Mean value property associated with the Dunkl Laplacian

0.13 MB·

by  Kods Hassine

Mean value property associated with the
Dunkl Laplacian
Kods Hassine
4 Department of Mathematics, Faculty of Sciences, University of Monastir
1
5019 Monastir, Tunisia
0
2
E-mail: [email protected]

n
a
J

9 Abstract

] Let ∆k be the Dunkl Laplacian on Rd. The main goal of this paper is
A
to characterize ∆ -harmonic functions by means of a mean value property.
k
C
.
h Keywords : DunklLaplacian, Mean value property, ∆ -harmonic func-
k
t
a tions.
m
[
MSC (2010): 31A05, 51F15, 42B99.

1
v
1 Introduction
9
4
9
1 Let R be a root system of Rd, d 1, k :R R be a multiplicity function
+
. ≥ →
1 and W be the group generated by the reflections σα, α R. The Dunkl
∈
0 Laplacian is defined in [1] for every function f C2(Rd) by
4 ∈
1 < f(x),α > α2f(x) f(σ (x))
: ∆ f(x)= ∆f(x)+2 k(α) ∇ | | − α ,
v k (cid:18) < α,x > − 2 < α,x >2 (cid:19)
Xi αX∈R+
r where∆ and denote respectively theusualLaplacian and gradient on Rd
a
∇
andR isapositivesubsystemofR. Clearly,ifkistheidenticallyvanishing
+
function, then ∆ is reduced to ∆.
k
It is well known that a locally bounded function f on an open subset
D of Rd, is ∆-harmonic (i.e., f C2(D) and ∆f = 0 on D) if and only if
∈
1
f(x)= f(y)dσ (y),
x,r
σ Z
x,r(S(x,r)) S(x,r)
for every x D and every r > 0 such that the closed ball B(x,r) of center
∈
x and radius r is contained in D. Here σ is the surface area measure on
x,r
the sphere S(x,r) with center x and radius r.
1
H. Mejjaolli and K. Trimèche showed in [4] that every infinitely differ-
entiable function f onRd is ∆ -harmonic onRd if and only if forall x Rd
k
∈
and r > 0,
1
f(x) = τxf(ry) y,α 2k(α)dσ0,1(y), (1)
dk ZS(0,1) αY∈R+|h i|
 
where d is a normalized constant and τ is the Dunkl translation. The
k x
maingoalofthispaperistoinvestigateameanvaluepropertywhichcharac-
terizes the ∆ -harmonicity of locally bounded functions on an open subset
k
of Rd.
Let D Rd be an open set which is W-invariant. We shall say that
⊂
a function f : D R satisfies the mean value property on D if for every
→
x D and r > 0 such that B(x,r) D,
∈ ⊂
f(x) = f(y)dσk (y),
x,r
ZRd
where σk (see [7]) is the unique probability measure on Rd such that the
x,r
right hand side of (1) coincides with
f(y)dσk (y).
x,r
ZRd
We shall prove that every locally bounded function f on D is ∆ -
k
harmonic if and only if it satisfies the mean value property on D. To that
end, we prove first the equivalence for infinitely differentiable functions on
D. Next, we show that for a locally bounded function f on D, if f satisfies
the mean value property then f is infinitely differentiable on D. Thus, f
is ∆ -harmonic provided it satisfies the mean value property on D. To
k
prove the converse, we need only show that if f is ∆ -harmonic then it is
k
infinitely differentiable on D. This will be proved once we have shown that
the operator ∆ is hypoelliptic. Thus, by means of convergence property
k
of ∆ -harmonic functions, we prove that the operator ∆ is hypoelliptic
k k
on D.
Note that the condition that D is W-invariant is nearly optimal. In
fact, in the case where d = 1, for every open set D R which is not W-
⊂
invariant, we can always construct a ∆ -harmonic function function f on
k
D which does not satisfy the mean value property on D.
2 Preliminaries and some lemmas
Let S(Rd) be the Schwartz space and C (Rd) be the set of all continuous
0
functions on Rd vanishing at infinity. For every open set U Rd, C(U)
⊂
and C (U) will denote respectively the set of all continuous functions on U
c
and the set of all continuous functions with compact support on U. The
2
set of all bounded functions in C(U) will be denoted by C (U). For every
b
α Rd 0 , let H be the hyperplane of Rd orthogonal to α and let σ be
α α
∈ \{ }
the reflection in H , i.e.,
α
< α,x >
σ (x) := x 2 α,
α
− α2
| |
where x,y = d x y and x := x,x . A finite subset R of Rd 0
h i i=1 i i | | h i \{ }
is called a rootPsystem if R Rα =p α and σα(R) = R for all α R.
∩ {± } ∈
For a given root system R, we denote by W the finite group generated by
all refections σ , α R. A function k : R R is called a multiplicity
α +
∈ →
function if it satisfies k(wα) = k(α), for every w W and every α R.
∈ ∈
Throughout this paper we fix aroot systemR, amultiplicity function k
and a W-invariant open subset D of Rd, that is, w(D) D for all w W.
⊂ ∈
Let w be the weight function on Rd defined by,
k
w (x) := x,α 2k(α),
k
|h i|
αY∈R+
where R := α R : α,β > 0 for some β Rd H . Note that
+ α∈R α
{ ∈ h i } ∈ \∪
w is homogeneous of degree 2γ, with γ := k(α). From now on, we
k α∈R+
assume that P
d
λ := γ + 1 > 0.
2 −
TheDunklLaplacianassociatedwiththerootsystemRandthemultiplicity
function k is the operator
d
∆ := T2,
k i
Xi=1
where for every 1 i d and f C1(D),
≤ ≤ ∈
f(x) f(σ (x))
α
T f(x):= ∂ f(x)+ k(α)α − , x D.
i i i
< α,x > ∈
αX∈R+
By [2], there exists a unique linear isomorphism V from the space of
k
homogenous polynomials of degree n on Rd into it self such that V 1 = 1
k
and T V = V ∂ . Later, it was shown in [9] that the intertwining operator
i k k i
V has an homeomorphism extension to C∞(Rd). The positivity of V (see
k k
[6]) yields the existence of a family of probability measures (µk) such that
x x
for every x Rd and every f C∞(Rd),
∈ ∈
V f(x) = f(y)dµk(y).
k x
ZRd
The support of µk is contained in the convex hull C(x) of the orbit of x
x
under the reflection group W,
C(x) := co wx, w W .
{ ∈ }
3
The Dunkl kernel associated with R and k is defined on Rd Rd by
×
E (x,y) := ehy,ξidµk(ξ).
k x
ZRd
It is well known that E is positive, symmetric and admits a unique holo-
k
morphic extension to Cd Cd satisfying E (ξz,ω) = E (z,ξω) for every
k k
×
z,ω Cd and every ξ C. The corresponding Dunkl transform is then
∈ ∈
given for every bounded measure µ on Rd by
(µ)(x) := c E ( iξ,x)dµ(ξ), x Rd,
D k k
F ZRd − ∈
where
ck := e−|y2|2wk(y)dy −1.
(cid:18)ZRd (cid:19)
If µ = fw dx where f S(Rd) and dx is the Lebesgue measure on Rd,
k
∈
then we shall write (f) instead of (µ). Note that is injective on
D D D
F F F
the space of all bounded Borel measures (Rd) on Rd (see [8]) and is a
b
M
topological isomorphism from S(Rd) into it self (see [3]). For each x Rd,
∈
the Dunkl translation τ is defined for every f S(Rd) by
x
∈
τ f = −1(E (ix, ) f),
x FD k · FD
where −1 denotes the inverse of on S(Rd). In [10], this translation
FD FD
was extended to C∞(Rd) by
τ f(y)= V−1f(z+η)dµk(z)dµk(η),
x ZRdZRd k x y
where V−1 is the inverse of V on C∞(Rd). It was shown that, for every
k k
f C∞(Rd), the function u : (x,y) τ f(y) is symmetric, infinitely
x
∈ 7→
differentiable on Rd Rd and for every x,y Rd,
× ∈
(T ) u(x,y) = (T ) u(x,y). (2)
i x i y
Moreover,τ f(0) = f(0),T τ f = τ T f andτ E (z, )(y) = E (x,z)E (y,z)
x i x x i x k k k
·
for every z Cd. Further, if the support of f (noted suppf) is in B(0,r)
∈
for some r > 0, then supp τ f B(0,r+ x ).
x
⊂ | |
According to [7], for each x Rd and r > 0, there exists a unique
∈
probability measure σk on Rd which is supported by B(wx,r)
x,r w∈W \
B(0, x r ) such that for every f C∞(Rd), S
|| |− | ∈
1
τ f(ry)w (y)dσ (y) = f(y)dσk (y). (3)
dk ZS(0,1) x k 0,1 ZRd x,r
where,
1
d := w (y)dσ (y) = .
k ZS(0,1) k 0,1 ck2λΓ(λ+1)
4
Lemma 2.1. Let ϕ S(Rd) be radial. Then for every Borel set A Rd
∈ ⊂
and every x Rd,
∈
∞
τ ϕ(y)w (y)dy = d ϕ(t)t2λ+1 dσk (y) dt. (4)
−x k k x,t
Z Z (cid:18)Z (cid:19)
A 0 A
Proof. Let x Rd and denote by µ(A) and ν(A) the left hand side and
∈
the right hand side respectively of (4). Clearly, both µ and ν are bounded
measures on Rd. For every y Rd,
∈
(µ)(y) = (τ ϕ)(y)
D D −x
F F
= ϕ(y)E ( iy,x)
D k
F −
= c E ( iy,x) ϕ(z)E ( iy,z)w (z)dz
k k k k
− ZRd −
= c τ E ( iy, )(z)ϕ(z)w (z)dz
k x k k
ZRd − ·
Using spherical coordinates and (3), we deduce that,
∞
(µ)(y) = c d t2λ+1ϕ(t) E (ξ, iy)dσk (ξ)dt
D k k k x,t
F Z0 ZRd −
= (ν)(y).
D
F
Finally, we use the injectivity of on (Rd) to conclude.
D b
F M
Let ϕ S(Rd) be a radial function with support in B(0,r), r > 0. We
∈
claim that for every x Rd,
∈
suppτ ϕ B(wx,r). (5)
x
⊂
w[∈W
Indeed, let A be a Borel subset of Rd B(wx,r). Then by (4),
w∈W
\∪
r
τ ϕ(y)w (y)dy = d ϕ(t)t2λ+1 dσk (y) dt.
−x k k x,t
Z Z (cid:18)Z (cid:19)
A 0 A
Since for every 0< t < r, supp σk B(wx,r) we deduce that,
x,t w∈W
⊂ ∪
τ ϕ(y)w (y)dy = 0.
−x k
Z
A
This proves the claim.
In the sequel we shall write
M (f)= f(y)dσk (y),
x,r x,r
ZRd
whenever the integral makes sense. A Borel function f : D R is said to
→
satisfy the mean value property on D if M (f) = f(x) for every x Rd
x,r
∈
and r > 0 such that B(x,r) D.
⊂
5
Lemma 2.2. Let f be a locally bounded function on D. If f satisfies the
mean value property on D, then f C∞(D).
∈
Proof. Without loss of generality we suppose that f is bounded on D. Let
φ be the function defined for every t R by φ(t) := ce−1tχ]0,∞[(t), where
∈
χ is the indicator function of ]0, [ and the constant c is chosen so
]0,∞[
∞
that
1
cd φ(1 t2)t2λ+1dt = 1.
k
Z −
0
For every n 1 we define the function φ by,
n
≥
φ (x) = n2λ+2φ(1 n2 x 2), x Rd. (6)
n
− | | ∈
Obviously φ is infinitely differentiable on Rd with support in B(0, 1).
n n
Thus, by (5), for every x Rd,
∈
1
suppτ φ B(wx, ).
x n w∈W
⊂ ∪ n
Let D := x D : B(x, 1) D and let
n { ∈ n ⊂ }
f (x) := f(y)τ φ (y)w (y)dy, x Rd.
n −x n k
Z ∈
D
Then f C∞(D ). On the other hand, it follows from (4) that for every
n n
∈
x D ,
n
∈
1
f (x) = d n φ (t)t2λ+1M (f)dt = f(x).
n k n x,t
Z
0
Hence f C∞(D ) and consequently f C∞(D) as desired.
n
∈ ∈
3 Main result
D will always denotes a W-invariant open subset of Rd. Our main result
is the following:
Theorem 3.1. Let f be a locally bounded function on D. The following
statements are equivalent:
(a) f C2(D) and ∆ f = 0 on D.
k
∈
(b) M (f) = f(x) for every x D and r > 0 such that B(x,r) D .
x,r
∈ ⊂
The following proposition shows the equivalence between (a) and (b)
whenever f is infinitely differentiable on D. First, let us recall the Green
formula associated with the Dunkl Laplacian (see [4]): For every f
∈
C2(B(0,t)), t > 0,
∂
∆ f(y)w (y)dy = f(y)w (y)dσ (y), (7)
k k k 0,t
Z Z ∂n
B(0,t) S(0,t)
where ∂ is the partial derivation operator in the direction of the exterior
∂n
unit normal.
6
Proposition 3.2. Assume that f is infinitely differentiable on D. Then f
is ∆ -harmonic on D if and only if f satisfies the mean value property on
k
D.
Proof. Let x D and r > 0 such that B(x,r) D. We claim that
∈ ⊂
t M (f) is derivable on ]0,r[ and for every t ]0,r[,
x,t
7→ ∈
d 1 t
M (f)= s2λ+1M (∆ f)ds. (8)
dt x,t t2λ+1 Z x,s k
0
Indeed,sinceforeverys ]0,r[,thesupportofσk iscontainedin B(wx,r),
x,s w∈W
∈ ∪
it suffices to prove (8) replacing f by a function h C∞(Rd) such that
∈
h = f on B(wx,r).
w∈W
∪
It is easily seen from (3) that for every t ]0,r[,
∈
d 1
M (h) = (τ h)(ty),y w (y)dσ (y)
x,t x k 0,1
dt d Z h∇ i
k S(0,1)
1 u
= (τ h)(u), w (u)dσ (u)
d t2λ+1 Z h∇ x ti k 0,t
k S(0,t)
1 ∂
= (τ h)(u)w (u)dσ (u).
dkt2λ+1 ZS(0,t) ∂n x k 0,t
Therefore, by the Green formula (7) and the fact that ∆ τ = τ ∆ ,
k x x k
d 1
M (h) = τ (∆ h)(u)w (u)du.
dt x,t dkt2λ+1 ZB(0,t) x k k
Hence, using spherical coordinates we deduce that,
d 1 t
M (h) = s2λ+1M (∆ h)ds.
dt x,t t2λ+1 Z x,s k
0
Thus the claim is proved. Now assume that ∆ f = 0 on D. Then for all
k
t ]0,r[, dM (f) =0, by (8). This yields that M (f) = lim M (f).
∈ dt x,t x,t s→0 x,s
On the other hand, it is known from [7] that the map (x,s) σk is
x,s
7→
continuous with respect to the weak topology on (Rd). Thus,
b
M
limM (f) = f(x). (9)
x,s
s→0
Whence M (f) = f(x) which yields the necessity . Conversely, assume
x,t
that f satisfies the mean value property on D. Then, using (8) we deduce
that M (∆ f) = 0 for all t ]0,r[. Letting t tend to 0 we obtain that
x,t k
∈
∆ f(x) =0.
k
We then conclude, in virtue of Lemma 2.2, that every locally bounded
functionf onD whichsatisfies themeanvalueproperty on D isnecessarily
∆ -harmoniconD. Theconversestatementwillbeprovedintheremainder
k
of this section.
7
Lemma 3.3. Let (h ) C∞(D) be a locally uniformly bounded se-
n n≥1
⊂
quenceof∆ -harmonicfunctionsonD withpointwiselimith. Thenh C∞(D)
k
∈
and ∆ h = 0 on D.
k
Proof. Let x D and let r > 0 such that B(x,r) D. Since for every
∈ ⊂
n 1 thefunction h is ∆ -harmonic on D, it follows from Proposition 3.2
n k
≥
that,
h (x) = h (y)dσk (y).
n n x,r
ZRd
Applying the dominated convergence theorem, we get h(x) = M (h).
x,r
Whence h satisfies the mean value property on D which finishes the proof,
by Lemma 2.2 and Proposition 3.2.
Let g be the fundamental solution of the Dunkl Laplacian. That is, for
k
every ϕ C∞(Rd),
c
∈
g (y)∆ ϕ(y)w (y)dy = ϕ(0). (10)
k k k
ZRd −
It is well known from [4] that,
g (y) = c Γ(λ)2λ−1 y −2λ. (11)
k k
| |
Theorem 3.4. Let h C(D) and f C∞(D). Assume that for every
∈ ∈
ϕ C∞(D),
c
∈
h(x)∆ ϕ(x)w (x)dx = f(x)ϕ(x)w (x)dx.
k k k
Z Z
D D
Then h C∞(D).
∈
Proof. It suffices to prove that h C∞(U), for every W- invariant open
∈
set U such that U D.
⊂
Step 1. Assume first that f = 0 on D. Choose n 1 such that for
0
≥
every x U, B(x, 1 ) D. For every n n , let φ be as in (6). Then,
∈ n0 ⊂ ≥ 0 n
the function h defined on U by
n
h (x) := h(y)τ φ (y)w (y)dy,
n −x n k
Z
D
is infinitely differentiable on U and by (2) for every x U,
∈
∆ h = h(y)∆ (τ ϕ )(y)w (y)dy.
k n k −x n k
Z
D
On the other hand, it follows from (4) that for every x U,
∈
1
h (x) = d n φ (t)t2λ+1M (h)dt
n k n x,t
Z
0
1
= cdk φ(1 u2)u2λ+1Mx,u(h)du.
Z − n
0
8
This yields that (h ) is uniformly bounded on U and converges point-
n n≥n0
wise to h on U. Hence, in view of Lemma 3.3, h C∞(U) and ∆ h = 0
k
∈
on U.
Step 2. We now turn to the general case where f is not trivial. Let
v C∞(Rd) such that v = f on U and define ψ on Rd by
c
∈
ψ(x) := g (y)τ v(y)w (y)dy,
k x k
ZRd
whereg isgivenby(11). Usingsphericalcoordinates,iteasilyseenthatthe
k
function g w is locally Lebesgue integrable on Rd. Thus, ψ C∞(Rd).
k k
∈
Furthermore, it follows from (2) and (10) that ∆ ψ = f on U. Then, for
k
−
every ϕ C∞(U),
c
∈
(h(x)+ψ(x))∆ ϕ(x)w (x)dx = (f(x)+∆ ψ(x))ϕ(x)w (x)dx = 0.
k k k k
ZRd ZRd
Whence, the first step yields that h + ψ is infinitely differentiable on U
which finishes the proof.
We note that the previous theorem was already proved by H. Mejjaolli
and K. Trimèche [5] using Sobolev spaces associated with the Dunkl oper-
ators.
Proof of Theorem 3.1 Statement (a) follows from (b) by means of
Lemma 2.2 and Proposition 3.2. Assume now that (a) holds. Then, by [?],
for every ϕ C∞(D),
c
∈
f(x)∆ ϕ(x)w (x)dx = ∆ f(x)ϕ(x)w (x)dx = 0.
k k k k
Z Z
D D
Use now Theorem 3.4 and Proposition 3.2 to finish the proof. ✷
In the following we shall give a counterexample proving that Theo-
rem 3.1 does not hold true if the open set D is not W-invariant. To that
end, let d= 1 and consider the root system R = √2 . Then, the corre-
{± }
sponding reflection group is given by W = idR . Therfore, an open set
{± }
U R is W-invariant if and only if it is symmetric.
⊂
Proposition 3.5. For every non symmetric open set U R there exists
⊂
a function h : R R which is ∆ -harmonic on U but does not satisfy the
k
→
mean value property on U.
Proof. To abbreviate the notation we write I :=]x r,x+r[. Let x U
x,r
− ∈
and r > 0 such that I U and I U = . Choose f C∞(Rd)
x,r −x,r c
⊂ ∩ ∅ ∈
such that f = 1 on I and f = 0 on U. Since g w is locally Lebesgue
−x,r k k
−
integrable, we deduce that the function h defined on R by
h(z) = g (y)τ f(y)w (y)dy,
k z k
ZR
9
is infinitely differentiable on R. Moreover, by (10), for every z R,
∈
∆ h(z) = τ f(0) = f(z).
k z
− −
Hence, ∆ h = 0 on U. On the other hand, for every t ]0,r[,
k
∈
M (∆ h) = f(y)dσk (y) = σk (I ).
x,t k x,t x,t −x,t
−ZR
Moreover, it follows from [7, Remarks 4.2] that
supp σk = I I .
x,t x,t −x,t
∪
Thus σk (I ) > 0 and consequently M (∆ h) > 0. Whence, by (8)
x,t −x,t x,t k
the function t dM (h) is positive on ]0,r[. Hence, M (h) = h(x)
7→ dt x,t x,t 6
for every t ]0,r[, which means that h does not satisfy the mean value
∈
property on U.
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