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

Images of some subspaces of L2(Rm) 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

6th Fourier Analysis Workshop in Fourier Analysis and Related Area, August 24-31, 2017

Outline

Introduction Preliminaries Grushin and hermite Semigroup Grushin Semigroup Image of

• L2(R) under the heat kernel transform

Image of

• L2(Rn+1) under Grushin semigroup

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 =

• C

|f (z)|2dλ(z) < ∞ where dλ(z) = 1

πe−|z|2dxdy, and the inner-product is defined by

f , g =

• C f (z)g(z)dλ(z).

Definition

Bargmann space or Fock space F is the Hilbert space of entire function on C such that f 2 =

• C

|f (z)|2dλ(z) < ∞ where dλ(z) = 1

πe−|z|2dxdy, and the inner-product is defined by

f , g =

• C f (z)g(z)dλ(z).

Theorem

[1] The Bargmann transform Bf (z) =

• R

f (x)e2xz−x2− 1

2 z2dx

is an isometry from L2(R) onto F.

Outline

Introduction Preliminaries Grushin and hermite Semigroup Grushin Semigroup Image of

• L2(R) under the heat kernel transform

Image of

• L2(Rn+1) under Grushin semigroup

Hermite Sobolev Space with positive order Grushin Sobolev space Of Positive Order Bibliography

Let N0 = N ∪ {0}, and k ∈ N0, then the normalized Hermite function, related to kth degree Hermite polynomial in R is, hk(x) = (2kk!π

1 2 ) −1 2 (−1)k dk

dxk (e−x2)e

x2 2 ,

and n-dimensional normalized Hermite function φα is given by φα(x) =

n

• i=1

hαi(x), where α = (αi)n

i=1 ∈ Nn 0. ◮ Hermite Operator: H = −∆x + x2

Let N0 = N ∪ {0}, and k ∈ N0, then the normalized Hermite function, related to kth degree Hermite polynomial in R is, hk(x) = (2kk!π

1 2 ) −1 2 (−1)k dk

dxk (e−x2)e

x2 2 ,

and n-dimensional normalized Hermite function φα is given by φα(x) =

n

• i=1

hαi(x), where α = (αi)n

i=1 ∈ Nn 0. ◮ Hermite Operator: H = −∆x + x2 ◮ φα’s are eigen vector of H corresponding to the eigen value

(2|α| + n)

Let N0 = N ∪ {0}, and k ∈ N0, then the normalized Hermite function, related to kth degree Hermite polynomial in R is, hk(x) = (2kk!π

1 2 ) −1 2 (−1)k dk

dxk (e−x2)e

x2 2 ,

and n-dimensional normalized Hermite function φα is given by φα(x) =

n

• i=1

hαi(x), where α = (αi)n

i=1 ∈ Nn 0. ◮ Hermite Operator: H = −∆x + x2 ◮ φα’s are eigen vector of H corresponding to the eigen value

(2|α| + n)

◮ Fourier transform Rn: Ff (ξ) = 1 √ 2π

• R f (x)e−iξxdx,

Outline

Introduction Preliminaries Grushin and hermite Semigroup Grushin Semigroup Image of

• L2(R) under the heat kernel transform

Image of

• L2(Rn+1) under Grushin semigroup

Hermite Sobolev Space with positive order Grushin Sobolev space Of Positive Order Bibliography

Grushin Semigroup

◮ Grushin operator on Rn+1:G = −(∆x + |x|2 ∂2 ∂t2 ), ◮ 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 L2(Rn+1)

◮ The solution: u(x, t; s) = e−sGf (x, t).

Grushin Semigroup

◮ Grushin operator on Rn+1:G = −(∆x + |x|2 ∂2 ∂t2 ), ◮ 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 L2(Rn+1)

◮ The solution: u(x, t; s) = e−sGf (x, t).

Grushin Semigroup

◮ Grushin operator on Rn+1:G = −(∆x + |x|2 ∂2 ∂t2 ), ◮ 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 L2(Rn+1)

◮ The solution: u(x, t; s) = e−sGf (x, t).

Grushin Semigroup

◮ Grushin operator on Rn+1:G = −(∆x + |x|2 ∂2 ∂t2 ), ◮ 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 L2(Rn+1)

◮ The solution: u(x, t; s) = e−sGf (x, t). ◮ Fourier transform with respect to last variable Notation:

f λ(x) =

1 √ 2π

• R f (x, t)e−iλtdt

Grushin Semigroup

◮ Grushin operator on Rn+1:G = −(∆x + |x|2 ∂2 ∂t2 ), ◮ 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 L2(Rn+1)

◮ The solution: u(x, t; s) = e−sGf (x, t). ◮ Fourier transform with respect to last variable Notation:

f λ(x) =

1 √ 2π

• R f (x, t)e−iλtdt

◮ Parametrized hermite function: φλ α(x) = |λ|

n 4 φα(

• |λ|x),

◮ λ = 0 then {φλ α} forms an orthonormal basis for L2(Rn) and

Hλφλ

α(x) = (2|α| + n)|λ|φλ α(x). ◮ Parametraized Hermite operator: Hλ = −∆x + |x|2λ2, λ = 0 ◮ Fourier transform on the last variable reduces the Grushin

heat equation to ∂ ∂s uλ = −Hλuλ with initial condition uλ(x; 0) = f λ(x).

◮ With the help of spectral resolution of Hλ we have

e−sHλf (x) =

• α

e−(2|α|+n)|λ|sf , φλ

αL2(Rn)φλ α(x).

=

• Rn K λ

s (x, y)f (y)dy (using Mehlar’s formula).

where K λ

s (x, y)

=(2π)

−n 2

• |λ|

sinh(2s|λ|)

n

2 e −|λ| 2

(x2+y2) coth(2s|λ|)e

|λ|xy sinh(2|λ|s) .

◮

e−sGf (x, t) = u(x, t; s) = 1 √ 2π

• R

uλ(x; s)eiλtdλ. = 1 √ 2π

• R

eiλt

• Rn K λ

s (x, y)f λ(y)dydλ ◮ e−sGf (x, t) can be extended to a holomorphic function in

both the variables.

◮ e−sG :

• L2(Rn+1) −

→ O(Cn+1), where O(Cn+1) is the vector space of holomorphic functions on Cn+1.

Let us consider the Hilbert space

• L2(Rn+1) =
• f ∈ L2(Rn+1) :
• Rn+1|f λ(x)|2eλ2dxdλ < ∞
• ,

where the inner product is f , g =

• Rn+1 f λ(x)gλ(x)eλ2dxdλ, We

wish to find a positive weight function Ws(z, w) where z ∈ Cn and w ∈ C such that,

• Cn+1|e−sGf (z, w)|2Ws(z, w)dzdw = f 2
• L2(Rn+1).

(2.2)

◮ we note that if u(x, t; s) is solution of (??) with initial

condition from

• L2(Rn+1), then for each x ∈ Rn, u(x, · ; ·) is

solution to the 1-dimensional heat equation with initial condition from the following space,

• L2(R) =
• f ∈ L2(R) :
• R

|Ff (λ)|2eλ2dλ

• ,

(2.3)

◮ i.e, for each x, u(x, t; s) = hs(f )(t) with f ∈

L2(R), where hs is the heat kernel transform So we will find first the image of hs in the following sub-section,

Outline

Introduction Preliminaries Grushin and hermite Semigroup Grushin Semigroup Image of

• L2(R) under the heat kernel transform

Image of

• L2(Rn+1) under Grushin semigroup

Hermite Sobolev Space with positive order Grushin Sobolev space Of Positive Order Bibliography

Let qt(x) = 1 √ 4πt e− x2

4t .

For f ∈ L2(R), f ∗ qt(x) has holomorphic extension on C, where f ∗ qt(x) =

• R f (y)qt(y − x)dy. The heat kernel tranform

ht : L2(R) − → O(C) such that f − → f ∗ qt, is one to one.

Lemma

ht( L2(R)) is reproducing kernel Hilbert space with kernel Kz(w) = Cte− z2+w2

4(2t+1) + zw 2(2t+1)

where Ct =

1

√

2(2t+1) is a constant depending on t.

Proposition

For F ∈ ht( L2(R)), |F(x + iy)| ≤ √CtFht(

L2(R))e

y2 2(2t+1) .

Proposition

For F ∈ ht( L2(R)), |F(x + iy)| ≤ √CtFht(

L2(R))e

y2 2(2t+1) .

◮ This gives a growth condition for the elements of ht(

L2(R))

Proposition

For F ∈ ht( L2(R)), |F(x + iy)| ≤ √CtFht(

L2(R))e

y2 2(2t+1) .

◮ This gives a growth condition for the elements of ht(

L2(R))

◮ Let us consider the following Hilbert Space of holomorphic

functions B2t+1(C) =

• f : C

holomorphic

− → C : f 2 =

• 4π

2t+1

• C|f (z)|2e

−y2 2t+1 dxdy < ∞

• .

Proposition

Let em(z) = zme

−z2 4(2t+1) , then the set A = {em : m = 0, 1, 2, · · · } is

a complete orthogonal set in B2t+1(C). Moreover A ⊂ B2t+1(C) ∩ ht( L2(R)).

Theorem

The image of L2(R) under heat kernel transform is the space of holomorphic functions B2t+1(C). Let’s define L2

+(R) = {f ∈

L2(R) : Ff (λ) = 0 when λ ≤ 0} and

• L2

−(R) = {f ∈

L2(R) : Ff (λ) = 0 when λ ≥ 0}, then since

• L2(R) =

L2

+(R) ⊕

L2

−(R) we can observe that,

B2t+1(C) := ht( L2

+(R)) ⊕ ht(

L2

−(R)) = B+ 2t+1(C) ⊕ B− 2t+1(C).

Theorem

[11] Let s > 0, λ = 0 and Uλ

s (z) = 4n

• |λ|

sinh 4s|λ| n

2

e|λ|(tanh(2s|λ|)x2−coth(2|λ|s)y2). Then the semigroup e−sHλ is isometric isomorphism from L2(Rn) to Hλ

s (Cn) where, Hλ s (Cn) is the space of holomorphic functions

• n Cn such that, F2 =
• Cn|F(z)|2Uλ

s (z)dz < ∞.

Lemma

Ws(z, w) satisfying the equation

• Cn+1|e−sGf (z, w)|2Ws(z, w)dzdw = f 2
• L2(Rn+1).

(3.1) with w = ξ + iη is independent of ξ.

Lemma

For each λ0 = 0, the non-negative weight function in (3.1) satisfies the following, Uλ0

s (z) =

• R

e−2ηλ0Ws(z, iη)e−λ2

0dη

(3.2) where Uλ0

s (z) is the weight of λ0-parametrized Hermite Bergman

space.

Theorem

There is no non-negative weight function Ws(z, w) satisfying (3.1).

◮ So it’s not possible to characterize the image of Grushin

semigroup as non-negative weighted Bergman space with isometry.

◮ Now we take λ > 0 and consider the following formal

expression, W +

s (z, w) =

• R

e(λ+ is

2 )2U

λ+ is

2

s

(z)e2η(λ+ is

2 )ds.

(3.3)

Proposition

The function W +

s (z, ξ + iη) defined in the above equation (3.3) is

well defined everywhere, independent of choice of ξ and having following properties,

• 1. W +

s

is independent of the choice of λ, in fact W +

s (z, ξ + iη) = lim λ→0+

• R

e(λ+ is

2 )2U

λ+ is

2

s

(z)e2η(λ+ is

2 )ds (3.4)

• 2. Let a > 0 and Q ⊆ Cn be compact set. Then there exist a

constant C = C(Q, a) > 0 such that for all ǫ ∈ [a−1, a] and ξ ∈ R we have, sup

z∈Q

• R

|e2ǫηW +

s (z, ξ + iη)|dη < C.

(3.5)

• 3. W +

s

satisfies (3.2) with λ > 0, i.e. Uλ

s (z) =

• R

e−2ηλW +

s (z, iη)e−λ2dη.

(3.6)

Let’s introduce the following subspaces of

• L2(Rn+1),
• L2

+(Rn+1) = {f ∈

• L2(Rn+1) : f λ = 0, when λ ≤ 0} and
• L2

−(Rn+1) = {f ∈

• L2(Rn+1) : f λ = 0, when λ ≥ 0}.

Then we have the decomposition,

• L2(Rn+1) =
• L2

+(Rn+1) ⊕

• L2

−(Rn+1).

Let R > 0, BR denotes the ball in Cn of radius R, centered at 0, and define KR = BR × C, then ∪R>0KR = Cn+1. Now we define the vector space V+

s (Cn+1) containing all

holomorphic functions F on Cn+1 such that the followings hold,

• 1. F|KR ∈ L2(KR, |W +

s (z, w)|dzdw),

• 2. limR→∞
• KR|F(z, w)|2W +

s (z, w)dzdw < ∞, and

• 3. F(z, ·) ∈ B+

2s+1(C).

And define the sesquilinear form on V+

s (Cn+1) by,

F, G+ = lim

R→∞

• KR

F(z, w)G(z, w)W +

s (z, w)dzdw,

(3.7) where F, G ∈ V+

s (Cn+1).

Lemma

The sesquilinear form ·, ·+ induces an innerproduct on V+

s (Cn+1).

Theorem

Let s > 0, B+

s (Cn+1) := e−sG(

• L2

+(Rn+1)) is the Hilbert

completion of V+

s (Cn+1) with respect to the inner product , +

defined in (3.7). In fact, B+

s (Cn+1) is isometrically isomorphic to

the completion of V+

s (Cn+1).

On the similar line of (3.3) we can define W −

s (z, w) when λ < 0,

then a proposition like (3.4) is valid, and as well V−

s (Cn+1) can be

constructed as above, consequently B−

s (Cn+1) = e−sG(

• L2

−(Rn+1))

is Hilbert completion of V−

s (Cn+1) will be followed. And hence we

have the direct sum decomposition e−sG( L2(Rn+1)) = B+

s (Cn+1) ⊕ B− s (Cn+1).

(3.8)

Hermite-Sobolev space of positive order

Definition

For µ > 0 the Hermite Soblolev space W µ,2

H (Rn) is defined as the

image of L2(Rn) under H−µ. That is f ∈ W µ,2

H (Rn) if and only if

f W µ,2

H

:=

• α

(2|α| + n)2µ|f , φα|2 < ∞.

◮ W µ,2 H (Rn) forms a Hilbert space under the inner product

f , g :=

• α

(2|α| + n)2µf , φαg, φα.

◮ Define holomorphic Sobolev space

W µ,2

t

(Cn) := e−tH(W µ,2

H (Rn)), t > 0.

this is a Hilbert space with the inner product, F, GW µ,2

t

:= f , gW µ,2

H

, where e−tHf = F and e−tHg = G.

◮ Clearly e−tH is an isometric isomorphism from W µ,2 H (Rn) to

the holomorphic space W µ,2

t

(Cn).

◮ Let us define Fµ t , vector space of holomorphic functions F on

Cn such that,

• Cn|F(z)|2

t (t − s)m−2µ−1

• dm

dsm (Us(z))

• dsdz < ∞.

◮ We use following Caputo fractional derivative

C∗Dα

t f (t) =

1 Γ(m − α) t f (m)(x) (t − x)α+1−m dx, where m is the positive integer with m − 1 < α < m and f (m) is the ordinary derivative of order m.

Image of Hermite Sobolev space under e−tH

Lemma

Fµ

t is a pre-Hilbert space with respect to the sesquilinear form,

F, G =

• Cn F(z)G(z)C∗D2µ

t (Ut(z))dz. Moreover there exist

constants C1, C2 > 0 such that, C1 FW µ,2

t

≤ FFµ

t ≤ C2 FW µ,2 t

, (4.1) when F = e−tHf with f in Schwartz space.

Theorem

The holomorphic Sobolev space W µ,2

t

(Cn) can be identified as the completion of Fµ

t (Cn).

Grushin Sobolev space Of Positive Order

Definition

For s > 0, we define Grushin Sobolev space W µ,2

G (Rn+1) as

sub-space of

• L2(Rn+1) such that an element f satisfies the

following condition, ∞

−∞

• α

(2|α| + n)2µ|λ|2µ|f λ, φλ

α|2eλ2dλ < ∞.

(5.1)

◮ We denote

W µ,2

G+(Rn+1) as the Hilbert space of all those

f ∈ W µ,2

G (Rn+1) such that,

∞

• α(2|α| + n)2µ|λ|2µ|f λ, φλ

α|2eλ2dλ < ∞,

The space W µ,2

s,+(Cn+1) := e−sG(

W µ,2

G,+(Rn+1)) is made into a

Hilbert space by defining the innerproduct in following way, F, G

W µ,2

s,+ (Cn+1)

= f , g

W µ,2

G,+(Rn+1)

= ∞

• α

(2|α| + n)2µ|λ|2µf λ, φλ

αφλ α, gλeλ2dλ,

where F = e−sGf and G = e−sGg. For λ > 0, let us consider the weight function W +,µ

s

(z, w) =

• R

e

• λ+ is

2

2 C∗D2µ

s

• U

λ+ is

2

s

(z)

• e2η
• λ+ is

2

• ds. (5.2)

Then this weight obeys all properties similar to the proposition (3.4). Define the vector space V+,µ

s

(Cn+1) of holomorphic functions F on Cn+1 such that

• 1. For all R > 0,
• KR
• R

s |F(z, w)|2|e(λ+ is

2 )2 dm

dθm U

(λ+ is

2 )

θ

(z)e2η(λ+ is

2 )|dsdθdzdw < ∞

(5.3)

• 2. limR→∞
• KR|F(z, w)|2W +,µ

s

(z, w)dzdw < ∞

• 3. F(z, ·) ∈ B2s+1(C) for all z ∈ Cn.

And the sesquilinear form in V+,µ

s

(Cn+1), F, G+,µ = lim

R→∞

• KR

F(z, w)g(z, w)W +,µ

s

(z, w)dzdw.

Proposition

The above sesqulinear form is an innerproduct and there exist positive constants C1 and C2 such that, C1F

W µ,2

s,+ (Cn+1) ≤ FV+,µ s

(Cn+1) ≤ C2F W µ,2

s,+ (Cn+1),

(5.4) where F = e−Gf , f is in suitable dense class of W µ,2

G,+(Rn+1).

Proposition

• W µ,2

s,+(Cn+1) is the Hilbert completion of V+,µ s

(Cn+1). When λ < 0, considering the weight function W −,µ

s

(z, w), similar way as above, we can construct V−,µ

s

(Cn+1) and conclude that

• W µ,2

s,−(Cn+1) is Hilbert completion of V−,µ s

(Cn+1).

Theorem

The image of Grushin Sobolev space under e−sG is identified as direct-sum of two weighted Bergman space. That is,

• W µ,2

s

(Cn+1) = W µ,2

s,+(Cn+1) ⊕

W µ,2

s,−(Cn+1).

Theorem

The image of Grushin Sobolev space under e−sG is identified as direct-sum of two weighted Bergman space. That is,

• W µ,2

s

(Cn+1) = W µ,2

s,+(Cn+1) ⊕

W µ,2

s,−(Cn+1).

References I

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