A Clifford deformation of the Hermite semigroupmachiang/PresentationHDBHKExtraSemin… · Cli ord algebras, Dirac operators and osp(1j2) Intertwining operators and function theory

IntroductionFourier transforms in harmonic analysis

Clifford deformation of the Hermite semigroup

A Clifford deformation of the Hermite semigroup

Hendrik De Bie

Ghent University

(joint work with B. Ørsted, P. Somberg and V. Soucek)

Hong Kong, March 2011

Hendrik De Bie A Clifford deformation of the Hermite semigroup



Introduction

Fourier transforms in harmonic analysisClassical FTNew realizations of sl2 in harmonic analysis

Clifford deformation of the Hermite semigroupClifford algebras, Dirac operators and osp(1|2)Intertwining operators and function theoryFurther research




Outline

Introduction






Problem:

Find complex-valued functions f (z ,w), g(z ,w) with(z ,w) ∈ R+ × [−1, 1], satisfying

(m − 2 + c)g + (1 + c)z∂zg +1

z∂w f + icf − iczwg = 0

cz∂z f − w∂w f − czwg − cz2w∂zg

+z(w 2 − 1)∂w g + icz2g = 0

Here: m ∈ N, c > 0

Observe: c = 1, g = 0 then f = e−izw ∼ Fourier kernel!

I Why is this system interesting?

I Can we find nice other solutions?

I Are there symmetries present?




Problem:


(m − 2 + c)g + (1 + c)z∂zg +1



+z(w 2 − 1)∂w g + icz2g = 0

Here: m ∈ N, c > 0








Problem:


(m − 2 + c)g + (1 + c)z∂zg +1



+z(w 2 − 1)∂w g + icz2g = 0

Here: m ∈ N, c > 0








Classical FTNew realizations of sl2 in harmonic analysis

Outline

Introduction







4 definitions of classical FT in Rm:

I F1

F(f )(y) =

∫Rm

e−i〈x ,y〉 f (x) dx

I F2 F(f )(y) =∫

Rm K (x , y) f (x) dx with K (x , y) uniquesolution of

∂yj K (x , y) = −ixjK (x , y), j = 1, . . . ,m.

I F3F = e

iπ4

(∆−|x |2)

I F4

K (x , y) =∞∑

k=0

(k + λ)(−i)k(z)−λJk+λ(z) Cλk (w)

with z = |x ||y |, w = 〈x ′, y ′〉 and λ = (m − 2)/2.

Each with its specific usesHendrik De Bie A Clifford deformation of the Hermite semigroup




F3 F = eiπ4

(∆−|x |2)

I easiest to generalize

I connects FT with representation theory of sl2:

∆ =∑m

i=1 ∂2xi, Laplace operator

r 2 = |x |2 =∑m

i=1 x2i

E =∑m

i=1 xi∂xi , Euler operator

∆, r 2 and E + m/2 generate the Lie algebra sl2:[∆, r 2

]= 4(E +

m

2)[

∆,E +m

2

]= 2∆[

r 2,E +m

2

]= −2r 2





Overview of possible deformations:

∆κ − |x |2

∆− |x |2

Dunkl deformation

OO

Clifford deformation

{{xxxxxxxxxxxxxxxxxx

a - deformation

""DDDDDDDDDDDDDDDDD

D2 + (1 + c)2|x |2 |x |2−a∆− |x |a





Dunkl operators:

I reduce O(m) symmetry to finite reflection group symmetry

I change the structure of functions on the sphere (e.g. sphericalharmonics → Dunkl harmonics)

? Natural question

I can we preserve the spherical symmetry?

I change the radial structure





Deforming the operators in Rm

Introduce a parameter a > 0 and substitute

r 2 −→ ra

∆ −→ r 2−a∆

E +m

2−→ E +

a + m − 2

2

The sl2 relations also hold for ra, r 2−a∆ and E + a+m−22 :[

r 2−a∆, ra]

= 2a (E +a + m − 2

2)[

r 2−a∆,E +a + m − 2

2

]= a r 2−a∆[

ra,E +a + m − 2

2

]= −a ra

Ben Saıd S., Kobayashi, T. and Ørsted B.,

Laguerre semigroup and Dunkl operators.Preprint. arXiv:0907.3749, 74 pages.





Deforming the operators in Rm

Introduce a parameter a > 0 and substitute

r 2 −→ ra

∆ −→ r 2−a∆

E +m

2−→ E +

a + m − 2

2

The sl2 relations also hold for ra, r 2−a∆ and E + a+m−22 :[

r 2−a∆, ra]

= 2a (E +a + m − 2

2)[

r 2−a∆,E +a + m − 2

2

]= a r 2−a∆[

ra,E +a + m − 2

2

]= −a ra

Ben Saıd S., Kobayashi, T. and Ørsted B.,

Laguerre semigroup and Dunkl operators.Preprint. arXiv:0907.3749, 74 pages.





The associated Fourier transform

Fa = eiπ2a

(r2−a∆k−ra)

Main question: write Fa as

Fa =

∫Rm

K (x , y)f (x)ha(x)dx

Here, ha(x) = ra−2 is the measure naturally associated with thedeformation

Resulting kernel:

I explicitly known if m = 1 or a = 1, 2

I other values: infinite series of Bessel functions timesGegenbauer polynomials −→ Boundedness?





New approach: try to find (and solve) system of PDEs for kernelK (x , y) (i.e. formulation F2)

equivalent with factorizing r 2−a∆ as

r 2−a∆ =m∑

i=0

D2i

with Di family of commuting differential operators (DiDj = DjDi )

Example (classical Fourier transform, formulation F2)

In the case a = 2, Di = ∂xi and one has

∂xj K (x , y) = −iyjK (x , y), j = 1, . . . ,m

leading toK (x , y) = e−i〈x ,y〉





New approach: try to find (and solve) system of PDEs for kernelK (x , y) (i.e. formulation F2)

equivalent with factorizing r 2−a∆ as

r 2−a∆ =m∑

i=0

D2i

with Di family of commuting differential operators (DiDj = DjDi )

Example (classical Fourier transform, formulation F2)

In the case a = 2, Di = ∂xi and one has

∂xj K (x , y) = −iyjK (x , y), j = 1, . . . ,m

leading toK (x , y) = e−i〈x ,y〉




Clifford algebras, Dirac operators and osp(1|2)Intertwining operators and function theoryFurther research

Outline

Introduction







Factorization of ∆ and r 2

Introduce orthogonal Clifford algebra Clm with generatorsei , i = 1, . . . ,m:

eiej + ejei = 0e2i = −1

We have Clm = ⊕mk=0Clkm with

Clkm := span{ei1ei2 . . . eik , i1 < . . . < ik}.

Then put

∂x =m∑

i=1

ei∂xi Dirac operator

x =m∑

i=1

eixi vector variable





We have

∂2x = −∆

x2 = −r 2

and x , ∂x generate the Lie superalgebra osp(1|2)

Questions:

I Can we factorize r 2−a∆ and ra?

I Do CA methods provide insight in radial deformation?





We have

∂2x = −∆

x2 = −r 2

and x , ∂x generate the Lie superalgebra osp(1|2)

Questions:

I Can we factorize r 2−a∆ and ra?

I Do CA methods provide insight in radial deformation?





Definition of the operatorsFactorization of ra:

xa = ra2−1x

Factorization of r 2−a∆:Ordinary Dirac operator given by

∂x = −1

2[x ,∆]

Hence first Ansatz:

Da = −1

2[xa, r

2−a∆]

= r 1− a2∂x +

1

2

(a

2− 1)(a

2+ m − 1

)r−

a2−1x +

(a

2− 1)

r−a2−1xE





Definition of the operatorsFactorization of ra:

xa = ra2−1x

Factorization of r 2−a∆:Ordinary Dirac operator given by

∂x = −1

2[x ,∆]

Hence first Ansatz:

Da = −1

2[xa, r

2−a∆]

= r 1− a2∂x +

1

2

(a

2− 1)(a

2+ m − 1

)r−

a2−1x +

(a

2− 1)

r−a2−1xE





We can work even slightly more general, by replacing

r 1− a2∂x +

1

2

(a

2− 1)(a

2+ m − 1

)r−

a2−1x +

(a

2− 1)

r−a2−1xE

⇓

D = r 1− a2∂x + br−

a2−1x + cr−

a2−1xE

with a > 0 and b, c ∈ C−→ a-deformed Dirac operator (depending on 3 parameters)

If a = 2, b = c = 0 then D = ∂x





We can work even slightly more general, by replacing

r 1− a2∂x +

1

2

(a

2− 1)(a

2+ m − 1

)r−

a2−1x +

(a

2− 1)

r−a2−1xE

⇓

D = r 1− a2∂x + br−

a2−1x + cr−

a2−1xE

with a > 0 and b, c ∈ C−→ a-deformed Dirac operator (depending on 3 parameters)

If a = 2, b = c = 0 then D = ∂x





Theorem

The operators D and xa

D = r 1− a2∂x + br−

a2−1x + cr−

a2−1xE, xa = r

a2−1x

generate for each value of a, b and c a copy of osp(1|2):

{xa,D} = −2(1 + c)(E + δ

2

) [E + δ

2 ,D]

= − a2D[

x2a,D

]= a(1 + c)xa

[E + δ

2 , xa

]= a

2 xa[D2, xa

]= −a(1 + c)D

[E + δ

2 ,D2]

= −aD2[D2, x2

a

]= 2a(1 + c)2

(E + δ

2

) [E + δ

2 , x2a

]= ax2

a,

with

δ =a

2+

2b + m − 1

1 + c.

−→ δ = dimension of theory!Hendrik De Bie A Clifford deformation of the Hermite semigroup




However, a computation shows

D2 6= −r 2−a∆

Hence, we have NOT found a factorization of r 2−a∆

Nevertheless,

I new operators interesting in their own right

I instead of radial deformation of Laplace, radial deformation ofunderlying Dirac

−→ we continue by developing related function theory





Intertwining operatorsWe want to reduce D to its simplest form

Let P and Q be two operators defined by

Pf (x) = rbf

((a

2

) 1a

xr2a−1

)Qf (x) = r−

ab2 f

((2

a

) 12

xra2−1

).

These two operators act as generalized Kelvin transformations:

QP = PQ =

(2

a

) b2

.





Intertwining operatorsWe want to reduce D to its simplest form

Let P and Q be two operators defined by

Pf (x) = rbf

((a

2

) 1a

xr2a−1

)Qf (x) = r−

ab2 f

((2

a

) 12

xra2−1

).

These two operators act as generalized Kelvin transformations:

QP = PQ =

(2

a

) b2

.





Proposition

One has the following intertwining relations

Q(∂x + br−2x + cr−2xE

)P = r 1− a

2∂x + βr−a2−1x + γr−

a2−1xE

Q x P = xa

with

β = 2b + 2c

γ =2

a(1 + c)− 1.

−→ we can get rid of r 1− a2

−→ similarly technique: we can make b = 0





So we are reduced to

D = ∂x + cr−2xE

Theorem

The operators D and x generate osp(1|2), with δ = 1 + m−11+c :

{x ,D} = −2(1 + c)(E + δ

2

) [E + δ

2 ,D]

= −D[x2,D

]= 2(1 + c)x

[E + δ

2 , x]

= x[D2, x

]= −2(1 + c)D

[E + δ

2 ,D2]

= −2D2[D2, x2

]= 4(1 + c)2

(E + δ

2

) [E + δ

2 , x2]

= 2x2.

c is deformation parameter





The operatorD = ∂x + cr−2xE,

also appears in a different context in

Cacao, I., Constales, D., and Krausshar, R. S.

On the role of arbitrary order Bessel functions in higher dimensional Dirac type equations.Arch. Math. (Basel) 87 (2006), 468–477.

Its square is complicated and NOT scalar:

D2 = −∆− (cm − c) r−1∂r −(c2 + 2c

)∂2

r

−cr−2∑i<j

eiej(xi∂xj − xj∂xi ).

RecallΓ =

∑i<j

eiej(xi∂xj − xj∂xi )

is Gamma operatorHendrik De Bie A Clifford deformation of the Hermite semigroup




The measure associated to D:

Proposition

If c > −1, then for suitable differentiable functions f and g one has∫Rm

(Df ) g h(r)dx =

∫Rm

f (Dg) h(r)dx

with h(r) = r 1− 1+mc1+c , provided the integrals exist.

. is the main anti-involution on Clm

Measure looks complicated. However, radial part:

h(r)dx −→ r δ−1dr





Function spaceWe use

L2c(Rm) = L2(Rm, h(r)dx)⊗ Clm

with inner product

〈f , g〉 =

[∫Rm

f c g h(r)dx

]0

satisfying

〈Df , g〉 = 〈f ,Dg〉〈xf , g〉 = −〈f , xg〉.

The related norm is defined by ||f ||2 = 〈f , f 〉.





We need a basis for L2c(Rm):

functions φt,`,m (t, ` ∈ N and m = 1, . . . dimM`) defined as

φ2t,`,m = Lγ`2−1

t (r 2)rβ`M(m)` e−r2/2

φ2t+1,`,m = Lγ`2

t (r 2)xrβ`M(m)` e−r2/2

with Lβα the Laguerre polynomials and

β` = − c

1 + c`

γ` =2

1 + c

(`+

µ− 2

2

)+

c + 2

1 + c

and with {M(m)` } a basis of M`

M` = ker ∂x ∩ P`: spherical monogenics of degree `





We need a basis for L2c(Rm):

functions φt,`,m (t, ` ∈ N and m = 1, . . . dimM`) defined as

φ2t,`,m = Lγ`2−1

t (r 2)rβ`M(m)` e−r2/2

φ2t+1,`,m = Lγ`2

t (r 2)xrβ`M(m)` e−r2/2

with Lβα the Laguerre polynomials and

β` = − c

1 + c`

γ` =2

1 + c

(`+

µ− 2

2

)+

c + 2

1 + c

and with {M(m)` } a basis of M`

M` = ker ∂x ∩ P`: spherical monogenics of degree `





Theorem (Orthogonality of basis)

After suitable normalization, one has

〈φt1,`1,m1 , φt2,`2,m2〉 = δt1t2δ`1`2δm1m2 .

Every f ∈ L2c(Rm) can hence be decomposed as

f =∑t,`,m

at,`,mφt,`,m, at,`,m ∈ R

with ∑t,`,m

|at,`,m|2 <∞





Creation and annihilation operators in this context:

A+ = D− (1 + c)x

A− = D + (1 + c)x

satisfying A+φt,`,m = φt+1,`,m and A−φt,`,m = φt−1,`,m.Lead to generalized harmonic oscillator:

Theorem

The functions φt,`,m satisfy the following second-order PDE(D2 − (1 + c)2x2

)φt,`,m = (1 + c)2(γ` + 2t)φt,`,m.

Recall

γ` =2

1 + c

(`+

m − 2

2

)+

c + 2

1 + c





We want to study the holomorphic semigroup attached to thisharmonic oscillator

FωD = e−ω

2(1+c)2 (D2−(1+c)2x2).

ω complex number, <ω ≥ 0

ω = iπ/2 −→ generalized Fourier transform

eω(∆−|x |2) is so-called Hermite semigroup





Theorem

Suppose c > −1. Then

1. {φt,`,m} is an eigenbasis of FωD:

FωD(φt,`,m) = e−ωte− ω`

(1+c)φt,`,m.

2. FωD is a continuous operator on L2c(Rm) for all ω with

<ω ≥ 0, in particular

||FωD(f )|| ≤ ||f ||

3. If <ω > 0, then FωD is a Hilbert-Schmidt operator on L2c(Rm).

4. If <ω = 0, then FωD is a unitary operator on L2c(Rm).





Explicit formula for semigroupWe want to find K (x , y ;ω) such that

FωD(f ) = e−ω

2(1+c)2 (D2−(1+c)2x2)f (x)

=

∫Rm

K (x , y ;ω)f (x) h(r)dx

Note that in general K (x , y ;ω) takes values in ClmMoreover, K (x , y ;ω) 6= K (y , x ;ω)

We use techniques developed inH. De Bie and Y. Xu,

On the Clifford-Fourier transformIMRN, arXiv:1003.0689, 30 pages.





Theorem (<ω > 0 and c > −1)

Put K (x , y ;ω) = e−cothω

2(|x |2+|y |2)

(A(z ,w) + x ∧ yB(z ,w)

)with

A(z ,w) =+∞∑k=0

(αk

k + 2λ

2λz

k1+c J γk

2−1

(iz

sinhω

)+

αk−1

4 sinhω

k

λz

k+c1+c J γk−1

2

(iz

sinhω

))Cλ

k (w)

and z = |x ||y |, w = 〈x , y〉/z and αk = 2eωδ2 (2 sinhω)−γk/2. These

series are convergent and the transform defined on L2c(Rm) by

Fωc (f ) =

∫Rm

K (x , y ;ω)f (x)h(rx)dx

coincides with the operator FωD = e−ω

2(1+c)2 (D2−(1+c)2x2)on {φt,`,m}.





If <ω = 0 the kernel follows by taking a limit

We are mostly interested in 1 specific value:

ω = iπ

2

−→ the ‘Clifford deformed Fourier transform’

We denote it by Fc





Theorem (Fourier transform; ω = iπ/2)

Put K (x , y) = A(z ,w) + x ∧ yB(z ,w) with

A(z ,w) =+∞∑k=0

z−δ−2

2

(αk

k + 2λ

2λJ γk

2−1(z)− iαk−1

k

2λJ γk−1

2(z)

)Cλ

k (w)

and z = |x ||y |, w = 〈x , y〉/z and αk = e− iπk

2(1+c) . These series areconvergent and the transform defined on L2

c(Rm) by

Fc(f ) =

∫Rm

K (x , y)f (x)h(rx)dx

coincides with the operator FD = e−iπ

4(1+c)2 (D2−(1+c)2x2)on {φt,`,m}.





Theorem (Properties of Fourier transform)

The operator Fc defines a unitary operator on L2c(Rm) and

satisfies the following intertwining relations on a dense subset:

Fc ◦D = i(1 + c)x ◦ Fc

Fc ◦ x =i

1 + cD ◦ Fc

Fc ◦ E = − (E + δ) ◦ Fc .

Moreover, Fc is of finite order if and only if 1 + c is rational.





Proposition (Bochner formulae)

Let M` ∈M` be a spherical monogenic of degree `. Letf (x) = f (r) be a radial function. Then:

Fc(f (r)M`) = e− iπ`

2(1+c) M`(y ′)

∫ +∞

0r `f (r)z−

δ−22 J γk

2−1(z)h(r)rm−1dr

Fc(f (r)xM`) = −ie− iπ`

2(1+c) y ′M`(y ′)

∫ +∞

0r `+1f (r)z−

δ−22 J γk

2(z)h(r)rm−1dr

with y = sy ′, y ′ ∈ Sm−1 and z = rs. h(r) is the measureassociated with D.

−→ connects deformed Fourier transform with Hankel transform





Proposition (Heisenberg inequality)

For all f ∈ L2c(Rm), the deformed Fourier transform satisfies

||x f (x)||.||x (Fc f ) (x)|| ≥ δ

2||f (x)||2.

The equality holds if and only if f is of the form f (x) = λe−r2/α.

Heisenberg inequality = statement about lowest eigenvalue ofD2 − (1 + c)2x2





Theorem (Master formula)

Let s > 0. Then one has∫Rm

K (y , x ; iπ

2)K (z , y ;−i

π

2)e−sr2

y h(ry )dy

= cst × K (z , x ;ω)e−|x|2+|z|2

21−coshω

sinhω

with 2s = sinhω.

I connects our semigroup with the fundamental solution of theheat equation

I defines a ‘generalized translation’

I order of variables is important





Theorem (Master formula)

Let s > 0. Then one has∫Rm

K (y , x ; iπ

2)K (z , y ;−i

π

2)e−sr2

y h(ry )dy

= cst × K (z , x ;ω)e−|x|2+|z|2

21−coshω

sinhω

with 2s = sinhω.

I connects our semigroup with the fundamental solution of theheat equation

I defines a ‘generalized translation’

I order of variables is important





Series representation of kernel is nice for L2-theory

If we want to go beyond that, we need more explicit expressions

Kernel satisfies system of PDEs:

Dy K (x , y) = −i(1 + c)K (x , y)x

(K (x , y)Dx) = −i(1 + c)yK (x , y)

As we have series representation, we have Ansatz for solution:

K (x , y) = f (z ,w) + x ∧ y g(z ,w)

with z = |x ||y | and w = 〈x , y〉/z





This leads us back to the beginning of the talk!

Problem:


(m − 2 + c)g + (1 + c)z∂zg +1



+z(w 2 − 1)∂w g + icz2g = 0

Here: m ∈ N, c > 0

Moreover, series representation learns that solution in Rm isdetermined by solution in Rm−2

−→ sufficient to solve PDEs for m = 2, 3!





Further research and outlook

I Clifford analysis study of new Dirac operatorI determination of the Fourier kernel for special values (easier

than in Dunkl case?)I heat equation and translation operatorI connection with Hecke algebras and DAHAI investigation of more general deformations

H. De Bie, B. Ørsted, P. Somberg and V. Soucek,

Dunkl operators and a family of realizations of osp(1|2).Preprint: arXiv:0911.4725, 25 pages.

H. De Bie, B. Orsted, P. Somberg and V. Soucek,

The Clifford deformation of the Hermite semigroup.Preprint, 27 pages, arXiv:1101.5551.


Documents

A Clifford deformation of the Hermite semigroupmachiang/PresentationHDBHKExtraSemin… · Cli ord algebras, Dirac operators and osp(1j2) Intertwining operators and function theory