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On the Images of Sobolev space under Schrodinger semigroup associated to the Dunkl operator

Sivaramakrishnan C

IIT Hyderabad 6 th Worshop on Fourier analysis and related fields, University of pecs. Joint work with Venku Naidu and Sukumar D

August 26, 2017

Sivaramakrishnan C (IIT Hyderabad) On the Images of Dunkl Sobolev spaces August 26, 2017 1 / 27

Overview

1

Basics

2

Heat semigroup

3

Schrodinger semigroup Image of L2

µ under Schrodinger semigroup

Image of Sobolev space under Schrodinger semigroup

4

References

Sivaramakrishnan C (IIT Hyderabad) On the Images of Dunkl Sobolev spaces August 26, 2017 2 / 27

Classical settings

For ∂i (i = i, 2, · · · , n) and ∆ we have semigroups on L2(Rn, du). Such as Heat kernel semigroup, Hermite semigroup, Special Hermite semigroup, Schrodinger semigroup and so on... The Fourier transform F : L2(R, du) → L2(R, du) is unitary. For x ∈ Rn the translation operator τx : L2(R, du) → L2(R, du). Where τxf (y) = f (x − y) . Convolution ...

Sivaramakrishnan C (IIT Hyderabad) On the Images of Dunkl Sobolev spaces August 26, 2017 3 / 27

Classical settings

For ∂i (i = i, 2, · · · , n) and ∆ we have semigroups on L2(Rn, du). Such as Heat kernel semigroup, Hermite semigroup, Special Hermite semigroup, Schrodinger semigroup and so on... The Fourier transform F : L2(R, du) → L2(R, du) is unitary. For x ∈ Rn the translation operator τx : L2(R, du) → L2(R, du). Where τxf (y) = f (x − y) . Convolution ...

Dunkl Setting

For given finite reflection group G on Rn , root system R and µ : R+ → R≥0 invariant under the group action C. F. Dunkl introduced

Sivaramakrishnan C (IIT Hyderabad) On the Images of Dunkl Sobolev spaces August 26, 2017 3 / 27

Classical settings

For ∂i (i = i, 2, · · · , n) and ∆ we have semigroups on L2(Rn, du). Such as Heat kernel semigroup, Hermite semigroup, Special Hermite semigroup, Schrodinger semigroup and so on... The Fourier transform F : L2(R, du) → L2(R, du) is unitary. For x ∈ Rn the translation operator τx : L2(R, du) → L2(R, du). Where τxf (y) = f (x − y) . Convolution ...

Dunkl Setting

For given finite reflection group G on Rn , root system R and µ : R+ → R≥0 invariant under the group action C. F. Dunkl introduced an operator which is having properties similar to the Differential

• perator.

Sivaramakrishnan C (IIT Hyderabad) On the Images of Dunkl Sobolev spaces August 26, 2017 3 / 27

Classical settings

For ∂i (i = i, 2, · · · , n) and ∆ we have semigroups on L2(Rn, du). Such as Heat kernel semigroup, Hermite semigroup, Special Hermite semigroup, Schrodinger semigroup and so on... The Fourier transform F : L2(R, du) → L2(R, du) is unitary. For x ∈ Rn the translation operator τx : L2(R, du) → L2(R, du). Where τxf (y) = f (x − y) . Convolution ...

Dunkl Setting

For given finite reflection group G on Rn , root system R and µ : R+ → R≥0 invariant under the group action C. F. Dunkl introduced an operator which is having properties similar to the Differential

• perator.

a transformation which is having properties similar to the Fourier transform.

Sivaramakrishnan C (IIT Hyderabad) On the Images of Dunkl Sobolev spaces August 26, 2017 3 / 27

Basics

• C. F. Dunkl, Reflection groups and orthogonal polynomials on the

sphere, Math. Z. 197, 33-60(1988).

• C. F. Dunkl, Differential-difference operators associated to reflection

groups, Trans. Amer. Math. Soc. 311 (1989)no. 1.

Sivaramakrishnan C (IIT Hyderabad) On the Images of Dunkl Sobolev spaces August 26, 2017 4 / 27

Basics

• C. F. Dunkl, Reflection groups and orthogonal polynomials on the

sphere, Math. Z. 197, 33-60(1988).

• C. F. Dunkl, Differential-difference operators associated to reflection

groups, Trans. Amer. Math. Soc. 311 (1989)no. 1. For µ > 0, the Dunkl operator associated to the reflection group Z2 is denoted by Dµ and it is given by

Dunkl Operator

(Dµf )(x) = df dx (x) + µ x (f (x) − f (−x)), x ∈ R.

Sivaramakrishnan C (IIT Hyderabad) On the Images of Dunkl Sobolev spaces August 26, 2017 4 / 27

Basics

Theorem

Now consider the equation, for x, y ∈ R, Dµf (x, y) = yf (x, y). The above equation has a unique real analytic solution Eµ : R × R → R and it can be extended as an analytic function Eµ : C × C → C.

Sivaramakrishnan C (IIT Hyderabad) On the Images of Dunkl Sobolev spaces August 26, 2017 5 / 27

Basics

Theorem

Now consider the equation, for x, y ∈ R, Dµf (x, y) = yf (x, y). The above equation has a unique real analytic solution Eµ : R × R → R and it can be extended as an analytic function Eµ : C × C → C.

Definition

The function Eµ(x, y) is called the Dunkl kernel.

Sivaramakrishnan C (IIT Hyderabad) On the Images of Dunkl Sobolev spaces August 26, 2017 5 / 27

Basics

Dunkl Kernel

Eµ(x, y) =

∞

• k=0

(xy)k γµ(k). Where for k ∈ N,

Generalized factorial function

γµ(2k) = 22kk!Γ(k + µ + 1

2)

Γ(µ + 1

2)

and γµ(2k + 1) = 22k+1k!Γ(k + µ + 3

2)

Γ(µ + 1

2)

.

Sivaramakrishnan C (IIT Hyderabad) On the Images of Dunkl Sobolev spaces August 26, 2017 6 / 27

Basics

Dunkl transform

For f ∈ L1(R, |u|2µdu), the Dunkl transform of f is defined by, ˆ f (y) = c−1

µ

• R

f (x)Eµ(−ix, y)|u|2µdx, y ∈ Rn. Where cµ is the constant chosen so that cµ =

• R e− |x|2

2 |u|2µdx. Sivaramakrishnan C (IIT Hyderabad) On the Images of Dunkl Sobolev spaces August 26, 2017 7 / 27

Basics

Dunkl transform

For f ∈ L1(R, |u|2µdu), the Dunkl transform of f is defined by, ˆ f (y) = c−1

µ

• R

f (x)Eµ(−ix, y)|u|2µdx, y ∈ Rn. Where cµ is the constant chosen so that cµ =

• R e− |x|2

2 |u|2µdx.

Theorem

The Dunkl transform : L2(R, |u|2µdu) → L2(R, |u|2µdu) is unitary.

Sivaramakrishnan C (IIT Hyderabad) On the Images of Dunkl Sobolev spaces August 26, 2017 7 / 27

Basics

Dunkl Translation

The generalized translation (or Dunkl translation) of a function f ∈ L2(R, |u|2µdu) is defined by τ µ

y f (x) = c−1 µ

• R

ˆ f (ξ)Eµ(ix, ξ)Eµ(−iy, ξ)|ξ|2µdξ, x, y ∈ R.

Genrealized convolution

Generalized convolution of f , g ∈ L2(R, |u|2µdu) is given by f ∗µ g(x) =

• R

f (y)τ µ

x ˇ

g(y)|y|2µdy, where ˇ g(u) = g(−u). Equivalently it can be written as f ∗µ g(x) =

• R

ˆ f (ξ)ˆ g(ξ)Eµ(ix, ξ)|ξ|2µdξ.

Sivaramakrishnan C (IIT Hyderabad) On the Images of Dunkl Sobolev spaces August 26, 2017 8 / 27

Heat semi group

Dunkl Laplacian on Rn

∆µ := D2

µ.

Theorem

Dunkl Laplacian generates a strongly continuous, positive prserving semigroup on L2(R, |u|2µdu). Where, et∆µf (x) :=

• R f (y)Γµ(t, x, u)|u|2µdu

if t > 0 f if t = 0. and Γµ(t, x, y) = c−1

µ 2−(µ+ n 2 )

tµ+ n

2

e− |x|2+|y|2

4t

Eµ( x

√ 2t , y √ 2t ), x, y ∈ R.

Note

Γµ(t, x, y) = τ µ

y Fµ(t, x), where Fµ(t, x) = c−1

µ 2−(µ+ 1 2 )

tνµ+ n

2

e− x2

4t . Sivaramakrishnan C (IIT Hyderabad) On the Images of Dunkl Sobolev spaces August 26, 2017 9 / 27

Heat Semigroup

et∆µ : L2(R, |u|2µdu) → L2(R, |u|2µdu) is injective bounded operator. et∆µf = f ∗ Fµ(t, .), for f ∈ L2(R, |u|2µdu). So et∆f can be extended as an entire function on C. et∆µ : L2(R, |u|2µdu) → O(C), where O(C) is the space of all analytic functions on C. Consider et∆µ(L2) =

• et∆µf : f ∈ L2(R, |u|2µdu)
• .

Sivaramakrishnan C (IIT Hyderabad) On the Images of Dunkl Sobolev spaces August 26, 2017 10 / 27

Heat Semigroup

et∆µ : L2(R, |u|2µdu) → L2(R, |u|2µdu) is injective bounded operator. et∆µf = f ∗ Fµ(t, .), for f ∈ L2(R, |u|2µdu). So et∆f can be extended as an entire function on C. et∆µ : L2(R, |u|2µdu) → O(C), where O(C) is the space of all analytic functions on C. Consider et∆µ(L2) =

• et∆µf : f ∈ L2(R, |u|2µdu)
• .

For f , g ∈ L2(R, |u|2µdu), define et∆µf , et∆µget∆µ(L2) := f , gL2(R,|u|2µdu). With respect to above inner product et∆µ(L2) becomes a Hilbert space and et∆µ : L2(R, |u|2µdu) → et∆µ(L2) is unitary.

Sivaramakrishnan C (IIT Hyderabad) On the Images of Dunkl Sobolev spaces August 26, 2017 10 / 27

Heat semigroup

Is there exists a a positive continuous function ρ(z) on C such that et∆µf , et∆µget∆µ(L2) =

• C

et∆µf (z)et∆µg(z)ρ(z)dz for all f , g ∈ L2(C, |u|2µdu). Weighted Bergman space: HL2

ρ := HL2(C, ρ(z)dz) =

• F ∈ O(C) :
• C

|F(z)|2ρ(z)dx < ∞

• .

The space HL2(C, ρ(z)dz) becomes a Hilber space with respect to the following inner product: F, GHL2

ρ :=

• C F(z)G(z)ρ(z)dz.

For µ = 0 , et∆(L2) = HL2(C, (4πt)− 1

2 e− y2 4t dz). Sivaramakrishnan C (IIT Hyderabad) On the Images of Dunkl Sobolev spaces August 26, 2017 11 / 27

Heat semigroup

Theorem

The operator et∆µ : L2(R, |u|2µdu) → Cµ,t is unitary. Where Cµ,t is the Hilbert space of analytic functions on C with reproducing kernel Kµ,t(z, w) := cµe−

• z2+w2

8t

• Eµ

z 2t

1 2

, w 2t

1 2

• , z, w ∈ C.
• S. B. Sontz, The µ-deformed Segal-Bargmann transform is a Hall type

transform, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 12 (2009)no. 2.

• S. B. Sontz, On Segal-Bargmann analysis for finite Coxeter groups and

its heat kernel, Math. Z. 269 (2011)no. 1-2.

Sivaramakrishnan C (IIT Hyderabad) On the Images of Dunkl Sobolev spaces August 26, 2017 12 / 27

Given any f ∈ O(C) it can be written as f = fe + fo. Cµ,t := {f ∈ O(C) : fe ∈ L2(C, νe,µ,t(z)dz) and fo ∈ L2(C, νo,µ,t(z)dz)}. Where, For z ∈ C, t > 0 and µ > 0, define the weight functions νe,µ,t(z) := π−12

1 2 +µ(2t)µ− 1 2 e z2+z2 8t Kµ− 1 2 (|

z (4t)

1 2

|2)| z (4t)

1 2

|2µ+1 (1) νo,µ,t(z) := π−12

1 2 +µ(2t)µ− 1 2 e z2+z2 8t Kµ+ 1 2 (|

z (4t)

1 2

|2)| z (4t)

1 2

|2µ+1. (2) The space Cµ,t is the Hilbert space with respect to the following inner product: For f , g ∈ Cµ,t, f , gCµ,t := fo, goL2(C,νo,µ,t(z)dz) + fe, geL2(C,νe,µt(z)dz).

Sivaramakrishnan C (IIT Hyderabad) On the Images of Dunkl Sobolev spaces August 26, 2017 13 / 27

Schrodinger semi group

Image

∆µ is self-adjoint. The operator i∆µ is skew-adjoint. By Stones theorem, i∆µ generates a strongly continuous unitary semigroup

• eit∆µ

t≥0 on L2(Rn, |u|2µdu).

Where, eit∆µf :=

• Rn Γµ(it, ., y)f (y)|u|2µdy

if t > 0 f if t = 0.

Sivaramakrishnan C (IIT Hyderabad) On the Images of Dunkl Sobolev spaces August 26, 2017 14 / 27

Schrodinger semi group

Image

∆µ is self-adjoint. The operator i∆µ is skew-adjoint. By Stones theorem, i∆µ generates a strongly continuous unitary semigroup

• eit∆µ

t≥0 on L2(Rn, |u|2µdu).

Where, eit∆µf :=

• Rn Γµ(it, ., y)f (y)|u|2µdy

if t > 0 f if t = 0. Moreover, eit∆µf solves the Schrodinger equation associated to the Dunkl Laplacian ∆µ on Rn i∂tu = ∆µu ; u(x, 0) = f (x), where f ∈ L2(R, |u|2µdu).

Sivaramakrishnan C (IIT Hyderabad) On the Images of Dunkl Sobolev spaces August 26, 2017 14 / 27

Image of L2

µ Schrodinger Semi group

The Schrodinger semigroup is unitary on L2(R, |u|2µdu). eitf cannot be extended as an entire function on C for all f ∈ L2(R, |u|2µdu).

Sivaramakrishnan C (IIT Hyderabad) On the Images of Dunkl Sobolev spaces August 26, 2017 15 / 27

Image of L2

µ Schrodinger Semi group

The Schrodinger semigroup is unitary on L2(R, |u|2µdu). eitf cannot be extended as an entire function on C for all f ∈ L2(R, |u|2µdu). So, for f ∈ L2

µ := L2(R, eu2|u|2µdu), it is easy to see that eit∆µf can

be extended as an analytic function on C.

Sivaramakrishnan C (IIT Hyderabad) On the Images of Dunkl Sobolev spaces August 26, 2017 15 / 27

Image of L2

µ Schrodinger Semi group

The Schrodinger semigroup is unitary on L2(R, |u|2µdu). eitf cannot be extended as an entire function on C for all f ∈ L2(R, |u|2µdu). So, for f ∈ L2

µ := L2(R, eu2|u|2µdu), it is easy to see that eit∆µf can

be extended as an analytic function on C. Hence eit∆µ(L2

µ) is a subspace of space of all analytic functions on C.

It is clear that eit∆µ : L2(R, eu2|u|2µdu) − → eit∆µ(L2

µ) is linear and

bijective. For f , g ∈ L2(R, eu2|u|2µdu) we define, eit∆µf , eit∆µgeit∆µ(L2

µ) := f , gL2 µ,

With respect to the above inner product eit∆µ(L2

µ) becomes a Hilbert

• space. Our aim is to identify this space as a weighted Bergman space.

Sivaramakrishnan C (IIT Hyderabad) On the Images of Dunkl Sobolev spaces August 26, 2017 15 / 27

Image of L2

µ Schrodinger Semi group

For µ = 0 case

eit∆ : L2(R, eu2du) − → HL2(C, wt(x + iy)dxdy) is unitary, where wt(x + iy) =

1 (2√πt)e

xy t − y2 4t2 .

• N. Hayashi and S. Saitoh, Analyticity and smoothing effect for the

Schr¨

• dinger equation, Ann. Inst. H. Poincar´

e Phys. Th´

• eor. 52 (1990),
• no. 2, 163–173.
• S. Parui, P. K. Ratnakumar and S. Thangavelu, Analyticity of the

Schr¨

• dinger propagator on the Heisenberg group, Monatsh. Math.

168 (2012), no. 2, 279–303.

Sivaramakrishnan C (IIT Hyderabad) On the Images of Dunkl Sobolev spaces August 26, 2017 16 / 27

Image of L2

µ Schrodinger Semi group

For f ∈ L2

µ set g(u) = f (u)e

u2 2 e i 4t u2, we have

eit∆µf (z) =

1 (2it)µ+ 1

2 e i 4t z2

• ˆ

g ∗µ Fµ(1 2, .)

• ( z

2t ), ∀z ∈ C. Define a linear map G : O(C) − → O(C) by GF(z) = (it)νµ+ n

2 e−itz2F(2tz).

Consider the space, Hµ,t :=

• F ∈ O(C) : G(F) ∈ Cµ, 1

2

• .

The space Hµ,t is Hilbert space with respect to the following inner product : For F, G ∈ Hµ,t, F, GHµ,t := GF, GGCµ, 1

2 . Sivaramakrishnan C (IIT Hyderabad) On the Images of Dunkl Sobolev spaces August 26, 2017 17 / 27

Image of L2

µ Schrodinger Semi group

Theorem

The operator eit∆µ : L2(R, |u|2µeu2du) − → Hµ,t is unitary.

Note

The space Hµ,t is a Reproduing Kernel Hilbert space with Reproducing kernel is given by, Kµ,t(w, z) = |(2ti)−(µ+ 1

2 )|2e i 4t (−z2+w2)Kµ, 1 2

• w

2 √ 2t , z 2 √ 2t

• .

Sivaramakrishnan C (IIT Hyderabad) On the Images of Dunkl Sobolev spaces August 26, 2017 18 / 27

Sobolev space

For m ∈ N,

Dunkl Sobolev space

W m,2

µ

(R) :=

• f ∈ L2

µ : Dk µf ∈ L2 µ, for k ∈ N with k ≤ m

• .

(3) For f , g ∈ W m,2

µ

(R), define f , gW m,2

µ

:=

m

• k=0

Dk

µf , Dk µgL2

µ.

The space W m,2

µ

(R) becomes a Hilbert space with respect to the above inner product.

Sivaramakrishnan C (IIT Hyderabad) On the Images of Dunkl Sobolev spaces August 26, 2017 19 / 27

Image of Dunkl Sobolev space under Schrodinger Semigroup

Define, eit∆µ(W m,2

µ

) :=

• eit∆µf ∈ O(C) : f ∈ W m,2

µ

(Rn)

• .

eit∆µ : W m,2

µ

(R) − → eit∆µ(W 2,m

µ

) is a bijective linear map. For f , g ∈ W m,2

µ

(Rn),

• eit∆µf , eit∆µg
• eit∆µ(W 2,m

µ

) := f , gW m,2

µ

. With respect to the above inner product, the space eit∆µ(W 2,m

µ

) becomes a Hilbert space. The operator eit∆µ : W m,2

µ

(R) − → eit∆µ(W m,2

µ

) becomes unitary.

Sivaramakrishnan C (IIT Hyderabad) On the Images of Dunkl Sobolev spaces August 26, 2017 20 / 27

Image of Dunkl Sobolev space under Schrodinger Semigroup

We cannot get a single weight function on C such that eit∆µ(W m,2

µ

) becomes a Hilbert space such that eit∆µ : W m,2

µ

→ eit∆µ(W m,2

µ

) becomes unitary.

Sivaramakrishnan C (IIT Hyderabad) On the Images of Dunkl Sobolev spaces August 26, 2017 21 / 27

Image of Dunkl Sobolev space under Schrodinger Semigroup

We cannot get a single weight function on C such that eit∆µ(W m,2

µ

) becomes a Hilbert space such that eit∆µ : W m,2

µ

→ eit∆µ(W m,2

µ

) becomes unitary. So we are looking inner product on eit∆µ(W m,2

µ

) such that eit∆µ(W m,2

µ

) becomes a Hilbert space such that eit∆µ : W m,2

µ

→ eit∆µ(W m,2

µ

) becomes bounded invertible map.

Sivaramakrishnan C (IIT Hyderabad) On the Images of Dunkl Sobolev spaces August 26, 2017 21 / 27

Image of Dunkl Sobolev space under Schrodinger Semigroup

We cannot get a single weight function on C such that eit∆µ(W m,2

µ

) becomes a Hilbert space such that eit∆µ : W m,2

µ

→ eit∆µ(W m,2

µ

) becomes unitary. So we are looking inner product on eit∆µ(W m,2

µ

) such that eit∆µ(W m,2

µ

) becomes a Hilbert space such that eit∆µ : W m,2

µ

→ eit∆µ(W m,2

µ

) becomes bounded invertible map. That is Image of W m,2

µ

under Schrodinger semigroup is norms equivalent some weighted Bergman space.

Sivaramakrishnan C (IIT Hyderabad) On the Images of Dunkl Sobolev spaces August 26, 2017 21 / 27

Image of sobolev space in L2(R, eu2du) under Scrodinger semigroup I

Let um

t (z) = m k=0

1

t

2k |zk|2wt(z) and consider the weighted Bergman space HL2(C, um

t (z)dz).

Theorem

The operator eit∆ : W m,2(R) − → HL2(C, um

t (z)dz) is bounded invertible.

Proof: Hermite polynomial: For k N, Hk(x) = k! [ k

2 ]

k=0 (−1)j(2x)k−2j j!(k−2j)!

. Consider ψk(u) =

• 1

√π2kk!

1

2 Hk(u)e−u2, u ∈ R.

The set {

ψk ψkW m,2(R) : k ∈ N} forms an orthonormal basis for

W m,2(R).

Sivaramakrishnan C (IIT Hyderabad) On the Images of Dunkl Sobolev spaces August 26, 2017 22 / 27

Image of sobolev space in L2(R, eu2du) under Scrodinger semigroup II

Consider the multiplication operator Sf (u) = f (u)e− i

4t u2 for

f ∈ wm,2(R). S : W m,2(R) − → W m,2(R) is bounded invertible map. eit∆µSΨµ

k(z) = i− 1

2 1

i

|α| Υ t

α for all k ∈ N and z ∈ C. Where

Υ t

k(z) = (4iπt)− 1

2 (√π) 1 2

• 1

2|k|k!

1

2 1

2ti

|k| zke( i

4t − 1 16t2 )z2.

The set

• Υk

ΥkHL2

um t

: k ∈ N

• form an orthonormal basis for

HL2(C, um

t (x + iy)dxdy).

ΥkHL2

um t

= ψkW m,2(R). for all k ∈ N. eit∆S : W m,2(R) → HL2(C, um

t (x + iy)dxdy) is unitary.

HL2(C, um

t (x + iy)dxdy) is equivalent to the space eit∆(W m,2).

Sivaramakrishnan C (IIT Hyderabad) On the Images of Dunkl Sobolev spaces August 26, 2017 23 / 27

References I

• V. Bargmann, On a Hilbert space of analytic functions and an

associated integral transform, Comm. Pure Appl. Math. 14 (1961), 187–214.

• N. Ben Salem and W. Nefzi, Images of some functions and functional

spaces under the Dunkl-Hermite semigroup, Comment. Math. Univ.

• Carolin. 54 (2013), no. 3, 345–365.
• F. M. Cholewinski, Generalized Fock spaces and associated operators,

SIAM J. Math. Anal. 15 (1984)no. 1.

• C. F. Dunkl, Differential-difference operators associated to reflection

groups, Trans. Amer. Math. Soc. 311 (1989)no. 1.

• S. B. Sontz, The µ-deformed Segal-Bargmann transform is a Hall type

transform, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 12 (2009)no. 2.

Sivaramakrishnan C (IIT Hyderabad) On the Images of Dunkl Sobolev spaces August 26, 2017 24 / 27

References II

• S. B. Sontz, On Segal-Bargmann analysis for finite Coxeter groups and

its heat kernel, Math. Z. 269 (2011)no. 1-2.

• N. Hayashi and S. Saitoh, Analyticity and smoothing effect for the

Schr¨

• dinger equation, Ann. Inst. H. Poincar´

e Phys. Th´

• eor. 52 (1990),
• no. 2, 163–173.
• S. Parui, P. K. Ratnakumar and S. Thangavelu, Analyticity of the

Schr¨

• dinger propagator on the Heisenberg group, Monatsh. Math.

168 (2012), no. 2, 279–303.

• M. Rosler, Positivity of Dunkl’s intertwining operator, Duke Math. J.

98 (1999)no. 3.

Sivaramakrishnan C (IIT Hyderabad) On the Images of Dunkl Sobolev spaces August 26, 2017 25 / 27

References III

• M. Rosenblum, Generalized Hermite polynomials and the Bose-like
• scillator calculus, in Nonselfadjoint operators and related topics (Beer

Sheva, 1992), 369–396, Oper. Theory Adv. Appl., 73, Birkh¨ auser, Basel.

• S. Ben Said and B.Orsted, Segal-Bargmann transforms associated

with finite Coxeter groups, Math. Ann. 334 (2006)no. 2.

• I. E. Segal, Mathematical problems of relativistic physics, With an

appendix by George W. Mackey. Lectures in Applied Mathematics (proceedings of the Summer Seminar, Boulder, Colorado, 1960), Vol. II, American Mathematical Society, Providence, R.I.(1963).

• S. Thangavelu and Y. Xu, Convolution operator and maximal function

for the Dunkl transform, J. Anal. Math. 97 (2005).

Sivaramakrishnan C (IIT Hyderabad) On the Images of Dunkl Sobolev spaces August 26, 2017 26 / 27

Thanks for your attention... K¨

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Sivaramakrishnan C (IIT Hyderabad) On the Images of Dunkl Sobolev spaces August 26, 2017 27 / 27