bandlimited on Σ for the Dunkl transform, then
2
q
−2
2
2
(4.23)
µk (S)µk (Σ) ≥ ck
1 − ε1 + ε2 .
Note that the Donoho-Stark’s uncertainty principle has recently been proved in [34] for the Dunkl
transform but our inequality (4.23) is a little stronger.
Let us now state how our results translate to the Fourier-Dunkl transform. These results are new
to our knowledge.
Theorem 4.4.
Let S, Σ be a pair of measurable subsets of Rd with finite measure, 0 < µk (S), µk (Σ) < ∞. Then the
following uncertainty principles hold.
(1) Local uncertainty principle for Fk :
(a) For 0 < s < γ + d/2, there is a constant c(s, k) such that for all f ∈ L2 (Rd , µk ),
s
h
i 2γ+d
s
kFk (f )kL2 (Σ,µk ) ≤ c(s, k) µk (Σ)
k|x| f kL2 (Rd ,µk ) .
(b) For s > γ + d/2, there is a constant c′ (s, k) such that for all f ∈ L2 (Rd , µk ),
h
i 12
2γ+d
1− 2γ+d
s
2s
kFk (f )kL2 (Σ,,µk ) ≤ c′ (s, k) µk (Σ) kf kL2 (R2s
d ,µ ) k|x| f kL2 (Rd ,µ ) .
k
k
(2) Benedicks-Amrein-Berthier’s uncertainty principle for Fk :
There exists a constant Ck (S, Σ) such that for all f ∈ L2 (Rd , µk ),
kf k2L2 (Rd ,µk ) ≤ Ck (S, Σ) kf k2L2 (S c ,µk ) + kFk (f )k2L2 (Σc ,µk ) .
(3) Global uncertainty principle for Fk :
For s, β > 0, there exists a constant Cs,β,k such that for all f ∈ L2 (Rd , µk ),
2β
s s+β
β
2s
|x| f 2 d
|ξ| Fk (f ) s+β
≥ Cs,β,k kf k2L2 (Rd ,µk ) .
L (R ,µk )
L2 (Rd ,µk )
A simple computation shows that
c(s, k) =
i
2γ + d h ck p
(2γ + d − 2s)dk
2γ + d − 2s 2s
2γ+d
2s
14
SAIFALLAH GHOBBER AND PHILIPPE JAMING
and
′
c (s, k) = ck
"
dk
2γ + d
2s
−1
2γ + d
2γ+d
2s −1
Γ
2γ + d
2s
#1/2
2γ + d
Γ 1−
.
2s
In the particular case s = β = 1 for the global uncertainty principle, we recover Heinsenberg’s
inequality for the Dunkl transform but with C1,1,k ≤ γ + d/2, where γ + d/2 is the optimal constant
in the Heisenberg uncertainty principle given in [43, 47].
4.3. The Fourier-Clifford transform.
Let us now introduce the basics of Clifford analysis that are needed to introduce the FourierClifford transform. Facts used here can be found e.g. in [7, 9]. We also follow as closely as possible
the presentation of Clifford analysis from [8, 14].
Throughout this section d ≥ 2 will be a fixed integer and the measure dµ(x) = db
µ(x) = (2π)−d/2 dx
d
is the Lebesgue measure on R . We first associate the Clifford algebra Cl0,d (C) generated by the
canonical basis ej , j = 1, . . . , d. For A = {j1 , j2 , . . . , jk } ⊂ {1, . . . , d} with j1 < j2 < ·· · < jk ,
we denoteby eA = ej1 ej2 · · · ejk . The basis of the Clifford algebra is then given by E = eA , A ⊂
{1, . . . , d} . The Clifford algebra is then the complex vector space generated by E endowed with the
multiplication rule given by
(i) e∅ = 1 is the unit element
(ii) e2j = −1, j = 1, . . . , d
(iii) ej ek + ek ej = 0, j, k = 1, . . . , d, j 6= k.
Conjugation is defined by the anti-involution for which ej = −ej , j = 1, . . . , d with the additional
rule ī = −i.
The scalars are then identified with span {e∅ } while we identify a vector x = (x1 , . . . , xd ) with
x=
d
X
ej xj .
j=1
The product of two vectors splits into a scalar part and a bivector part
xy = − x, y + x ∧ y
and
x∧y =
d
d
X
X
ej ek (xj yk − xk yj ).
j=1 k=i+1
Note that x2 = −|x|2 .
The functions defined in this section are defined on Rd and take their values in the Clifford algebra
Cl0,d (C). We can now introduce the so-called Dirac operator, a first order vector differential operator
defined by
d
X
∂x =
∂xj ej .
j=1
Its square equals, up to a minus sign, the Laplace operator on Rd , ∂x2 = −∆. The central notion
in Clifford analysis is the notion of monogenicity, the higher-dimensional analogue of holomorphy: a
function is called (left)-monogenic if ∂x f = 0.
We will denote by Mk the space of all spherical monogenics of degree k, that is, homogeneous poly(ℓ)
nomials of degree k that are null-solutions of the Dirac operator. We fix a basis {Mk }ℓ=1,2,...,dim Mk of
α
Mk . Further, the Laguerre polynomials are denoted by Lj . We then consider the following functions,
called the Clifford-Hermite functions
(4.24)
ψ2j,k,ℓ (x)
ψ2j+1,k,ℓ (x)
=
d
γ2j,k,ℓ Lj2
+k−1
d
2 +k
= γ2j+1,k,ℓ Lj
(ℓ)
(|x|2 )Mk (x)e−|x|
(ℓ)
2
(|x|2 )xMk (x)e−|x|
/2
2
/2
,
UNCERTAINTY PRINCIPLES FOR INTEGRAL OPERATORS
15
where j, k ∈ N and ℓ ∈ {1, . . . , dim Mk }. Provided the γj,k,ℓ ’s are properly chosen, this is an orthonormal basis of L2 (Rd ) (see [6]).
Next, introducing spherical coordinates in Rd : x = rω, r = |x| ∈ R+ , ω ∈ Sd−1 , the Dirac operator
takes the form
1 ∂x = ω ∂r + Γx
r
where
d
d
X
X
Γ = x ∧ ∂x = −
ej ek (xj ∂xk − xk ∂xj )
j=1 k=j+1
is the so-called angular Dirac operator.
We are now in position to define the Clifford-Fourier transforms on S(Rd ). This can be done in
three equivalent ways:
2
π
π
– F± [f ] = eid 4 ei 4 (∆−|x| ∓2Γ) f ;
– via an integral kernel
Z
F± [f ](η) =
f (x)K± (x, η) dµ(x)
Rd
iπ
id π
2 Γη −ihx,η i
4
where K± (x, η) = e e
– via its eigenfunctions
e
;
F± [ψ2j,k,ℓ ] = (−1)j+k (∓1)k ψ2j,k,ℓ
and F± [ψ2j+1,k,ℓ ] = id (−1)j+1 (∓1)k+d−1 ψ2j,k,ℓ .
The third definition immediately shows that F± extend to unitary operators on L2 (Rd , µ).
The fact that the integral operator definition makes sense on S(Rd ) and that the kernel of the inverse
transform is indeed K± (x, η) has been proved respectively in [14, Theorem 6.3 and Proposition 3.4].
Finally, the kernel is not known to be polynomially bounded, excepted when the dimension d is
even [14, Theorem 5.3] and then
|K(x, η)| ≤ C(1 + |x|)(d−2)/2 (1 + |η|)(d−2)/2 .
Thus m = m
b = (d − 2)/2, cT = C and a = d/2.
It remains to notice that all results from the first part of the paper extend with no change to
Clifford-valued functions. More precisely, we obtain the following results:
Theorem 4.5.
Let d be even and dν(x) = (1 + |x|)d−2 dµ(x). Let S, Σ be a pair of measurable subsets of Rd . Then
the Clifford-Fourier transform satisfies the following uncertainty principles.
(1) Donoho-Stark’s uncertainty principle for F± :
If f ∈ L2 (Rd , µ) of unit L2 -norm is ε1 -concentrated on S and ε2 -bandlimited on Σ for the
Clifford-Fourier transform, then
2
q
−2
2
2
ν(S)ν(Σ) ≥ C
1 − ε1 + ε2 .
(2) Local uncertainty principle for F± :
If Σ is subset of finite measure 0 < ν(Σ) < ∞, then
(a) for 0 < s < d/2, there is a constant c(s) such that for all f ∈ L2 (Rd , µ),

h
i s

c(s) ν(Σ) 2(d−1) k|x|s f k 2 d , if ν(Σ) ≤ 1;
L (R ,µ)
kF± (f )kL2 (Σ,bµ) ≤
h
i ds s

c(s) ν(Σ) |x| f 2 d ,
if ν(Σ) > 1;
L (R ,µ)
(b) for d/2 ≤ s ≤ d − 1 then, for every ε > 0 there is a constant c(s, ε) such that for all
f ∈ L2 (Rd , µ),

h
i 1 −ε d
d

c(s, ε) ν(Σ) 4(1−1/d) f 1−2 2sd+ε |x|s f 2s2−εd , if ν(Σ) ≤ 1;
L (R ,µ)
L (R ,µ)
kF± (f )kL2 (Σ,bµ) ≤
h
i 12 −ε d
d
1−
+ε
−ε

s
c(s, ε) ν(Σ)
f 2 2sd |x| f 2s2 d ,
if ν(Σ) > 1;
L (R ,µ)
L (R ,µ)
16
SAIFALLAH GHOBBER AND PHILIPPE JAMING
(c) for s > d − 1, there is a constant c′ (s) such that for all f ∈ L2 (Rd , µ),
h
i 12
s
kF± (f )kL2 (Σ,µ) ≤ c′ (s) ν(Σ) k(1 + |x| )f kL2 (Rd ,µ) .
(3) Benedicks-Amrein-Berthier’s uncertainty principle for F± :
If S, Σ are subsets of finite measure 0 < ν(S), ν(Σ) < ∞, then there exists a constant C(S, Σ)
such that for all f ∈ L2 (Rd , µ),
kf k2L2(Rd ,µ) ≤ C(S, Σ) kf k2L2 (S c ,µ) + kF± (f )k2L2 (Σc ,µ) .
(4) Global uncertainty principle for F± :
For s, β > 0, there exists a constant Cs,β such that for all f ∈ L2 (Rd , µ),
2β
s s+β
2s
2
|x| f 2 d |ξ|β F± (f ) s+β
≥ Cs,β kf kL2 (Rd ,µ) .
L (R ,µ)
L2 (Rd ,µ)
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S. G : Département de Mathématiques Appliquées, Institut Préparatoire Aux Études D’ingénieurs de
Nabeul, Université de Carthage, Campus Universitaire, Merazka, 8000, Nabeul, Tunisie
E-mail address: [email protected]
P. J : Univ. Bordeaux, IMB, UMR 5251, F-33400 Talence, France.
Talence, France.
E-mail address: [email protected]
CNRS, IMB, UMR 5251, F-33400