On the Images of Sobolev space under Schrodinger semigroup - PowerPoint PPT Presentation

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 Basics 1 Heat semigroup 2 Schrodinger semigroup 3 Image of L 2 µ under Schrodinger semigroup Image of Sobolev space under Schrodinger semigroup References 4 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 L 2 ( R n , du ). Such as Heat kernel semigroup, Hermite semigroup, Special Hermite semigroup, Schrodinger semigroup and so on... The Fourier transform F : L 2 ( R , du ) → L 2 ( R , du ) is unitary. For x ∈ R n the translation operator τ x : L 2 ( R , du ) → L 2 ( R , du ). Where τ x f ( 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 L 2 ( R n , du ). Such as Heat kernel semigroup, Hermite semigroup, Special Hermite semigroup, Schrodinger semigroup and so on... The Fourier transform F : L 2 ( R , du ) → L 2 ( R , du ) is unitary. For x ∈ R n the translation operator τ x : L 2 ( R , du ) → L 2 ( R , du ). Where τ x f ( y ) = f ( x − y ) . Convolution ... Dunkl Setting For given finite reflection group G on R n , 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 L 2 ( R n , du ). Such as Heat kernel semigroup, Hermite semigroup, Special Hermite semigroup, Schrodinger semigroup and so on... The Fourier transform F : L 2 ( R , du ) → L 2 ( R , du ) is unitary. For x ∈ R n the translation operator τ x : L 2 ( R , du ) → L 2 ( R , du ). Where τ x f ( y ) = f ( x − y ) . Convolution ... Dunkl Setting For given finite reflection group G on R n , 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 operator. 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 L 2 ( R n , du ). Such as Heat kernel semigroup, Hermite semigroup, Special Hermite semigroup, Schrodinger semigroup and so on... The Fourier transform F : L 2 ( R , du ) → L 2 ( R , du ) is unitary. For x ∈ R n the translation operator τ x : L 2 ( R , du ) → L 2 ( R , du ). Where τ x f ( y ) = f ( x − y ) . Convolution ... Dunkl Setting For given finite reflection group G on R n , 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 operator. 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 Z 2 is denoted by D µ and it is given by Dunkl Operator ( D µ f )( 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 , D µ f ( x , y ) = yf ( x , y ) . The above equation has a unique real analytic solution E µ : R × R → R and it can be extended as an analytic function E µ : 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 , D µ f ( x , y ) = yf ( x , y ) . The above equation has a unique real analytic solution E µ : R × R → R and it can be extended as an analytic function E µ : C × C → C . Definition The function E µ ( 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 ∞ ( xy ) k � E µ ( x , y ) = γ µ ( k ) . k =0 Where for k ∈ N , Generalized factorial function γ µ (2 k ) = 2 2 k k !Γ( k + µ + 1 and γ µ (2 k + 1) = 2 2 k +1 k !Γ( k + µ + 3 2 ) 2 ) . Γ( µ + 1 Γ( µ + 1 2 ) 2 ) Sivaramakrishnan C (IIT Hyderabad) On the Images of Dunkl Sobolev spaces August 26, 2017 6 / 27

Basics Dunkl transform For f ∈ L 1 ( R , | u | 2 µ du ), the Dunkl transform of f is defined by, � f ( y ) = c − 1 ˆ f ( x ) E µ ( − ix , y ) | u | 2 µ dx , y ∈ R n . µ R R e − | x | 2 2 | u | 2 µ dx . � Where c µ is the constant chosen so that c µ = Sivaramakrishnan C (IIT Hyderabad) On the Images of Dunkl Sobolev spaces August 26, 2017 7 / 27

Basics Dunkl transform For f ∈ L 1 ( R , | u | 2 µ du ), the Dunkl transform of f is defined by, � f ( y ) = c − 1 ˆ f ( x ) E µ ( − ix , y ) | u | 2 µ dx , y ∈ R n . µ R R e − | x | 2 2 | u | 2 µ dx . � Where c µ is the constant chosen so that c µ = Theorem The Dunkl transform : L 2 ( R , | u | 2 µ du ) → L 2 ( R , | u | 2 µ du ) 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 ∈ L 2 ( R , | u | 2 µ du ) is defined by � ˆ τ µ y f ( x ) = c − 1 f ( ξ ) E µ ( ix , ξ ) E µ ( − iy , ξ ) | ξ | 2 µ d ξ, x , y ∈ R . µ R Genrealized convolution Generalized convolution of f , g ∈ L 2 ( R , | u | 2 µ du ) is given by � g ( y ) | y | 2 µ dy , f ( y ) τ µ f ∗ µ g ( x ) = x ˇ R where ˇ g ( u ) = g ( − u ). Equivalently it can be written as � ˆ g ( ξ ) E µ ( ix , ξ ) | ξ | 2 µ d ξ. f ∗ µ g ( x ) = f ( ξ )ˆ R Sivaramakrishnan C (IIT Hyderabad) On the Images of Dunkl Sobolev spaces August 26, 2017 8 / 27

Heat semi group Dunkl Laplacian on R n ∆ µ := D 2 µ . Theorem Dunkl Laplacian generates a strongly continuous, positive prserving semigroup on L 2 ( R , | u | 2 µ du ) . Where, �� R f ( y )Γ µ ( t , x , u ) | u | 2 µ du if t > 0 e t ∆ µ f ( x ) := f if t = 0 . µ 2 − ( µ + n e − | x | 2+ | y | 2 and Γ µ ( t , x , y ) = c − 1 2 ) y E µ ( x 2 t ) , x , y ∈ R . √ 2 t , √ 4 t t µ + n 2 Note µ 2 − ( µ + 1 e − x 2 y F µ ( t , x ) , where F µ ( t , x ) = c − 1 2 ) Γ µ ( t , x , y ) = τ µ 4 t . t νµ + n 2 Sivaramakrishnan C (IIT Hyderabad) On the Images of Dunkl Sobolev spaces August 26, 2017 9 / 27

Heat Semigroup e t ∆ µ : L 2 ( R , | u | 2 µ du ) → L 2 ( R , | u | 2 µ du ) is injective bounded operator. e t ∆ µ f = f ∗ F µ ( t , . ), for f ∈ L 2 ( R , | u | 2 µ du ). So e t ∆ f can be extended as an entire function on C . e t ∆ µ : L 2 ( R , | u | 2 µ du ) → O ( C ), where O ( C ) is the space of all analytic functions on C . Consider e t ∆ µ ( L 2 ) = � e t ∆ µ f : f ∈ L 2 ( R , | u | 2 µ du ) � . Sivaramakrishnan C (IIT Hyderabad) On the Images of Dunkl Sobolev spaces August 26, 2017 10 / 27

Heat Semigroup e t ∆ µ : L 2 ( R , | u | 2 µ du ) → L 2 ( R , | u | 2 µ du ) is injective bounded operator. e t ∆ µ f = f ∗ F µ ( t , . ), for f ∈ L 2 ( R , | u | 2 µ du ). So e t ∆ f can be extended as an entire function on C . e t ∆ µ : L 2 ( R , | u | 2 µ du ) → O ( C ), where O ( C ) is the space of all analytic functions on C . Consider e t ∆ µ ( L 2 ) = � e t ∆ µ f : f ∈ L 2 ( R , | u | 2 µ du ) � . For f , g ∈ L 2 ( R , | u | 2 µ du ), define � e t ∆ µ f , e t ∆ µ g � e t ∆ µ ( L 2 ) := � f , g � L 2 ( R , | u | 2 µ du ) . With respect to above inner product e t ∆ µ ( L 2 ) becomes a Hilbert space and e t ∆ µ : L 2 ( R , | u | 2 µ du ) → e t ∆ µ ( L 2 ) is unitary. Sivaramakrishnan C (IIT Hyderabad) On the Images of Dunkl Sobolev spaces August 26, 2017 10 / 27