Images of some subspaces of L 2 ( R m ) under Grushin and Hermite - PowerPoint PPT Presentation

Images of some subspaces of L 2 ( R m ) under Grushin and Hermite semigroup Partha Sarathi Patra (This is a joint work with Dr. D Venku Naidu) Department of Mathematics Indian Institute of Technology Hyderabad, India 6 th Fourier Analysis Workshop in Fourier Analysis and Related Area, August 24-31, 2017

Outline Introduction Preliminaries Grushin and hermite Semigroup Grushin Semigroup � L 2 ( R ) under the heat kernel transform Image of � L 2 ( R n +1 ) under Grushin semigroup Image of Hermite Sobolev Space with positive order Grushin Sobolev space Of Positive Order Bibliography

Definition Bargmann space or Fock space F is the Hilbert space of entire function on C such that � � f � 2 = | f ( z ) | 2 d λ ( z ) < ∞ C π e −| z | 2 dxdy , and the inner-product is defined by where d λ ( z ) = 1 � � f , g � = C f ( z ) g ( z ) d λ ( z ) .

Definition Bargmann space or Fock space F is the Hilbert space of entire function on C such that � � f � 2 = | f ( z ) | 2 d λ ( z ) < ∞ C π e −| z | 2 dxdy , and the inner-product is defined by where d λ ( z ) = 1 � � f , g � = C f ( z ) g ( z ) d λ ( z ) . Theorem [1] The Bargmann transform � f ( x ) e 2 xz − x 2 − 1 2 z 2 dx B f ( z ) = R is an isometry from L 2 ( R ) onto F .

Outline Introduction Preliminaries Grushin and hermite Semigroup Grushin Semigroup � L 2 ( R ) under the heat kernel transform Image of � L 2 ( R n +1 ) under Grushin semigroup Image of Hermite Sobolev Space with positive order Grushin Sobolev space Of Positive Order Bibliography

Let N 0 = N ∪ { 0 } , and k ∈ N 0 , then the normalized Hermite function, related to k th degree Hermite polynomial in R is, 2 ( − 1) k d k x 2 1 − 1 dx k ( e − x 2 ) e h k ( x ) = (2 k k ! π 2 ) 2 , and n -dimensional normalized Hermite function φ α is given by n � h α i ( x ) , where α = ( α i ) n i =1 ∈ N n φ α ( x ) = 0 . i =1 ◮ Hermite Operator: H = − ∆ x + x 2

Let N 0 = N ∪ { 0 } , and k ∈ N 0 , then the normalized Hermite function, related to k th degree Hermite polynomial in R is, 2 ( − 1) k d k x 2 1 − 1 dx k ( e − x 2 ) e h k ( x ) = (2 k k ! π 2 ) 2 , and n -dimensional normalized Hermite function φ α is given by n � h α i ( x ) , where α = ( α i ) n i =1 ∈ N n φ α ( x ) = 0 . i =1 ◮ Hermite Operator: H = − ∆ x + x 2 ◮ φ α ’s are eigen vector of H corresponding to the eigen value (2 | α | + n )

Let N 0 = N ∪ { 0 } , and k ∈ N 0 , then the normalized Hermite function, related to k th degree Hermite polynomial in R is, 2 ( − 1) k d k x 2 1 − 1 dx k ( e − x 2 ) e h k ( x ) = (2 k k ! π 2 ) 2 , and n -dimensional normalized Hermite function φ α is given by n � h α i ( x ) , where α = ( α i ) n i =1 ∈ N n φ α ( x ) = 0 . i =1 ◮ Hermite Operator: H = − ∆ x + x 2 ◮ φ α ’s are eigen vector of H corresponding to the eigen value (2 | α | + n ) � ◮ Fourier transform R n : F f ( ξ ) = 1 R f ( x ) e − i ξ x dx , √ 2 π

Outline Introduction Preliminaries Grushin and hermite Semigroup Grushin Semigroup � L 2 ( R ) under the heat kernel transform Image of � L 2 ( R n +1 ) under Grushin semigroup Image of Hermite Sobolev Space with positive order Grushin Sobolev space Of Positive Order Bibliography

Grushin Semigroup ◮ Grushin operator on R n +1 : G = − (∆ x + | x | 2 ∂ 2 ∂ t 2 ) , ◮ Heat Equation Corresponding to this Grushin operator: ∂ ∂ s u ( x , t ; s ) = − Gu ( x , t ; s ) , (2.1) with the initial condition u ( x , t , 0) = f ( x , t ) where f is a function in L 2 ( R n +1 ) ◮ The solution: u ( x , t ; s ) = e − sG f ( x , t ) .

Grushin Semigroup ◮ Grushin operator on R n +1 : G = − (∆ x + | x | 2 ∂ 2 ∂ t 2 ) , ◮ Heat Equation Corresponding to this Grushin operator: ∂ ∂ s u ( x , t ; s ) = − Gu ( x , t ; s ) , (2.1) with the initial condition u ( x , t , 0) = f ( x , t ) where f is a function in L 2 ( R n +1 ) ◮ The solution: u ( x , t ; s ) = e − sG f ( x , t ) .

Grushin Semigroup ◮ Grushin operator on R n +1 : G = − (∆ x + | x | 2 ∂ 2 ∂ t 2 ) , ◮ Heat Equation Corresponding to this Grushin operator: ∂ ∂ s u ( x , t ; s ) = − Gu ( x , t ; s ) , (2.1) with the initial condition u ( x , t , 0) = f ( x , t ) where f is a function in L 2 ( R n +1 ) ◮ The solution: u ( x , t ; s ) = e − sG f ( x , t ) .

Grushin Semigroup ◮ Grushin operator on R n +1 : G = − (∆ x + | x | 2 ∂ 2 ∂ t 2 ) , ◮ Heat Equation Corresponding to this Grushin operator: ∂ ∂ s u ( x , t ; s ) = − Gu ( x , t ; s ) , (2.1) with the initial condition u ( x , t , 0) = f ( x , t ) where f is a function in L 2 ( R n +1 ) ◮ The solution: u ( x , t ; s ) = e − sG f ( x , t ) . ◮ Fourier transform with respect to last variable Notation: � 1 f λ ( x ) = R f ( x , t ) e − i λ t dt √ 2 π

Grushin Semigroup ◮ Grushin operator on R n +1 : G = − (∆ x + | x | 2 ∂ 2 ∂ t 2 ) , ◮ Heat Equation Corresponding to this Grushin operator: ∂ ∂ s u ( x , t ; s ) = − Gu ( x , t ; s ) , (2.1) with the initial condition u ( x , t , 0) = f ( x , t ) where f is a function in L 2 ( R n +1 ) ◮ The solution: u ( x , t ; s ) = e − sG f ( x , t ) . ◮ Fourier transform with respect to last variable Notation: � 1 f λ ( x ) = R f ( x , t ) e − i λ t dt √ 2 π � n ◮ Parametrized hermite function: φ λ 4 φ α ( α ( x ) = | λ | | λ | x ) ,

◮ λ � = 0 then { φ λ α } forms an orthonormal basis for L 2 ( R n ) and H λ φ λ α ( x ) = (2 | α | + n ) | λ | φ λ α ( x ) . ◮ Parametraized Hermite operator: H λ = − ∆ x + | x | 2 λ 2 , λ � = 0 ◮ Fourier transform on the last variable reduces the Grushin heat equation to ∂ ∂ s u λ = − H λ u λ with initial condition u λ ( x ; 0) = f λ ( x ) .

◮ With the help of spectral resolution of H λ we have � e − sH λ f ( x ) = e − (2 | α | + n ) | λ | s � f , φ λ α � L 2 ( R n ) φ λ α ( x ) . α � R n K λ = s ( x , y ) f ( y ) dy (using Mehlar’s formula). where K λ s ( x , y ) � � n 2 e | λ | xy −| λ | − n | λ | ( x 2 + y 2 ) coth(2 s | λ | ) e sinh(2 | λ | s ) . =(2 π ) 2 2 sinh(2 s | λ | )

◮ e − sG f ( x , t ) = u ( x , t ; s ) � 1 u λ ( x ; s ) e i λ t d λ. √ = 2 π R � � 1 e i λ t R n K λ s ( x , y ) f λ ( y ) dyd λ √ = 2 π R ◮ e − sG f ( x , t ) can be extended to a holomorphic function in both the variables. � ◮ e − sG : L 2 ( R n +1 ) − → O ( C n +1 ) , where O ( C n +1 ) is the vector space of holomorphic functions on C n +1 .

Let us consider the Hilbert space � � � � R n +1 | f λ ( x ) | 2 e λ 2 dxd λ < ∞ L 2 ( R n +1 ) = f ∈ L 2 ( R n +1 ) : , � R n +1 f λ ( x ) g λ ( x ) e λ 2 dxd λ, We where the inner product is � f , g � = wish to find a positive weight function W s ( z , w ) where z ∈ C n and w ∈ C such that, � C n +1 | e − sG f ( z , w ) | 2 W s ( z , w ) dzdw = � f � 2 (2.2) L 2 ( R n +1 ) . �

◮ we note that if u ( x , t ; s ) is solution of ( ?? ) with initial � L 2 ( R n +1 ), then for each x ∈ R n , u ( x , · ; · ) is condition from solution to the 1-dimensional heat equation with initial condition from the following space, � � � � |F f ( λ ) | 2 e λ 2 d λ f ∈ L 2 ( R ) : L 2 ( R ) = (2.3) , R ◮ i.e, for each x , u ( x , t ; s ) = h s ( f )( t ) with f ∈ � L 2 ( R ) , where h s is the heat kernel transform So we will find first the image of h s in the following sub-section,

Outline Introduction Preliminaries Grushin and hermite Semigroup Grushin Semigroup � L 2 ( R ) under the heat kernel transform Image of � L 2 ( R n +1 ) under Grushin semigroup Image of Hermite Sobolev Space with positive order Grushin Sobolev space Of Positive Order Bibliography

Let 1 e − x 2 4 t . q t ( x ) = √ 4 π t For f ∈ � L 2 ( R ) , f ∗ q t ( x ) has holomorphic extension on C , where � f ∗ q t ( x ) = R f ( y ) q t ( y − x ) dy . The heat kernel tranform h t : � L 2 ( R ) − → O ( C ) such that f �− → f ∗ q t , is one to one. Lemma h t ( � L 2 ( R )) is reproducing kernel Hilbert space with kernel K z ( w ) = C t e − z 2+ w 2 zw 4(2 t +1) + 2(2 t +1) 1 √ where C t = 2(2 t +1) is a constant depending on t.

Proposition L 2 ( R )) , | F ( x + iy ) | ≤ √ C t � F � h t ( � y 2 For F ∈ h t ( � 2(2 t +1) . L 2 ( R )) e

Proposition L 2 ( R )) , | F ( x + iy ) | ≤ √ C t � F � h t ( � y 2 For F ∈ h t ( � 2(2 t +1) . L 2 ( R )) e ◮ This gives a growth condition for the elements of h t ( � L 2 ( R ))

Proposition L 2 ( R )) , | F ( x + iy ) | ≤ √ C t � F � h t ( � y 2 For F ∈ h t ( � 2(2 t +1) . L 2 ( R )) e ◮ This gives a growth condition for the elements of h t ( � L 2 ( R )) ◮ Let us consider the following Hilbert Space of holomorphic functions B 2 t +1 ( C ) � � � � − y 2 holomorphic C : � f � 2 = 4 π f : C C | f ( z ) | 2 e 2 t +1 dxdy < ∞ = − → . 2 t +1 Proposition − z 2 4(2 t +1) , then the set A = { e m : m = 0 , 1 , 2 , · · · } is Let e m ( z ) = z m e a complete orthogonal set in B 2 t +1 ( C ) . Moreover A ⊂ B 2 t +1 ( C ) ∩ h t ( � L 2 ( R )) .