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Advances in Pure and Applied Mathematics

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On the images of Sobolev spaces under the Schrödinger semigroup

Sivaramakrishnan C / Sukumar D / Venku Naidu Dogga
  • Corresponding author
  • Department of Mathematics, Indian Institute of Technology Hyderabad, Telangana – 502285, India
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Published Online: 2018-01-20 | DOI: https://doi.org/10.1515/apam-2016-0116

Abstract

In this article, we consider the Schrödinger semigroup for the Laplacian Δ on n, and characterize the image of a Sobolev space in L2(n,eu2du) under this semigroup as weighted Bergman space (up to equivalence of norms). Also we have a similar characterization for Hermite Sobolev spaces under the Schrödinger semigroup associated to the Hermite operator H on n.

Keywords: Bargmann transform; Schrödinger semigroup; weighted Bergman space; Sobolev space, holomorphic Sobolev space

MSC 2010: 46E35; 46F12; 47D06; 35B65; 35J10

1 Introduction

It is well known that the classical Bargmann transform is an isometric isomorphism of L2(n) onto the space of square integrable holomorphic functions 2(n,e-|z|2dz); we refer to [1, 3] for further details. In [4], Hall generalized the concept of the Bargmann transform for the general compact Lie group K. He called this transform a heat-kernel transform. Also Hall characterized the image of L2(K) under the heat-kernel transform as a weighted Bergman space. Later, the topic became popular and attracted many mathematicians; see for instance [8, 7]. For the heat kernel transform associated to the Hermite and special Hermite semigroups, we refer to [2, 8]. The same kind of results are known for Sobolev spaces; we refer to [5, 10, 11].

In [6], Hayashi and Saitoh considered the Schrödinger semigroup for the Laplacian Δ on and showed that eitΔf can be extended as an entire function for fL2(,eu2du). Also they characterized the range of eitΔ as a weighted Bergman space. In [9], Parui et al. generalized the result of Huyashi and Saitoh to L2(n,eu2du) and gave a completely different and simple proof for the image characterization with the help of the classical Bargmann transform. The Schrödinger equation associated to the Laplacian on n is given by

iut=-Δu,u(x,0)=f(x)L2(n).

The solution of the Schrödinger equation is given by the unitary group eitΔ, which is unitary on L2(n). For a non-zero real number t, on a dense subspace the operator eitΔ is given by

eitΔf(x)=ctnnf(u)ei4t(x-u)2𝑑u.

Here the constant ct is given by ct=(4πit)-1/2. Since eitΔ is unitary, eitΔf cannot be extended as an entire function for all fL2(n). Further, if we assume enough decay on f, then we can extend eitΔf as an entire function on n. The following theorem is one in this direction and it also gives the image characterization.

Theorem 1.1 ([9]).

For a non-zero real number t and fL2(Rn,ev2dv), the function eitΔ(f) can be extended as an analytic function. Moreover,

n|eitΔ(f)(x+iy)|2exyte-y24t2𝑑x𝑑y=(2πt)nn|f(v)|2ev2𝑑v.

Conversely, if

F2(n,exyt-y24t2dxdy),

then there exists fL2(Rn,ev2dv) such that eitΔ(f)=F.

Remark 1.2.

From Theorem 1.1 we have the following:

  • (i)

    eitΔ:L2(n,ev2dv)2(n,wt(x+iy)dxdy) is unitary, where

    wt(x+iy)=1(2πt)nexyt-y24t2.

  • (ii)

    Given any F2(n,wt(x+iy)dxdy), there exists a non-negative constant c (depends only on t,n,F) such that

    |F(z)|ce-xy2tey28t2for all z=x+iyn.(1.1)

In this article, we wish to characterize the images of Sobolev spaces under the Schrödinger semigroup associated to the Laplacian Δ and the Hermite operator H on n as a weighted Bergman space (up to equivalence of norms).

We have organized the article as follows: Section 2 contains preliminaries. In Section 3, we will give some results on Sobolev spaces, which will help us to prove the main results. We present our main result associated to the Laplacian in Section 4. In Section 5, we will characterize the images of Hermite Sobolev spaces under the Schrödinger semigroup associated to the Hermite operator (up to equivalence of norms).

2 Preliminaries

We begin with the definition of Hermite polynomials Hk(u), where k is a non-negative integer and u. These are defined by

Hk(u)=(-1)keu2dkduk(e-u2).

By setting

ϕk(u)=(2kk!π12)-12Hk(u)e-u22

for k=0,1,2,, the family {ϕk:k=0,1,2,} forms an orthonormal basis for L2(). It is easy to see that the ϕk’s satisfy the following relations:

dduϕk(u)=(k2)12ϕk-1(u)-(k+12)12ϕk+1(u),(2.1)uϕk(u)=(k+12)12ϕk+1(u)+(k2)12ϕk-1(u).(2.2)

Definition 2.1.

For s>0 the scaled Hermite polynomial is defined by

Hks(u)=(-1)kdkduk(e-su2)esu2

for u and k.

So by definition, Hks(u) is a polynomial in u of degree k. Let us write Hks(u)=r=0kar(k),sur. From the definition it is clear that in Hks(u) the only even powers of u occur if k is even, and only odd powers of u occur if k is odd. The following proposition gives the coefficients of Hks.

Proposition 2.2.

The coefficient ar(k),s of ur in the scaled Hermite polynomial Hks is given by the relation

ak-2p(k),s=(2s)k-2psp(-1)pk!(k-2p)!p!for p{{0,1,,k2}if k is even,{0,1,,k-12}if k is odd.

Proof.

This follows from the Rodrigues formula for Hermite polynomials. ∎

Let us introduce some multi-index notations:

  • n=×× (n times), where :={0,1,2,}.

  • For k{1,2,,n}, define

    ek:=(0,,0,1k-th place,0,,0)n.

  • For α=(α1,α2,,αn)n, define |α|:=α1++αn.

  • α!:=α1!αn!.

  • For α,βn, we say αβ if αkβk for k=1,2,,n.

  • zα:=z1α1znαn and z2:=z12++zn2 for z=(z1,,zn)n and αn.

  • zw:=z1w1+z2w2++znwn for z=(z1,,zn),w=(w1,,wn)n.

  • Dα:=Dα1Dα2Dαn. Here

    Dαk=αkxkαk

    for k=1,2,,n.

Definition 2.3.

Let F:n be a function and let a=(a1,,an)n. Denote

Fa,j(z)=F(a1,,aj-1,z,aj+1,,an)for z.

We say that F is holomorphic if for every an the map Fa,j: is holomorphic for j=1,,n.

Let us denote 𝒪(n)={F:n:F is analytic}. Let ρ(z) be a strictly positive continuous function on n. We define the weighted Bergman space as follows:

ρ2:=2(n,ρ(z)dz)={f𝒪(n):n|f(z)|2ρ(z)𝑑x<}.

The inner product on 2(n,ρ(z)dz) is given by

F,Gρ2:=nF(z)G(z)¯ρ(z)𝑑z.

It is known that 2(n,ρ(z)dz) is the reproducing kernel Hilbert space. The well-known reproducing kernel Hilbert space is a Fock space and it is defined by

r2:=2(n,rπe-r|z|2dxdy)={F𝒪(n):Fr,22<},r>0,

where

Fr,22=(rπ)nn|F(z)|2e-r|z|2𝑑z.

The set

{𝐞αr(z)=(r|α|α!)12zα:αn}

forms an orthonormal basis for r2. The reproducing kernel for r2 is given by Kr(z,w)=erzw¯.

3 Sobolev space and its properties

We define Sobolev spaces in L2(n,eu2du) in the following way.

Definition 3.1.

Let m>0 be an integer. The Sobolev space Wm,2(n) in L2(,eu2du) is defined by

Wm,2(n)={fL2(n,eu2du):DαfL2(n,eu2du) for |α|m}.

The inner product on Wm,2(n) is defined for f,gWm,2(n) by

f,gWm,2(n)=|α|mDαf,DαgL2(n,eu2du).

With respect to the above inner product, Wm,2(n) becomes a Hilbert space. From now onwards, we consider for any f,gL2(n,eu2du) the inner product f,g and the norm f given by

f,g:=f,gL2(n,eu2du)andf:=fL2(n,eu2du),

respectively. Also we assume the following notation: for fWm,2(n), the Sobolev norm of f is denoted by fWm=fWm,2(n).

Note 3.2.

For m1 and fWm,2(), we have

lim|x||Dkf(x)p(x)|=0(3.1)

for any polynomial p on and k with km-1.

This can be seen as follows: First, we consider the case m=1; the general case follows similarly. Using the Sobolev embedding lemma, we have

(f(t)et22)=tf(t)et22+f(t)et22.

Integrate both sides from a to b and use the Schwarz inequality to estimate the integral on the right:

(|f(t)et22|)ab(b3-a33ab|f(t)|2et2𝑑t)12+((b-a)ab|f(t)|2et2𝑑t)12.

From the above equation we can find B~>0 such that

|f(u)|eu22B~(1+|u|)32 for all u.

Lemma 3.3.

The map P:L2(Rn,du)L2(Rn,eu2du) defined by P(f)(u)=f(u)e-u2/2 is unitary.

Proof.

This is obvious. ∎

Let ψk(u)=ϕk(u)e-u2/2 for k. Since {ϕk:k} forms an orthonormal basis for L2(), from Lemma 3.3 we can see that {ψk:k} is an orthonormal basis for L2(,eu2du). The relation between the ψk’s and their derivatives is as follows: Using equations (2.1) and (2.2), we have

dduψk(u)=dduϕk(u)e-u22-uϕk(u)e-u22={(k2)12ϕk-1(u)-(k+12)12ϕk+1(u)}e-u22-{(k2)12ϕk-1(u)+(k+12)12ϕk+1(u)}e-u22=-2(k+12)12ϕk+1(u)e-u22=-2(k+12)12ψk+1(u).

In general, for r,

drdurψk(u)=(-2)r(k+12)12(k+r2)12ψk+r(u).(3.2)

Also for k,

uψk=(k+12)12ψk+1+(k2)12ψk-1.(3.3)

For α=(α1,α2,,αn)n and u=(u1,u2,,un)n, we define

Ψα(u)=k=1nψαk(uk).

Then for any βn,

DβΨα(u)=(-2)|β|k=1np=1βk(αk+p2)12Ψα+β(u).

Thus we have the following theorem.

Theorem 3.4 ([12]).

The set {Ψα:αNn} forms an orthonormal basis for L2(Rn,eu2du).

Hence given any fL2(n,eu2du), we have

f=αnf,ΨαΨα=k=0|α|=kf,ΨαΨα.

Lemma 3.5.

If fWm,2(Rn), then Dβf=αNnf,ΨαDβΨα for all |β|m.

Proof.

For k,x and by using equations (3.2) and (3.3), we have

D(ψk(x)ex2)=(D(ψk)(x)+2xψk(x))ex2=((-2)(k+12)12ψk+1(x)+2(k+12)12ψk+1(x)+2(k2)12ψk-1(x))ex2=2(k2)12ψk-1(x)ex2.

For each r, inductively we can see that

Dr(ψk(x)ex2)=Dr-1(D(ψk)(x)ex2)+Dr-1(2xψk(x)ex2)=2(k2)12Dr-1(ψk-1(x)ex2)=2r(k2)12(k-(r-1)2)12ψk-r(x)ex2.(3.4)

Let βn with |β|m, fWm,2(n) and i{1,2,,n}. Then from equations (3.1) and (3.4) we have

Dβieif,Ψα=nDβieif(u)Ψα(u)eu2𝑑u=(-1)βinf(u)Dβiei(Ψα(u)eu2)𝑑u=(-1)βinf(u){jiψαj(uj)euj2}Dβiei(ψαi(ui)eui2)𝑑u=(-2)βip=1βi(αi-(p-1)2)12f,Ψα-βiei.

Hence,

Dβieif=αnDβieif,ΨαΨα=αn(-2)βip=1βi(αi-(p-1)2)12f,Ψα-βieiΨα=αn(-2)βip=1βi(αi+p2)12f,ΨαΨα+βi=αnf,ΨαDβieiΨα.

If we apply the above step repeatedly, we have

Dβf=αnf,ΨαDβΨα.

With the help of Lemma 3.5 we can see an important result which does not hold for classical Sobolev spaces.

Lemma 3.6.

If fW1,2(R), then ufL2(R,eu2du).

Proof.

Let fW1,2(). From Lemma 3.5 we can see that

dduf=k=0f,ψkdduψk=k=0-2(k+12)12f,ψkψk+1.(3.5)

Now let us consider the N-th partial sum of k=0f,ψkψk, that is,

k=0Nf,ψkuψk=k=1N(k2)12f,ψkψk-1+k=0n(k+12)12f,ψkψk+1.

For N, let us define

ANf=k=1N(k2)12f,ψkψk-1andBNf=k=0N(k+12)12f,ψkψk+1.

Since fW1,2() and by equation (3.5), ANf converges to

k=1(k2)12f,ψkψk-1,

and BNf converges to

k=1(k+12)12f,ψkψk+1

as N in L2(,eu2du). So k=0f,ψkuψk converges in L2(,eu2du). Since f=k=0f,ψkψk converges in L2(,eu2du), there is a subsequence, say k=0Nkf,ψkψk converges to f pointwise a.e. on . This gives us

limkk=0Nkf,ψkuψk(u)=limkuk-0Nkf,ψkψk(u)=uf(u)a.e. on .

This shows that uf=k=0f,ψkuψk and ufL2(,eu2du) for fW1,2(). ∎

The above result can be extended to higher dimensions, and the result is as follows.

Theorem 3.7.

If mN, αNn with |α|m, and fWm,2(Rn), then uαfWm-|α|,2(Rn).

The orthonormal basis {Ψα:αn} for L2(n,eu2du) is also an orthonormal basis for Wm,2(n) after multiplying each function with an appropriate normalizing constant.

Theorem 3.8.

The set {Ψα/ΨαWm:αNn} forms a complete orthonormal system in Wm,2(Rn).

Proof.

Let αβ. Then

Ψα,ΨβWm,2(n)=|γ|mDγΨα,DγΨβ=|γ|m(2)2|γ|k=1np=1γk(αk+p2)12(βk+p2)12Ψα+γ,Ψβ+γ=0.

For completeness, let fWm,2(n) such that f,ΨαWm,2(n)=0 for all αn. Then

0=f,ΨαWm,2(n)=|γ|mDγf,DγΨα=|γ|mβnf,ΨβDγΨβ,DγΨα=f,Ψα|γ|m(2)2|γ|k=1np=1γk(αk+p2).

The above relation forces that f,Ψα=0 for all αn. This implies that f=0. Thus the theorem follows. ∎

Note 3.9.

Here for αn,

ΨαWm2=|γ|m22|γ|k=1nq=1γk(αk+q2).

Now let us take

Ψ~α(u)=Ψα(u)e-i4tu2

for αn and un. Set ={Ψ~α:αn} and 𝒜={Ψα:αn}. Observe that 𝒜 and form an orthonormal basis for L2(n,eu2du). The reason to consider the collection is as follows: this collection behaves nicely with the Schrödinger semigroup, and in fact, the Schrödinger semigroup associated to Δ on n takes the collection into some nice functions in 𝒪(n). Those nice functions will help us to characterize the image of a Sobolev space under the Schrödinger semigroup. We will discuss the above ideas in the next section. The set 𝒜 is an orthogonal system in Wm,2(n), but is not an orthogonal system in Wm,2(). For instance, if m=n=1, then

ψ~k(u)=ψk(u)e-i4tu2-i2tuψk(u)e-i4tu2=(-1)(2+i2t)ψ~k+1(u)-i2t(k2)12ψ~k-1(u).

So for kp and k<p,

ψ~k,ψ~pW1,2()=ψ~k,ψ~p+ψ~k,ψ~p={(-1)(2+i2t)(i2t)(k+12)12(k+22)12if p=k+2,0if pk+2.

Remark 3.10.

Let m1 and fWm,2(n). Then we have the following:

  • (i)

    For each k{1,2,,n},

    Dek(f(u)e-i4tu2)={Dekf(u)-i2tukf(u)}e-i4tu2.

    By Lemma 3.6 we have ukf(u)L2(n,eu2du). So f(u)e-i/(4t)u2W1,2(n).

  • (ii)

    Similarly, we can see that f(u)e-i/(4t)u2Wm,2(n).

  • (iii)

    For αβn, we have Dαf2Dβf2.

Using Remark 3.10, we can define the transformation 𝒮:Wm,2(n)Wm,2(n) by 𝒮(f)(u)=f(u)e-i/(4t)u2. Let us take

𝒟n={|α|kaαΨα:aα and k}.

Then, by Theorem 3.8, 𝒟n forms a dense subspace of Wm,2(n).

Theorem 3.11.

The map S is a bounded invertible map.

Proof.

We use mathematical induction on m to prove this theorem. First let m=1 and f𝒟n with

f=|α|paαΨα,

where aα and p.

For k{1,2,,n},

ukf2=|α|paαukΨα2=|α|paα(αk+12)12Ψα+ek+1|α|p(αk2)12aαΨα-ek2{|α|paα(αk+12)12Ψα+ek+1|α|p(αk2)12aαΨα-ek}2={(|α|pαk+12|aα|2)12+(1|α|pαk2|aα|2)12}2{2(|α|pαk+12|aα|2)12}2=Dekf2.

Using the above inequality, we have

𝒮fW12=k=0nDek(f(u)e-i4tu2)2=f2+k=1nDekf(u)e-i4tu2-i2tukf(u)e-i4tu22f2+k=0n(Dekf(u)e-i4tu2+|12t|ukf)2f2+(1+|12t|)2k=1nDekf2B1fW12,

where B1=(1+|12t|)2.

So, 𝒮:W1,2(n)W1,2(n) is a bounded map. Now let m. For every α with |α|=m there exist k(α){1,2,,n} such that αk(α)0. Then observe that for f𝒟n,

uk(α)fWm-12=|β|m-1Dβ(uk(α)f)2=|β|m-1uk(α)Dβ(f)+βk(α)Dβ-ek(α)f2|β|m-1{uk(α)Dβ(f)+βk(α)Dβ-ek(α)f}2|β|m-1{Dβ+ek(α)(f)+βk(α)Dβ+ek(α)f}2m|β|m-1Dβ+ek(α)(f)2mfWm2.(3.6)

Also we can see that

Dα(f(u)e-i4tu2)=Dα-ek(α){(Dek(α)f)(u)e-i4tu2-i2t(uk(α)f(u)e-i4tu2)}=Dα-ek(α){𝒮(Dek(α)f)(u)-i2t𝒮(uk(α)f)(u)}.(3.7)

Using induction on m and equations (3.6) and (3.7), we have

𝒮fWm2=|α|m-1Dα(S(f))2+|α|=mDα(f(u)e-i4tu2)2=S(f)Wm-12+|α|=mDα-ek(α){𝒮(Dek(α)f)-i2t𝒮(uk(α)f)}2S(f)Wm-12+|α|=m{Dα-ek(α)𝒮(Dek(α)f)+12tDα-ek(α)𝒮(uk(α)f)}2S(f)Wm-12+{𝒮(Dek(α)f)Wm-1+12t𝒮(uk(α)f)Wm-1}2Bm-1fWm-12+Bm-1{Dek(α)fWm-1+12tuk(α)fWm-1}2Bm-1fWm-12+Bm-1(1+m12t)2fWm2Bm-1{1+(1+m12t)2}fWm2.

This implies that 𝒮:Wm,2(n)Wm,2(n) is a bounded map. It is clear that 𝒮 is one-to-one and onto from the definition. Hence the theorem is proved. ∎

Note 3.12.

We have just proved that if fWm,2(n), then ukfDekf for k=1,2,,n. In a similar way, it is easy to show that uαfDαf for all |α|m.

Remark 3.13.

Using Theorem 3.11, we can see that for each m the set forms a Schauder basis for Wm,2(n).

4 Image of Sobolev spaces under the Schrödinger semigroup

In this section, we will prove our main result. First we define the holomorphic Sobolev space Wtm,2(n) to be the image of Wm,2(n) under eitΔ. The space Wtm,2(n) is made into a Hilbert space by transferring the Hilbert space structure of Wm,2(n) to Wtm,2(n) so that the Schrödinger semigroup eitΔ is an isometric isomorphism from Wm,2(n) onto Wtm,2(n). This means that

F,GWtm,2(n):=f,gWm,2(n)

whenever F=eitΔf and G=eitΔg. We wish to characterize Wtm,2(n) as a weighted Bergman space. The following theorem gives a relationship between 2(n,wt(x+iy)dxdy) and the classical Fock space.

Theorem 4.1.

For r=18t2, the map

U:2(n,wt(x+iy)dxdy)18t22

defined for FHL2(Cn,wt(z)dz) by

U(F)(z)=(4tπ)n2F(z)e(-i4t+116t2)z2

is a unitary map.

Proof.

This is an obvious verification. ∎

Remark 4.2.

  • (i)

    From Theorem 4.1 we can conclude that the set

    {Υαt(z)=(4tπ)-n2𝐞α18t2(z)e(i4t-116t2)z2:αn}

    forms an orthonormal basis for 2(,wt(x+iy)dxdy).

  • (ii)

    Also for γ,αn,

    zγΥαt(z)=(4t)|γ|j=1nq=1γj(αj+q2)12Υα+γt(z).(4.1)

The following lemma gives the pointwise estimate of eitΔ(f) for fWm,2(n).

Lemma 4.3.

If fWm,2(Rn), then there exists C>0 (depends only on m,n,t and f) such that for any zCn,

|eitΔ(f)(z)|2Ce-xyt+y24t21+|zα|2

for |α|m.

Proof.

For fWm,2(), k{1,2,,n} and zn, consider

eitΔ(Dekf)(z)=ctnnDekf(u)ei4t(z-u)2𝑑u=-ictn2tn(uk-zk)f(u)ei4t(z-u)2𝑑u=ctn{i2tzkeitΔ(f)(z)-i2teitΔ(ukf)(z)}.

From the above equation we obtain

zkeitΔ(f)(z)=2tieitΔ(1ctnDekf+i2tukf)(z).

Now the assumption on f tells us that for |α|m,

(1ctnDe1+i2tu1)α1(1ctnDe1+i2tun)αnfL2(n,eu2du).

This implies that zαeitΔ(f) is in 2(n,wt(z)dz). Moreover, using pointwise estimates of functions in 2(n,wt(z)dz) given in (1.1), there exists a constant (depends only on f,m,n and t) C>0 such that

|eitΔ(f)(z)|2Ce-xyt+y24t21+|zα|2

for all zn. ∎

Let

utm(z)=|γ|m(12t)2|γ||zγ|2wt(z)

and consider the weighted Bergman space 2(n,utm(z)dz). Using pointwise estimate given in Lemma 4.3, we can compare the spaces Wtm,2(n) and 2(n,utm(z)dz).

Lemma 4.4.

The set

Ω={ΥαtΥαtutm2:αn}

forms a complete orthonormal basis for HL2(Cn,utm(z)dz).

Proof.

We know that {Υαt:αn} forms an orthonormal basis for 2(n,wt(z)dz). Using this, we can prove that Ω is a complete orthonormal set in 2(n,utm(z)dz). Since wt(z)utm(z), this implies that 2(n,utm(z)dz)2(,wt(z)dz). Now for αβn and using equation (4.1),

Υαt,Υβtutm2=nΥαt(z)Υβt(z)¯|γ|m(12t)2|γ||zγ|2wt(z)dz=|γ|m(12t)2|γ|nzγΥαt(z)zγΥβt(z)¯wt(z)𝑑z=|γ|m22|γ|j=1nq=1γj(αj+q2)12(βj+q2)12Υα+γt,Υβ+γtwt2=0.

This proves the orthogonality. The completeness of Ω in m,2(n,utm(z)dz) easily follows from the completeness of Ω in m,2(n,wt(z)dz). ∎

The following proposition and theorem show that the norms of the spaces Wtm,2(n) and 2(n,utm(z)dz) are equivalent.

Proposition 4.5.

For pN and xR, we have

eitΔ(upe-u2e-i4tu2)(x)=(2ti)pct(-1)pdpdxp(e-116t2x2)ei4tx2.

Furthermore,

eitΔ(ψk(u)e-i4tu2)(x)=ctπ(12kk!π)12(12ti)kxke(i4t-116t2)x2.

Proof.

For x and s=116t2,

eitΔ(upe-u2-i4tu2)(x)=ctupe-u2-i4tu2ei4t(x-u)2𝑑u=ct(-2ti)pei4tx2e-u2dpdxp(e-i2txu)𝑑u=ct(-2ti)pei4tx2dpdxp(e-u2e-i2txu𝑑u)=π(2ti)pct(-1)pdpdxp(e-116t2x2)ei4tx2=π(2ti)pctHps(x)e-116t2x2+i4tx2.

Now for k,

eitΔ(ψk(u)e-i4tu2)(x)=(12kk!π)12eitΔ(Hk(u)e-u2e-i4tu2)(x).

First assume that k is even. So,

eitΔ(Hk(u)e-u2e-i4tu2)(x)=p=0k2ak-2p(k)eitΔ(uk-2pe-u2e-i4tu2)(x)=ctπp=0k2ak-2p(k)(2ti)k-2pHk-2ps(x)e-116t2x2+i4tx2=ctπ{p=0k2bk-2pxk-2p}e-116t2x2+i4tx2.

Here bk-2p is a coefficient of xk-2p in the above polynomial, and it is given by

bk-2p=r=0pak-2r(k)(2ti)k-2rak-2p(k-2r),s.(4.2)

Claim: bk-2p=0 for all 1p. From Proposition 2.2 we have

ak-2r(k)=2k-2r(-1)rk!(k-2r)!r!=ak(k)2-2r(-1)rk!(k-2r)!r!

and

ak-2p(k-2r),s=(2s)k-2p(s)p-r(-1)p-r(k-2r)!(k-2p)!(p-r)!=ak-2p(k),ss-r(-1)-r(p-r+1)p(k-2r+1)k.

Using the above equations, we can see that equation (4.2) is equal to zero. In fact,

bk-2p=ak(k)(2ti)k{r=0p(2ti)-2r(116t2)-r2-2r(p-r+1)pr!}ak-2p(k),s=ak(k)(2ti)k{r=0pi2r(p-r+1)pr!}ak-2p(k),s=0

because

r=0pi2r(p-r+1)pr!=(1-1)p=0

for all 1pk2. This implies that

eitΔ(Hk(u)e-u2e-i4tu2)(x)=ctπak(k)(2ti)kak(k),sxke-116t2x2+i4tx2=ctπ2k(2ti)k(2116t2)kxke-116t2x2+i4tx2=ctπ(1i)k(12t)kxke-116t2x2+i4tx2.

If k is odd, the proof follows in similar steps. So, for each k, we have

eitΔ(ψk(u)e-i4tu2)(x)=ctπ(12kk!π)12(12ti)kxke-116t2x2+i4tx2.

Now define another operator 𝒯t:Wm,2(n)2(n,utm(z)dz) by

𝒯t(f)(z)=eitΔ(f(u)e-i4tu2)(z).

This will help us to prove norm equivalence between the spaces Wm,2(n) and 2(n,utm(z)dz).

Theorem 4.6.

The operator Tt is a unitary map. Furthermore, for any mN and fWm there exist M1,M2>0 depending on m,t and n such that

M1fWm2eitΔfutm2M2fWm2.

Proof.

We know that eitΔ:L2(n,eu2du)2(n,wt(z)dz) is unitary. Therefore, {eitΔ(Ψ~α):αn} is a complete orthonormal set in 2(n,wt(z)dz), as ={Ψ~α:αn} is a complete orthonormal set in L2(n,eu2du). Since Ψα=k=1nψαk, by Proposition 4.5 we have

eitΔ(Ψ~α)(x)=ctnπn(12|α|α!(π)n2)12(12ti)|α|xαe(-116t2+i4t)x2

for every xn and αn. Extending this analytically to n, for αn and zn we get

eitΔ(Ψ~α)(z)=(4iπt)-n2(π)n2(12|α|α!)12(12ti)|α|zαe(i4t-116t2)z2.

The function eitΔ(Ψ~α) is nothing but a constant multiple of Υαt, that is,

𝒯t(Ψα)=eitΔ(Ψ~α)=i-n2(1i)|α|Υαt.

But we have

ΨαWm2=|γ|mDγΨα2=|γ|m22|γ|k=1nq=1γk(αk+q2)

and

eitΔ(Ψ~α)utm22=n|eitΔ(Ψ~α)(z)|2|γ|m(12t)2|γ||zγ|2wt(z)dz=|γ|m(12t)2|γ|zγeitΔ(Ψ~α)wt22=|γ|m(1t)2|γ|(2t)|γ|j=1nq=1γj(αi+q2)12Υα+γtwt22=|γ|m(2)2|γ|j=1nq=1γj(αi+q2)=ΨαWm2.

So 𝒯t takes a complete orthonormal set in Wm,2(n) into a complete orthonormal set in 2(n,utm(z)dz). This implies that

𝒯tfutm22=fWm2for every fWm,2(n).(4.3)

By Theorem 3.11, for any fWm,2(n) there exist M1,M2>0 such that

M1f(u)e-i4tu2Wm2fWm2M2f(u)e-i4tu2Wm2.(4.4)

From equations (4.3) and (4.4) we can see that

M1f(u)e-i4tu2Wm2eitΔ(f(u)e-i4tu2)utm22M2f(u)e-i4tu2Wm2.

The above equation is true for any fWm,2(n). Hence the theorem follows. ∎

So the image of the Sobolev space Wm,2(n) under eitΔ is identified with the weighted Bergman space 2(n,utm(z)dz) up to equivalence of norms.

5 Image of Sobolev spaces under the Schrödinger semigroup associated to the Hermite operator

Consider the Schrödinger group eitH associated to the Hermite operator H=-Δ+|x|2 on n. The spectral decomposition of H is given by

H=k=0(2k+n)Pk,

where Pkf=|α|=kf,ΦαΦα with Φα being the normalized Hermite functions on n. Using Mehler’s formula, we have

eitHf(x)=nKit(x,u)f(u)𝑑u,

where

Kit(x,u)=(2πisin2t)-n2ei4(cot2t)(|x|2+|u|2)-i(csc2t)xu.

This Kit is well defined for 2tkπ for k. In [9], Parui et al. proved the following theorem.

Theorem 5.1.

For any tk2, kZ, eitH defines an isometric isomorphism between L2(Rn,eu2du) and

2(n,(2π|sin2t|)-ne-(csc22t)y2+(cot2t)xydxdy).

The Hermite operator H can be written as

H=12k=1n(AkAk*+Ak*Ak).

Here Ak=uk+uk and Ak*=-uk+uk for k=1,2,,n. For an integer m1, we define the Hermite Sobolev space WHm,2(n) as follows:

WHm,2(n)={fL2(n,eu2du):AαA¯βfL2(n,eu2du),|α|+|β|m}.

Here Aα=A1α1Anαn and A¯β=(A1*)α1(An*)αn. For f,gWHm,2(n), we define the inner product and norm on the space WHm,2(n) as follows:

f,gWHm,2(n):=|α|+|β|mAαA¯βf,AαA¯βg,fWHm,2(n)2:=|α|+|β|mAαA¯βf2.

Then WHm,2(n) becomes a Hilbert space with respect to the above inner product. Note that Ak+Ak*=2uk and Ak-Ak*=2uk for k=1,2,,n. By using this relation, it is easy to see that WHm,2(n) and Wm,2(n) represent the same vector space. From Note 3.12 it is easy to see that there exists B>0 (depending only on m) such that

fWHm,2(n)BfWm,2(n)

for any fWm,2(n). That is, the identity map i:Wm,2(n)WHm,2(n) is continuous and onto. From the bounded inverse theorem it follows that the norms WHm,2(n) and Wm,2(n) are equivalent. Hence characterizing the image of WHm,2(n) under eitH is equivalent to characterizing the image of Wm,2(n) under eitH.

Since the linear map eitH:Wm,2(n)𝒪(n) is one to one, we can give an inner product on

Wt,Hm,2(n):=eitH(Wm,2(n))

in the following way: For f,gWm,2(n), we define eitHf,eitHg:=f,gWm,2(n). From this we can see that Wt,Hm,2(n) is a Hilbert space and

eitH:Wm,2(n)Wt,Hm,2(n)

is unitary.

Let 𝒮~f(u)=f(u)e-i/4cot2tu2 for fWm,2(n). Then 𝒮~ becomes a bounded invertible operator on Wm,2(n). For z=x+iyn, let us denote

wt,H(z)=(2π|sin2t|)-ne-(csc22t)y2+(cot2t)xy,ut,Hm(z)=|γ|m(csc2t)2|γ||zα|2wt,H(z).

Consider the weighted Bergman space

ut,Hm2:=2(n,ut,Hm(z)dz).

The proofs of the following proposition and theorem are similar to the ones of Proposition 4.5 and Theorem 4.6, respectively.

Proposition 5.2.

For zCn and αNn, we have

(eitH𝒮~Ψα)(z)=(2πisin2t)-n2πn4(12|α|α!)12(-icsc2t)|α|zαe(i4cot2t-14csc22t)z2.

Moreover, the transform eitHS~:Wm,2(Rn)HL2(Cn,ut,Hm(z)dz) is unitary.

Theorem 5.3.

For any fWm,2(Rn), there exists M~1>0 and M~2>0 such that

M~1fWm2eitHfut,Hm2M~2fWm2.

So the image of the Sobolev space WHm,2(n) under eitH is identified with the weighted Bergman space 2(n,ut,Hm(z)dz) up to equivalence of norms.

Acknowledgements

The authors wish to thank G. B. Folland for giving clarification to their questions related to weighted Sobolev spaces.

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About the article

Received: 2016-12-03

Accepted: 2017-12-27

Published Online: 2018-01-20

Published in Print: 2019-01-01


The first author thanks University Grant Commission, India, for financial support.


Citation Information: Advances in Pure and Applied Mathematics, Volume 10, Issue 1, Pages 65–79, ISSN (Online) 1869-6090, ISSN (Print) 1867-1152, DOI: https://doi.org/10.1515/apam-2016-0116.

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