THE MULTIPLIER OF THE INTERVAL [−1, 1] FOR THE DUNKL
TRANSFORM OF ARBITRARY ORDER ON THE REAL LINE
ÓSCAR CIAURRI, LUZ RONCAL, AND JUAN L. VARONA
Abstract. We study the boundedness of the multiplier of the interval [−1, 1]
for the Dunkl transform of order α > −1 on weighted Lp spaces, with 1 < p <
∞. In particular, we get that it is bounded from Lp (R, |x|2α+1 dx) into itself
if and only if 4(α + 1)/(2α + 3) < p < 4(α + 1)/(2α + 1) when α > −1/2 or if
and only if 1 < p < ∞ when −1 < α ≤ −1/2.
1. Introduction and main result
For α > −1, let Jα denote the Bessel function of order α:
∞
x α X
(−1)n (x/2)2n
Jα (x) =
2 n=0 n! Γ(α + n + 1)
(a classical reference on Bessel functions is [1]). Throughout this paper, by
we denote the even function
∞
1 X (−1)n (z/2)2n
, z ∈ C.
2α n=0 n! Γ(α + n + 1)
Jα (z)
zα
In this way, for complex values of the variable z, let
Iα (z) = 2α Γ(α + 1)
∞
X
(z/2)2n
Jα (iz)
=
Γ(α
+
1)
(iz)α
n! Γ(n + α + 1)
n=0
(the function Iα is a small variation of the so-called modified Bessel function of the
first kind and order α, usually denoted by Iα ). Moreover, let us take
z
Eα (z) = Iα (z) +
Iα+1 (z), z ∈ C.
2(α + 1)
The Dunkl operators on Rn are differential-difference operators associated with
some finite reflection groups (see [2]). We consider the Dunkl operator Λα , α ≥
−1/2, associated with the reflection group Z2 on R given by
2α + 1 f (x) − f (−x)
d
f (x) +
.
(1.1)
Λα f (x) =
dx
x
2
For α ≥ −1/2 and λ ∈ C, the initial value problem
(
Λα f (x) = λf (x), x ∈ R,
(1.2)
f (0) = 1
has Eα (λx) as its unique solution (see [3] and [4]); this function is called the Dunkl
kernel. For α = −1/2, it is clear that Λ−1/2 = d/dx, and E−1/2 (λx) = eλx .
THIS PAPER HAS BEEN ACCEPTED IN: Integral Transforms Spec.Funct..
2000 Mathematics Subject Classification. Primary 42A45; Secondary 42A38.
Key words and phrases. Dunkl operator, Dunkl transform, Bessel functions, multipliers.
1
2
Ó. CIAURRI, L. RONCAL, AND J. L. VARONA
In a similar way to the Fourier transform (which is the particular case α = −1/2),
the Dunkl transform of order α ≥ −1/2 on the real line is
Z
(1.3)
Fα f (y) =
Eα (−ixy)f (x) dµα (x),
y ∈ R,
R
were dµα denotes the measure
dµα (x) =
1
2α+1 Γ(α+1)
|x|2α+1 dx.
The study of the properties of the Dunkl transform has a great interest, and many
papers about this subject have been published during the last years. See, for
instance, [5–14] and the references therein.
The behaviour of the Bessel functions is very well known. For instance, for real
values of the variable, they verify the asymptotics
(1.4)
Jα (x) =
xα
2α Γ(α
+ 1)
+ O(xα+2 ),
x → 0+ ;
and
(1.5)
Jα (x) =
2
πx
1/2 h
i
απ π cos x −
−
+ O(x−1 ) ,
2
4
x → +∞.
From these and similar results, and noticing that
Jα+1 (x)
Jα (x)
+ 2α Γ(α + 1)
xi,
xα
xα+1
it is easy to check that |Eα (ix)| ≤ 1 for every x ∈ R. Then, (1.3) is well defined for
every f ∈ L1 (R, dµα ), and
Eα (ix) = 2α Γ(α + 1)
(1.6)
kFα f k∞,α ≤ kf k1,α ,
(1.7)
where we use k · kp,α as a shorthand for k · kLp (R,dµα ) .
Just as the Fourier transform, Fα is an isomorphism of the Schwartz class S into
itself, and Fα2 f (x) = f (−x). Fubini’s theorem implies the multiplication formula
Z
Z
Fα f (x)g(x) dµα (x) =
Fα g(x)f (x) dµα (x), f, g ∈ S.
R
R
Taking g(x) = Fα f (x) we get kFα f k2,α = kf k2,α , f ∈ S. By density, this can be
extended to functions in L2 (R, dµα ).
Via S, the [−1, 1]-multiplier Mα is defined as
(1.8)
Mα f (x) = Fα (χ[−1,1] Fα f )(−x)
or, equivalently,
Fα (Mα f )(x) = χ[−1,1] (x)Fα f (x),
which is the usual notation.
In the paper [15], the authors studied the boundedness of the operator Mα for
α ≥ −1/2 in weighted Lp spaces.
But the Dunkl transform Fα can also be defined in L2 (R, dµα ) for α > −1, although some properties like (1.7) are no longer valid for −1 < α < −1/2. However,
it preserves the same properties in L2 (R, dµα ); see [16] for details. The aim of this
paper is to extend the case α ≥ −1/2 in [15] to the whole range α > −1.
Thus, the main result of this paper is the following:
Theorem. Let α > −1, 1 < p < ∞, and wa,b (x) = |x|a (1 + |x|)b−a . Then, there
exists a constant C such that
(1.9)
kwa,b Mα f kp,α ≤ Ckwa,b f kp,α
THE MULTIPLIER OF [−1, 1] FOR THE DUNKL TRANSFORM
3
if and only if
2α + 2
1
−
< a < (2α + 2) 1 −
p
p
(1.10)
and
(1.11)
2α + 1 2α + 2
2α + 1
1
−
<b<−
+ (2α + 2) 1 −
.
2
p
2
p
As a simple consequence, it is easy to check that, in the unweighted case a =
b = 0, we have

 4(α + 1) < p < 4(α + 1) , if α ≥ −1/2,
2α + 3
2α + 1
kMα f kp,α ≤ Ckf kp,α ⇐⇒

1 < p < ∞,
if −1 < α < −1/2.
In Figure 1 we show the region of boundedness of the multiplier Mα for α > −1
and 1 < p < ∞; there, the filled region consists of the values (α, 1/p) such that
Mα is bounded from Lp (R, dµα ) into itself.
1/p
1
1/2
α
0
0
− 12
1
2
3
4
5
6
Figure 1. Region of boundedness of Mα .
Although the theorem presented in this paper is very similar to the theorem
in [15], the proof is not the same. It is interesting to explain the difference.
The usual procedure in harmonic analysis to study the boundedness of an operator on Lp spaces is to write it as an integral operator with a kernel and then
analyse such kernel. For our multiplier Mα , by using (1.8) and (1.3), we have
Z 1
Mα f (x) = Fα (χ[−1,1] (r)Fα f (r))(−x) =
Eα (irx)Fα f (r) dµα (r)
−1
Z
1
=
Z
Eα (irx)
Eα (−iyr)f (y) dµα (y)
−1
dµα (r)
R
Z Z
1
=
R
Eα (irx)Eα (−iry) dµα (r) f (y) dµα (y),
−1
where in the last step we have used Fubini’s theorem (which is justified for suitable
functions and extended in the usual way). Then, Mα can be written as
Z
(1.12)
Mα f (x) =
Kα (x, y)f (y) dµα (y)
R
with kernel
Z
(1.13)
1
Kα (x, y) =
Eα (irx)Eα (−iry) dµα (r).
−1
4
Ó. CIAURRI, L. RONCAL, AND J. L. VARONA
The next step is to get a good expression for the kernel. In [15] we see that, for
x, y ∈ R, x 6= y, one has
(1.14)
Z 1
1
Eα (ix)Eα (−iy) − Eα (−ix)Eα (iy)
Eα (ixr)Eα (−iyr) dµα (r) = α+1
2
Γ(α + 1)
i(x − y)
−1
(this expression has been useful not only in [15], but also in [6] in connection with
a sampling theorem related to the Dunkl transform).
Then, the operator Mα can be written as
Z
(1.15) Mα f (x) =
Kα (x, y)f (y) dµα (y)
R
Z
Eα (ix)Eα (−iy)
1
f (y) dµα (y)
= α+1
2
Γ(α + 1) R
i(x − y)
Z
1
Eα (−ix)Eα (iy)
− α+1
f (y) dµα (y)
2
Γ(α + 1) R
i(x − y)
1
= 2α+2
Tα1 f (x) − Tα2 f (x) ,
2
2
Γ(α + 1)
with
Tα1 f (x) = Eα (ix)H (Eα (−iy)/i)f (y)|y|2α+1 (x),
Tα2 f (x) = Eα (−ix)H (Eα (iy)/i)f (y)|y|2α+1 (x),
where H denotes the Hilbert transform.
In [15], the process continues by proving the Lp -boundedness of the operators
1
Tα and Tα2 by means of the Ap theory of weights, what holds for α ≥ −1/2 in the
corresponding range of p’s.
But this method is not longer valid for the case −1 < α < −1/2 because, if we try
to follow it, we find that neither Tα1 nor Tα2 are bounded operators in the requires Lp
spaces. Therefore, such decomposition is not useful in the range −1 < α < −1/2.
Thus, in this paper we will use a different decomposition of the kernel, that will
lead to another decomposition of Mα as the sum of two operators. Then, we will
make a clever use of the Ap theory of weights to prove the Lp -boundedness of these
new operators. Note that the decomposition shown in this paper works both for
the cases α ≥ −1/2 and −1 < α < −1/2, so the present proof also covers the result
previously stated in [15].
2. Proof of the theorem
The method to prove that the conditions (1.10) and (1.11) are necessary for (1.9)
is as in [15]; here, we will not repeat it. Then, let us see that, provided α > −1
and 1 < p < ∞, the conditions (1.10) and (1.11) are sufficient for (1.9).
Given p ∈ (1, ∞), a weight w in R is said to belong to the Ap class if
Z
Z
p−1
w(x) dx
w(x)−1/(p−1) dx
≤ C|I|p
I
I
for every interval I ⊆ R, with C independent of I. An important application of Ap
theory lies on its relation with the boundedness of the Hilbert transform
Z
g(y)
dy.
Hg(x) =
R x−y
Indeed, in [17] (see also [18] for further information) it is proved that
H : Lp (R, w) → Lp (R, w) bounded ⇐⇒ w ∈ Ap .
THE MULTIPLIER OF [−1, 1] FOR THE DUNKL TRANSFORM
5
For radial weights, it is well known that
|x|β ∈ Ap ⇐⇒ −1 < β < p − 1.
If we take Φ(x) = |x|r for x ∈ [−1, 1] and Φ(x) = |x|s for x ∈ (−∞, 1) ∪ (1, ∞),
(2.1)
Φ ∈ Ap ⇐⇒ −1 < r < p − 1 and − 1 < s < p − 1
(the intuitive behaviour is clear; see [19] for details and a proof).
In what follows, we will use C (perhaps with subindex) to denote a positive
constant independent of x or f (and all other variables), which can assume different
values in different occurrences. Moreover, for non-negative functions u and v defined
on an interval, u(x) ∼ v(x) means that there exist two positive constants C1 and
C2 such that C1 ≤ u(x)/v(x) ≤ C2 .
As explained in the introduction, we are going to find a decomposition of Mα f
different to (1.15). To do this, we will manipulate the right hand side in (1.14). By
substituting (1.6), we obtain
Eα (ix)Eα (−iy) − Eα (−ix)Eα (iy)
Jα+1 (x) Jα (y) Jα (x) Jα+1 (y)
2α+1
2
ix α −
iy
=2
Γ(α + 1)
xα+1
y
xα
y α+1
and then the kernel can be expressed, for x 6= y, as
Jα+1 (x)
Z 1
Jα (y)
Jα (x) Jα+1 (y)
xα+1 x y α − xα
y α+1 y
(2.2)
Eα (ixr)Eα (−iyr) dµα (r) = 2α Γ(α + 1)
.
x−y
−1
Now, to write Mα f in terms of Hilbert transforms to apply the Ap theory of
weights, let us use (1.12), (1.13) and (2.2). In this way, we get
Z
(2.3)
Mα f (x) =
Kα (x, y)f (y) dµα (y)
R
= 2α Γ(α + 1)
Jα+1 (x)
xα+1
Z
x Jαy(y)
α
f (y) dµα (y)
x−y
R
− 2α Γ(α + 1)
Z
Jα (x) Jα+1 (y)
xα
y α+1
R
y
x−y
f (y) dµα (y)
= Uα1 f (x) − Uα2 f (x),
with
Jα+1 (x)
Jα (y)
2α+1
=x
H
f (y)|y|
(x),
xα+1
yα
Jα (x)
Jα+1 (y)
Uα2 f (x) =
H y
f (y)|y|2α+1 (x).
α
x
y α+1
Uα1 f (x)
With this decomposition, (1.9) follows if we prove that there exists a constant
C independent of f ∈ Lp (R, wa,b (x)p |x|2α+1 dx) such that
kUαj f (x)wa,b (x)kLp (R,|x|2α+1 dx) ≤ Ckf (x)wa,b (x)kLp (R,|x|2α+1 dx) ,
α
2α+1
Taking g(y) = (Jα (y)/y )f (y)|y|
equivalent to
(2.4)
j = 1, 2.
, the inequality corresponding to j = 1 is
kHg(x)kLp (R,|xJα+1 (x)/xα+1 |p wa,b (x)p |x|2α+1 dx)
≤ Ckg(x)kLp (R,|Jα (x)/xα |−p |x|−(2α+1)p wa,b (x)p |x|2α+1 dx) ;
similarly, the corresponding to j = 2 is equivalent to
(2.5)
kHg(x)kLp (R,|Jα (x)/xα |p wa,b (x)p |x|−(2α+1)p |x|2α+1 dx)
≤ Ckg(x)kLp (R,|xJα+1 (x)/xα+1 |−p |x|−(2α+1)p wa,b (x)p |x|2α+1 dx) .
6
Ó. CIAURRI, L. RONCAL, AND J. L. VARONA
Let us start with (2.4). It is enough to prove that there is a weight Φ ∈ Ap with
Jα (x) −p −(2α+1)p
Jα+1 (x) p
p
2α+1
wa,b (x) |x|
≤ Φ(x) ≤ C2 α |x|
wa,b (x)p |x|2α+1 .
C1 x
xα+1 x
Remember that we are here using the even functions Jα (x)/xα and Jα+1 (x)/xα+1
defined on the whole real line, so we can write this condition as
C1 |Jα+1 (|x|)|p |x|−αp wa,b (x)p |x|2α+1
(2.6)
≤ Φ(x) ≤ C2 |Jα (|x|)|−p |x|−(α+1)p wa,b (x)p |x|2α+1 .
We have wa,b (x) ∼ |x|a in [−1, 1] and wa,b (x) ∼ |x|b in (−∞, −1] ∪ [1, ∞).
Moreover, from the estimates (1.4) and (1.5) it is clear that, for α > −1,
(
Cα |x|α ,
if x ∈ [−1, 1],
|Jα (|x|)| ≤
−1/2
Cα |x|
, if x ∈ (−∞, −1] ∪ [1, ∞),
with a Cα constant depending only on α. According to these bounds, we have
(
C|x|p+ap+2α+1 ,
if |x| ∈ (0, 1),
p
−αp
p
2α+1
|Jα+1 (|x|)| |x|
wa,b (x) |x|
≤
−p/2−αp+bp+2α+1
C|x|
, if |x| ∈ (1, ∞),
and
(
−p
|Jα (|x|)|
|x|
−(α+1)p
p
wa,b (x) |x|
2α+1
≥
C|x|−(2α+1)p+ap+2α+1 ,
C|x|−p/2−αp+bp+2α+1 ,
if |x| ∈ (0, 1),
if |x| ∈ (1, ∞).
Let us write
(
Φ(x) =
|x|r ,
|x|−p/2−αp+bp+2α+1 ,
if |x| ∈ (0, 1),
if |x| ∈ (1, ∞).
By (2.1), Φ ∈ Ap will hold and fulfil (2.6) if


−(2α + 1)p + ap + 2α + 1 ≤ r ≤ p + ap + 2α + 1,
(2.7)
−1 < r < p − 1,


−1 < −p/2 − αp + bp + 2α + 1 < p − 1.
The third line is equivalent to
2α + 1 2α + 2
2α + 1
1
−
<b<−
+ (2α + 2) 1 −
,
2
p
2
p
which is the condition (1.11) of the theorem. For the inequalities in (2.7) involving
r, let us first note that −(2α + 1)p + ap + 2α + 1 ≤ p + ap + 2α + 1 is equivalent
to 0 ≤ (2α + 2)p, and this is true for any α > −1. Then, for the existence of r it is
enough to show that −(2α + 1)p + ap + 2α + 1 < p − 1 and −1 < p + ap + 2α + 1.
Even more, the existence of r will be also guaranteed if we substitute the condition
−1 < p + ap + 2α + 1 by −1 < ap + 2α + 1, that is stronger. Then, it suffices to
note that the inequalities −(2α + 1)p + ap + 2α + 1 < p − 1 and −1 < ap + 2α + 1
are equivalent to
1
2α + 2
< a < (2α + 2) 1 −
,
−
p
p
that is the condition (1.10) of the theorem.
The study of (2.5) is completely similar. Instead of (2.7) we get the conditions


−2(α + 1)p + ap + 2α + 1 ≤ r ≤ ap + 2α + 1,
(2.8)
−1 < r < p − 1,


−1 < −p/2 − αp + bp + 2α + 1 < p − 1,
THE MULTIPLIER OF [−1, 1] FOR THE DUNKL TRANSFORM
7
and, now, the inequality −2(α+1)p+ap+2α+1 ≤ ap+2α+1 is true for any α > −1.
With small variations of the arguments used for Uα1 we deduce the boundedness of
Uα2 , and consequently the theorem is proved.
Remark. If we try to follow the method in [15] with the operators Tα1 and Tα2 in
the decomposition (1.15), the first lines in (2.7) and (2.8) become, instead, −(2α +
1)p + ap + 2α + 1 ≤ r ≤ p + ap + 2α + 1 (twice the same condition). But, this time,
the inequality −(2α + 1)p + ap + 2α + 1 ≤ ap + 2α + 1 is not true in general for any
α > −1 (only for α ≥ −1/2), so the method fails (actually, it is possible to prove
that neither Tα1 nor Tα2 are bounded operators in the whole range given by (1.10)
and (1.11) when −1 < α < −1/2, but this is not of interest).
Funding. The research of the authors is supported by the DGI (Ministerio de
Economı́a y Competitividad, Spain) under Grant MTM2012-36732-C03-02.
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8
Ó. CIAURRI, L. RONCAL, AND J. L. VARONA
Departamento de Matemáticas y Computación, Universidad de La Rioja, 26004 Logroño, Spain
E-mail address: [email protected]
Departamento de Matemáticas y Computación, Universidad de La Rioja, 26004 Logroño, Spain
E-mail address: [email protected]
URL: http://www.unirioja.es/cu/luroncal
Departamento de Matemáticas y Computación, Universidad de La Rioja, 26004 Logroño, Spain
E-mail address: [email protected]
URL: http://www.unirioja.es/cu/jvarona