Leibniz's rule on two-step nilpotent Lie groups

0.19 MB·English

by  Krystian Bekała

LEIBNIZ’S RULE
ON TWO-STEP NILPOTENT LIE GROUPS
KRYSTIAN BEKAŁA
6
1
Abstract. Let g be a nilpotent Lie algebra which is also regarded as a homogeneous
0
Lie group with the Campbell-Hausdorff multiplication. This allows to define a generalized
2
multiplication f#g =(f∨∗g∨)∧ of two functions in the Schwartzclass S(g∗), where ∨ and
y ∧ are the Abelian Fourier transforms on the Lie algebra g and on the dual g∗.
a
M IntheoperatoranalysisonnilpotentLiegroupsanimportantnotionistheoneofsymbolic
calculus which can be viewed as a higher order generalization of the Weyl calculus for

4 pseudodifferentialoperatorsofHörmander.Theideaofsuchacalculusconsistsindescribing
2 the product f#g for some classes of symbols.
We find a formula for Dα(f#g) for Schwartzfunctions f,g in the case of two-stepnilpo-
]
T tent Lie groups, that includes the Heisenberg group. We extend this formula to the class
R of functions f,g such that f∨,g∨ are certain distributions acting by convolutionon the Lie
group, that includes usual classes of symbols. In the case of the Abelian group Rd we have
.
h f#g =fg, so Dα(f#g) is given by the Leibniz rule.
t
a
m
[

2 1. Statement of the result
v
8
Let g be a nilpotent Lie algebra of the dimension d which is endowed with a family of
9
4 dilations (δ ) . We also regard the vector space g as a Lie group with the multiplication
0 t t>0
0 law given by the Campbell-Hausdorff formula (see Corwin-Greenleef [2])
.
2
0 x◦y = x+y +r(x,y),
5
1 where r(x,y) is the (finite) sum of the commutator terms of order at least 2 in the Campbell-
:
v Hausdorff series for g.
i
X This allows to define a generalized multiplication f#g = (f∨∗g∨)∧ of two functions in the
r
a Schwartz class S(g∗), where ∨ and ∧ are the Abelian Fourier transforms on the Lie algebra
g and on the dual g∗. In the case of the Abelian group Rd, one gets f#g = fg.
In the operator analysis on nilpotent Lie groups an important notion is the one of sym-
bolic calculus which can be viewed as a higher order generalization of the Weyl calculus for
pseudodifferential operators of Hörmander [7]. The calculus was created in Melin [10] and
developed in Manchon [9] and Głowacki [3], [6], [4]. The idea of such a calculus consists in
describing the product f#g for some classes of symbols. One of the obstacles in extending
Weyl calculus to the ground of general nilpotent Lie groups is the lack of formula allowing
to calculate the derivatives of the product f#g.
2010 Mathematics Subject Classification. Primary 22E25; Secondary 22E15.
Key words and phrases. Leibniz’s rule, Heisenberg group, Fourier transform, homogeneous groups, sym-
bolic calculus, convolution.
1
2 K.BEKAŁA
In the Abelian case, we have the multidimensional Leibniz rule
α
(1.1) Dα(fg) = DβfDγg, α ∈ Nd.
(cid:18)β(cid:19)
X
β+γ=α
Let h = R2n+1 be the Heisenberg Lie algebra with the commutator
n
[x,y] = (0,....,0,{x,y}), x,y ∈ h ,
n
n
where {x,y} = (x y − x y )), and the Heisenberg group with the Campbell-
i=1 i n+i n+i i
Hausdorff multiplPication. In that case there is a simpler form of f#g (cf. Głowacki [3],
Example 3.3)
(1.2) f#g(w,λ) = cn f(w+λ12u,λ)g(w+λ12v,λ)ei{u,v}dudv, w ∈ R2n,λ > 0.
Z Z
By the chain rule and integration by parts one gets
n
1
(1.3) D (f#g) = D f#g +f#D g + (D f#D g −D f#D g).
2n+1 2n+1 2n+1 i n+i n+i i
2
Xi=1
A general formula for Dα(f#g), α ∈ N2n+1, seems to be more complicated.
The purpose of this note is to find such a "Leibniz’s formula" in the case of two-step
nilpotent Lie groups, that includes the Heisenberg group. By the Fourier transform this is
equivalent to find a formula for Tα(f ∗g), where Tαf(x) = xαf(x) and ∗ is the convolution
on the group g.
In the Abelian case, there is a formula for the convolution product corresponding to (1.1)
α
(1.4) Tα(f ∗ g) = Tβf ∗ Tγg,
0 0
(cid:18)β(cid:19)
X
β+γ=α
where ∗ is the standard convolution on Rd.
0
In the general case of nilpotent Lie groups Głowacki [5] showed that
(1.5) Tα(f ∗g) = Tαf ∗g +f ∗Tαg + c Tβf ∗Tγg,
β,γ
X
l(β)+l(γ)=l(α)
0<l(β)<l(α)
for α 6= 0 and Schwartz functions f, g on g. Here, c are real constants and l(α) is the
β,γ
homogeneous length of a multiindex α (see Section 2). Notice that this formula does not give
exact values of c , and the condition l(β)+l(γ) = l(α) does not characterize precisely the
β,γ
pairs (β,γ) which appear in (1.5) with a nonzero constant term c .
β,γ
Inorder to formulatethe main result we introduce some notation.Let X ,...,X be a base
1 d
of the vector space g. Suppose that A = (a ) is the matrix of the structure constants
i,j,k i,j,k
of g which are given by
d
[X ,X ] = a X , 1 ≤ i,j ≤ d.
i j i,j,k k
Xk=1
LEIBNIZ’S RULE ON TWO-STEP NILPOTENT LIE GROUPS 3
Let D = {(i,j,k) : a 6= 0} and σ ∈ ND. By σ ,σ ,σ ,∈ Nd we denote the multiindices
i,j,k [0] [1] [2]
σ = σ , σ = σ , σ = σ .
[0],k (i,j,k) [1],i (i,j,k) [2],j (i,j,k)
Xi,j Xj,k Xi,k
For α,β ∈ Nd,σ ∈ ND and β +σ ≤ α we define the generalized multinomial coefficient
[0]
α α!
(1.6) = .
(cid:18)β(cid:19) β!σ!(α−β −σ )!
σ [0]
Note that in the case of the Abelian group we have σ = σ = σ = 0 and α = α .
[0] [1] [2] β σ β
Our main result is the following. (cid:0) (cid:1) (cid:0) (cid:1)
Theorem 1.7. Suppose that g is a two-step nilpotent Lie group with the Campbell-Hausdorff
multiplication. For any Schwartz functions f, g on g and every multiindex α ∈ Nd,
α
(1.8) Tα(f ∗g) = c Tβ+σ[1]f ∗Tγ+σ[2]g,
σ
(cid:18)β(cid:19)
β+γX+σ =α σ
[0]
where the (nonzero) constants c are given by
σ
cσ = 2−Pi,j,kσ(i,j,k) aiσ,(ji,,kj,k), σ ∈ ND.
iY,j,k
An analogous formula for more than two functions is given in Proposition 3.11 below.
Moreover, in Corollary 3.15, we show that the above formula is still valid for tempered
distributions whose convolution with the Schwartz class functions is the Schwartz class.
Applying the Fourier transform to (1.8) we get an equivalent formula for Dα(f#g) for
Schwartz functions f,g on the dual g∗. We extend this formula to the certain class of func-
tions, that includes the classes of symbols Sm(g∗,g) which are admissible in calculus of
Głowacki [6] (see Subsection 3.4).
In Subsection 3.5 we illustrate results in the case of the Heisenberg group.
2. Two-step nilpotent Lie group
Let g be a Lie algebra of the dimension d endowed with a family of one-parameter group
automorphisms (δ ) which are called dilations. Let p = 1, p = 2 be the exponents of
t t>0 1 2
homogeneity of the dilations. Let
g = {x ∈ g : δ (x) = tp1x}, g = {x ∈ g : δ (x) = tp2x}.
1 t 2 t
Then g = g ⊕g and g is a two-step nilpotent Lie algebra. Let d = dimg .
1 2 1 1
The vector space g is also regarded as a Lie group with the multiplication
1
x◦y = x+y + [x,y].
2
The exponential map is then the identity map. From the antisymmetry and the Jacobi
identity,
a +a = 0, (a a +a a +a a ) = 0.
i,j,k j,i,k i,j,k k,l,m j,l,k l,i,m l,i,k k,j,m
Xk
4 K.BEKAŁA
Moreover, the homogeneous structure of g gives that a = 0 if any of the conditions i = j,
i,j,k
max(i,j) ≥ k, max(i,j) > d , k ≤ d is satisfied. For every k > d we have (x ◦ y) =
1 1 1 k
x +y +r (x,y), where
k k k
d d
1
r (x,y) = a x y .
k i,j,k i j
2
Xi=1 Xj=1
Let T f(x) = x f(x), D f(x) = i∂ f(x) and
j j j j
Tαf(x) = xα1...xαdf(x), Dαf(x) = Dα1...Dαdf(x).
1 d 1 d
Let |α| = d α be the length of α ∈ Nd. Let us also denote by l(α) the homogeneous
i=1 i
length of αP, i.e.
l(α) = p (α +...+α )+p (α +...+α ).
1 1 d1 2 d1+1 d
The Schwartz space is denoted by S(g). Let Lebesgue measures dx,dξ on g and g∗ be
normalized so that the relationship between a function f ∈ S(g) and its Abelian Fourier
transform f ∈ S(g∗) is given by
b
f(ξ) = e−ixξf(x)dx, f(x) = eixξf(ξ)dξ.
Z Z
g g∗
b b
The Fourier transform extends by duality to the space of tempered distributions.
A normalized Lebesgue measure on the vector space g is a Haar measure on the group g.
The convolution ∗ on g is given by
(2.1) f ∗g(x) = f(x◦y−1)g(y)dy.
Z
g
Recall some notation that we have already introduced in Section 1. For the group g and
σ ∈ ND we defined the d-dimensional multiindices σ ,σ ,σ ∈ Nd. We also defined the
[0] [1] [2]
generalized multinomial coefficient α for α,β ∈ Nd and σ ∈ ND. Let us also denote by c
β σ σ
the constants which appeared in (1(cid:0).7)(cid:1), i.e.
(2.2) cσ = 2−Pi,j,kσ(i,j,k) aiσ,(ji,,kj,k), σ ∈ ND.
iY,j,k
3. Leibniz’s rule
3.1. Multinomial theorem. The following proposition is a generalization of the multino-
mial theorem on Rd. This will be crucial in the proof of Theorem 1.7.
Proposition 3.1. For any x,y ∈ g and every multiindex α ∈ Nd,
α
(3.2) (x◦y)α = c xβ+σ[1]yγ+σ[2]g,
σ
(cid:18)β(cid:19)
β+γX+σ =α σ
[0]
where the (nonzero) constants c are given by (2.2).
σ
LEIBNIZ’S RULE ON TWO-STEP NILPOTENT LIE GROUPS 5
Proof. Let α ∈ Nd. We have
d d1 d
(3.3) (x◦y)α = (x◦y)αk = (x +y )αl (x +y +r (x,y))αk
k l l k k k
kY=1 Yl=1 k=Yd1+1
d1 α d α
= l xβlyγl k xβkyγkr (x,y)τk
(cid:18)β (cid:19) l l (cid:18)β γ τ (cid:19) k k k
Yl=1βl+Xγl=αl l k=Yd1+1 βXk+γk k k k
+τk=αk
d1 α d α
l k
=
(cid:18)β (cid:19) (cid:18)β γ τ (cid:19)
{βl+Xγl=αl:{βk+γXk+τk=αk:Yl=1 l k=Yd1+1 k k k
1≤l≤d1} d1+1≤k≤d}
d1 d d1 d d
× xβl xβk yγl yγk r (x,y)τk.
l k l k k
Yl=1 k=Yd1+1 Yl=1 k=Yd1+1 k=Yd1+1
Let D = {(i,j) : (i,j,k) ∈ D}. Clearly, (i,j) ∈ D if a 6= 0. Thus,
k k i,j,k
τk
1
(3.4) r (x,y)τk =  a x y 
k i,j,k i j
2
X
 (i,j)∈Dk 
τ
= 2−τk k (a x y )τk,i,j
i,j,k i j
(cid:18)...τ ...(cid:19)
X k,i,j Y
P(i,j)∈Dkτk,i,j=τk (i,j)∈Dk
τ
= 2−τk k aτk,i,j xτk,i,jyτk,i,j.
(cid:18)...τ ...(cid:19) i,j,k i j
X k,i,j Y Y
P(i,j)∈Dkτk,i,j=τk (i,j)∈Dk (i,j)∈Dk
Here, τk denotes a multinomial coefficient
...τk,i,j...
(cid:0) (cid:1)
τ τ !
k k
= .
(cid:18)...τ ...(cid:19) τ !
k,i,j (i,j)∈Dk k,i,j
Q
By using (3.4), the expression from (3.3) is equal to
(3.5)
X X X
{β1l≤+lγ≤ld=1α}l:{βdk1++γ1k≤+kτ≤k=d}αk:P(i,j)∈Dkτk,i,j=τk
d1 α d α τ
l  k k 2−τk aτk,i,j
(cid:18)β (cid:19) (cid:18)β γ τ (cid:19)(cid:18)...τ ...(cid:19) i,j,k
Yl=1 l k=Yd1+1 k k k k,i,j (i,jY)∈Dk 
d1 d d1 d
× xβl xβk xτk,i,j yγl yγk yτk,i,j.
l k i l k j
Yl=1 k=Yd1+1 (i,jY)∈Dk Yl=1 k=Yd1+1 (i,jY)∈Dk 
If we denote σ = τ , then σ ∈ ND. Moreover,
(i,j,k) k,i,j
d1 α d α σ α! α
l k k
= = .
(cid:18)β (cid:19) (cid:18)β γ σ (cid:19)(cid:18)...σ ...(cid:19) β!γ!σ! (cid:18)β(cid:19)
Yl=1 l k=Yd1+1 k k k (i,j,k) σ
6 K.BEKAŁA
The conditions β + γ = α , l = 1,2,...,d and τ = τ , β + γ + τ = α ,
l l l 1 (i,j)∈Dk k,i,j k k k k k
k = d + 1,...,d we can simply write as β + γ + σP = α. Recall that the numbers c are
1 [0] σ
given by (2.2). Thus, (3.5) is equal to
d d
α
c xβk xσ(i,j,k) yγk yσ(i,j,k)
(cid:18)β(cid:19) σ k i k j
β+γX+σ[0]=α σ kY=1 (i,jY)∈Dk kY=1 (i,jY)∈Dk 
d d
= α c xβi+Pj,k:(i,j)∈Dkσ(i,j,k) yγj+Pi,k:(i,j)∈Dkσ(i,j,k)
(cid:18)β(cid:19) σ i j
β+γX+σ =α σ Yi=1 Yj=1
[0]
α
= c xβ+σ[1]yγ+σ[2].
σ
(cid:18)β(cid:19)
β+γX+σ =α σ
[0]
(cid:3)
3.2. Convolution rule.
Proof of Theorem 1.7. By (2.1) we have
(3.6) Tα(f ∗g)(x) = xα(f ∗g)(x) = xαf(x◦y−1)g(y)dy.
Z
g
Applying the formula (3.2) we get
α
(3.7) xα = ((x◦y−1)◦y)α = c (x◦y−1)β+σ[1]yγ+σ[2].
σ
(cid:18)β(cid:19)
β+γX+σ =α σ
[0]
The thesis follows from combining (3.7) and (3.6). (cid:3)
As a consequence, we get the relationship between exponents β+σ and γ +σ on the
[1] [2]
right hand side in (1.8) in terms of homogeneous length, as in the formula (1.5).
Corollary 3.8. The formula (1.5) holds.
Proof. Let β +γ +σ = α, where α,β,γ ∈ Nd, σ ∈ ND. By a direct calculation,
[0]
d1 d
l(β +σ )+l(γ +σ ) = (β + σ )+2 β
[1] [2] i (i,j,k) k
Xi=1 Xj,k k=Xd1+1
d1 d d1 d
+ (γ + σ )+2 γ = α +2 α = l(α).
j (i,j,k) k k k
Xj=1 Xi,k k=Xd1+1 Xi=k i=Xd1+1
(cid:3)
IfwecomparethecoefficientsonthebothsidesoftheformulasTα1+α2(f∗g) = Tα1(Tα2(f∗
g)), obtained from Theorem 1.7, we get the following identity.
LEIBNIZ’S RULE ON TWO-STEP NILPOTENT LIE GROUPS 7
Corollary 3.9. For any α1,α2,β ∈ Nd, σ ∈ ND,
α +α α1 α2
1 2
(3.10) = ,
(cid:18) β (cid:19) (cid:18)β1(cid:19) (cid:18)β2(cid:19)
σ b(α1X,α2,β,σ) σ1 σ2
where b(α ,α ,β,σ) is the set
1 2
{(β1,β2,σ1,σ2) : β1 +β2 = β,σ1 +σ2 = σ,β1 +σ1 ≤ α1,β2 +σ2 ≤ α2}.
[0] [0]
Notice that this is the analog of the combinatorial identity
n +n n n
1 2 = 1 2 , n ,n ,k ∈ N.
1 2
(cid:18) k (cid:19) (cid:18)k (cid:19)(cid:18)k (cid:19)
k1+Xk2=k 1 2
k1≤n1,k2≤n2
In the similar way as in Theorem 1.7 we can find a convolution rule for more than two
functions. Before that, we extend a bit our notation. For n ∈ N let
D(n) = {(i,j,k,r,s) : a 6= 0, 1 ≤ r < s ≤ n}.
i,j,k
Notice that if n = 2, then D(2) is essentially the same as D. For τ ∈ ND(n) we denote the
multiindices in Nd
τ = τ , k = 1,...,d
[0],k (i,j,k,r,s)
iX,j,r,s
τ = τ + τ , m = 1,...,n,l = 1,...,d.
[m],l (l,j,k,m,s) (i,l,k,r,m)
Xj,k,s Xi,k,r
For α,β1,...,βn ∈ Nd,τ ∈ ND(n) and n βm +τ = α we denote also
m=1 [0]
P
α α!
(cid:18)β1...βn(cid:19) = β1!...βn!τ!, c˜τ = 2−Pi,j,k,r,sτ(i,j,k,r,s) aτi,(ji,,kj,k,r,s).
τ i,jY,k,r,s
Proposition 3.11. Let f ,...,f be Schwartz functions on g. For every α ∈ Nd,
1 n
α
(3.12) Tα(f ∗...∗f ) = c˜ Tβ1+τ[1]f ∗...∗Tβn+τ[n]f .
1 n (cid:18)β1...βn(cid:19) τ 1 n
β1+...+Xβn+τ =α τ
[0]
Proof. In a similar fashion as in the proof of Proposition 3.1 we find a formula for (y1◦y2◦
...◦yn)α, where y1,...,yn ∈ g. We get
d
1
(3.13) (y1◦y2 ◦...◦yn)α = (y1+...+yn + a yrys)αk
k k 2 i,j,k i j
kY=1 aiX,j,k6=0 Xr<s
α
= c˜ (y1)β1+τ[1]...(yn)βn+τ[n].
(cid:18)β1...βn(cid:19) τ
β1+...+Xβn+τ =α τ
[0]
If we apply (3.13) to the elements y1 = x1 ◦(x2)−1,...,yn−1 = xn−1 ◦(xn)−1, yn = xn, where
x1,...,xn are integral variables in the convolution, we get the thesis. (cid:3)
8 K.BEKAŁA
3.3. S-convolvers. Let A be a tempered distribution on g, i.e. a linear, continuous func-
tional on S(g). The convolution on the right by a tempered distribution A with a Schwartz
function f on g is defined by
f ∗A(x) = hA,f i,
x
where f (y) = f(xy−1). A denotes the distributionegiven by hA,fi = hA,fi. We say that
x
a distribution A ∈ S′(g) is a right S-convolver on a nilpotent Lie group g if f ∗ A ∈ S(g),
e e e e
whenever f ∈ S(g). We define the space of left S-convolvers in a similar way. A is called
an S-convolver if it is both left and right S-convolver. By Proposition 2.5 in Corwin [1],
the space of S-convolvers is closed under convolution and multiplication by polynomials. We
have
f ∗(A∗B) = (f ∗A)∗B, hA∗B,fi = hB,A∗fi.
Theformula(1.8)isvalidalsoforS-convolvers insteadofSchwaretzfunctionsonatwo-step
nilpotent Lie group.
Corollary 3.14. If A, B are S-convolvers on g, then,
α
(3.15) Tα(A∗B) = c Tβ+σ[1]A∗Tγ+σ[2]B.
σ
(cid:18)β(cid:19)
β+γX+σ =α σ
[0]
Proof. We prove (3.15) by the induction on the length of α. Let Tek = T ,k = 1,...,d .
k 1
Suppose at first that A is a Schwartz function. Then,
hT (A∗B),fi = hA∗B,T fi = hB,A∗T fi.
k k k
e
By (1.8), it is equal to
hB,T (A∗f)−T A∗fi = hT B,A∗fi−hB,T A∗fi.
k k k k
e e e e
As T A = −T A, the first step is done, when A is a Schwartz function. If Ais an S-convolver,
k k
thengwe can repeat the same reasoning using the just proven formula
e
T (f ∗A) = T f ∗A+f ∗T A, f ∈ S(h),
k k k
instead of the case α = e in (1.8).
k
Now, let Tek = T , k = d +1,...,d. If A is a Schwartz function, then
k 1
hT (A∗B),fi = hA∗B,T fi = hB,A∗T fi
k k k
1
= hB,T (A∗f)−T Ae∗f − a T A∗T fi
k k i,j,k i j
2
X
e e (i,j)∈Dk e
1
= hT B,A∗fi+hT A∗Bi+ a hT A∗B,T fi.
k k i,j,k i j
2
X
e (i,j)∈Dk
LEIBNIZ’S RULE ON TWO-STEP NILPOTENT LIE GROUPS 9
We get a T T A = 0 from the antisymmetry of the structure constants on g and
(i,j)∈Dk i,j,k j i
then P
1
(3.16) T (A∗B) = T A∗B +A∗T B + a T A∗T B,
k k k i,j,k i j
2
X
(i,j)∈Dk
whenever A is a Schwartz function. Similarly to the case Tek, for k = 1,...,d , we obtain
1
that (3.16) also holds when A is an S-convolver.
Now, assume that the formula (3.15) holds for a multiindex α. The inductive step follows
from the formula (3.10). (cid:3)
3.4. Leibniz’s rule for the product f#g. Applying the Fourier transform to (1.8) we get
an equivalent formula for the derivatives of the product f#g as follows.
Corollary 3.17. If α ∈ Nd and f,g ∈ S(g∗), then
α
(3.18) Dα(f#g) = c Dβ+σ[1]f#Dγ+σ[2]g,
σ
(cid:18)β(cid:19)
β+γX+σ =α σ
[0]
where the constants c are given by (2.2).
σ
The above formula is valid under some weaker smoothness conditions for functions, what
is essential for applying these results for a better understanding of the symbolic calculus on
two-step nilpotent Lie groups.
Let m , m be g-weights on g∗ (for more details see Głowacki [6]) and
1 2
Sm(g∗,g) = {a ∈ C∞(g∗) : |Dαa(x)| ≤ m(x)ρ(x)−l(α)},
where ρ(x) = 1+kxk, k·k being the homogeneous norm on g. A typical example of weight is
m(x) = ρ(x)N,N ∈ R.Noticethatifadistribution Asatisfies A ∈ Sm(g∗,g)forsomeweight
m, then one can write A as a sum of a tempered distribution with compact support and a
b
Schwartz function. Thus A isan S-convolver ong. If a ∈ Sm1(g∗,g) andb ∈ Sm2(g∗,g),then,
by the calculus by Głowacki [6], we have a#b ∈ Sm1m2(g∗,g) and a certain continuity of the
product #, which is sufficient to draw as a conclusion from Corollary 3.14 the following.
Corollary 3.19. The formula (3.18) holds for functions a,b such that a∨, b∨ are S-colvolvers
on g. In particular, if a ∈ Sm1(g∗,g) and b ∈ Sm2(g∗,g), then Dα(a#b) is given by (3.18),
which also can be understood pointwise.
3.5. Heisenberg group. The Heisenberg group/algebra h was introduced in Section 1.
n
Let us recall that the multiplication on h is given by
n
1
(3.20) x◦y = (x +y ,...,x +y ,x +y + {x,y}).
1 1 2n 2n 2n+1 2n+1
2
There is the remarkable relationship between the convolution structure of the Heisenberg
group and the Weyl calculus for pseudodifferential operators, which was explained in, e.g.,
10 K.BEKAŁA
Howe [8]. For λ = 1 in (1.2) one obtains the Weyl formula for the symbol of the composition
of two pseudodifferential operators (cf. Głowacki [3], Example 3.3)
a# b(ξ) = a(ξ +u)b(ξ +v)ei{u,v}dudv.
W
Z Z
It is easy to see that a formula for Dα(a# b) is given by (noncommutative) Leibniz’s rule
W
α
(3.21) Dα(a# b) = Dαa# Dγb.
W W
(cid:18)β(cid:19)
β+Xγ=α
Let f,g ∈ S(h ). It is directly checked that for i = 1,...,2n
n
T (f ∗g) = T f ∗g +f ∗T g.
i i i
If α ∈ N2n+1 and α = 0, then
2n+1
α
Tα(f ∗g) = Tαf ∗Tγg,
(cid:18)β(cid:19)
β+Xγ=α
which is corresponding to (3.21). On the other hand, by the relation
n
1
x = (x◦y−1) +y + ((x◦y−1) y −(x◦y−1) y )
2n+1 2n+1 2n+1 i n+i n+i i
2
Xi=1
we get also that (cf. the formula (1.3))
n
1
T (f ∗g) = T f ∗g +f ∗T g + (T f ∗T g −T f ∗T g).
2n+1 2n+1 2n+1 i n+i n+i i
2
Xi=1
Higher order formulas are more complicated, for instance
T2 (f ∗g) =T2 f ∗g +f ∗T2 g +2T f ∗T g
2n+1 2n+1 2n+1 2n+1 2n+1
n
+ (T T f ∗T g +T f ∗T T g
2n+1 i n+i i 2n+1 n+i
Xi=1
−T T f ∗T g −T f ∗T T g)
2n+1 n+i i n+i 2n+1 i
n n
1
+ (T T f ∗T T g −T T f ∗T T g
j i n+j n+i n+j i j n+i
4
Xi=1 Xj=1
−T T f ∗T T g +T T f ∗T T g).
j n+i n+j i n+j n+i j i
We find a general formula for Tk (f ∗ g), k ∈ N, as a conclusion from Theorem 1.7.
2n+1
Let us first illustrate the notation by using them in the case of the Heisenberg group. The
matrix A is given by
a = 1, a = −1, i = 1,...,n,
i,n+i,2n+1 n+i,i,2n+1
and a = 0, otherwise. We have
i,j,k
D = {(1,n+1,2n+1),...,(n,2n,2n+1),(n+1,1,2n+1),...,(2n,n,2n+1)}.