InverseSemigroupLap.dvi


CUBO A Mathematical Journal

Vol.12, No¯ 03, (83–97). October 2010

The Semigroup and the Inverse of the Laplacian

on the Heisenberg Group1

APARAJITA DASGUPTA

Department of Mathematics, Indian Institute of Science,

Bangalore–560012, India

email: adgupta@math.iisc.ernet.in

AND

M.W. WONG

Department of Mathematics and Statistics, York University,

4700 Keele Street, Toronto, Ontario M3J 1P3, Canada

email: mwwong@mathstat.yorku.ca

ABSTRACT

By decomposing the Laplacian on the Heisenberg group into a family of parametrized par-

tial differential operators L̃τ,τ ∈ R \ {0}, and using parametrized Fourier-Wigner trans-
forms, we give formulas and estimates for the strongly continuous one-parameter semi-

group generated by L̃τ, and the inverse of L̃τ. Using these formulas and estimates, we

obtain Sobolev estimates for the one-parameter semigroup and the inverse of the Lapla-

cian.

1This research has been supported by the Natural Sciences and Engineering Research Council of Canada.



84 Aparajita Dasgupta & M.W. Wong
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12, 3 (2010)

RESUMEN

Mediante descomposición del Laplaceano sobre el grupo de Heisenberg en una familia de

operadores diferenciales parciales parametrizados L̃τ,τ ∈ R\{0}, y usando transformada de
Fourier-Wigner parametrizada, damos fórmulas y estimativas para la continuidad fuerte

del semigrupo generado por L̃τ, y la inversa de L̃τ. Usando esas fórmulas y estimati-

vas obtenemos estimativas de Sobolev para el semigrupo a un parámetro y la inversa del

Laplaceano.

Key words and phrases: Heisenberg group, Laplacian, parametrized partial differential

operators, Hermite functions, Fourier-Wigner transforms, heat equation, one parameter semi-

group, inverse of Laplacian, Sobolev spaces.

Math. Subj. Class.: 47F05, 47G30, 35J70.

1 The Laplacian on the Heisenberg Group

If we identify R2 with the complex plane C via

R
2 ∋ (x, y) ↔ z = x + i y ∈ C

and let

H = C×R,

then H becomes a non-commutative group when equipped with the multiplication · given by

(z, t) · (w, s) =
(
z + w, t + s +

1

4
[z, w]

)
, (z, t), (w, s) ∈ H,

where [z, w] is the symplectic form of z and w defined by

[z, w] = 2 Im(zw).

In fact, H is a unimodular Lie group on which the Haar measure is just the ordinary Lebesgue

measure d z dt.

Let h be the Lie algebra of left-invariant vector fields on H. A basis for h is then given by

X , Y and T, where

X =
∂

∂x
+

1

2
y
∂

∂t
,

Y =
∂

∂y
−

1

2
x
∂

∂t
,



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The Semigroup and the Inverse of the Laplacian ... 85

and

T =
∂

∂t
.

The Laplacian ∆H on H is defined by

∆H = −(X 2 + Y 2 + T 2).

A simple computation gives

∆H = −∆−
1

4
(x

2 + y2)
∂2

∂t2
+

(
x
∂

∂y
− y

∂

∂x

)
∂

∂t
−

∂2

∂t2
,

where

∆=
∂2

∂x2
+

∂2

∂y2
.

Let g be the Riemannian metric on R3 given by

g(x, y, t) =




1 0 y/2

0 1 −x/2

y/2 −x/2 1
4
(x2 + y2)




for all (x, y, t) ∈ R3. Then ∆H is also given by

−∆H =
1

√
det g

∑

1≤ j,k≤3
∂j (

√
det g g j,k∂k),

where ∂1 = ∂/∂x, ∂2 = ∂/∂y, ∂3 = ∂/∂t. Since the symbol σ(∆H) of ∆H is given by

σ(∆H)(x, y, t;ξ,η,τ) =
(
ξ+

1

2
yτ

)2
+

(
η−

1

2
xτ

)2
+τ2

for all (x, y, t) and (ξ,η,τ) in R3, it is easy to see that ∆H is an elliptic partial differential

operator on R3 but not globally elliptic in the sense of Shubin [11]. Let us recall that ∆H is

globally elliptic if there exist positive constants C and R such that

|σ(∆H)(x, y, t;ξ,η,τ)| ≥ C
(
1 +|x|+|y|+|t|+|ξ|+|η|+|τ|

)2

whenever

|x|+|y|+|t|+|ξ|+|η|+|τ| ≥ R.

The aim of this paper is to give new estimates for the strongly continuous one-parameter

semigroup e−u∆H , u > 0, generated by ∆H and the inverse ∆−1H of ∆H. More precisely, we use the
Sobolev spaces L2s (H), s ∈ R, as in [1, 2] to estimate ‖e

−u∆H f ‖
L2s (H)

, u > 0, in terms of ‖f ‖L2 (H)
for all f in L2(H), and to give an estimate for ‖e−u∆H f ‖L2(H) in terms of ‖f ‖L2s (H). These Sobolev
spaces are also used to estimate ||∆−1

H
f ||

L2
s+2 (H)

in terms of ||f ||
L2s (H)

for all f in L2s (H).



86 Aparajita Dasgupta & M.W. Wong
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12, 3 (2010)

The function F on H× (0,∞) given by

F(z, t, u) = (e−u∆H f )(z, t), (z, t) ∈ H, u > 0,

is in fact the solution of the initial value problem





∂F
∂u

(z, t, u) = −(∆HF)(z, t, u),

F(z, t, 0) = f (z, t),

(z, t) ∈ H, u > 0,

(z, t) ∈ H,

for the Laplacian ∆H.

Using the same techniques as in [1], we get for all f ∈ L2(H) and u > 0,

(e
−u∆H f )(z, t) = (2π)−1/2

ˆ ∞

−∞
e
−itτ

(e
−uL̃τ f τ)(z) dτ, (z, t) ∈ H, (1.1)

where L̃τ, τ ∈ R \ {0}, is given by

L̃τ = −∆+
1

4
(x

2 + y2)τ2 − i
(
x
∂

∂y
− y

∂

∂x

)
τ+τ2

and f τ is the function on C given by

f
τ
(z) = (2π)−1/2

ˆ ∞

−∞
e

itτ
f (z, t) dt, z ∈ C,

provided that the integral exists. In fact, f τ(z) is the inverse Fourier transform of f (z, t) with

respect to t evaluated at τ. In this paper, the nonzero parameter τ can be looked at as Planck’s

constant.

To obtain the estimates in this paper, we use formulas for e−uL̃τ and L̃−1τ in terms of the τ-

Weyl transforms and the τ-Fourier–Wigner transforms of Hermite functions, τ ∈ R\ {0}, which
we recall in, respectively, Section 2 and Section 3. The L2-boundedness and the Hilbert–

Schimdt property of τ-Weyl transforms are instrumental in obtaining the estimates.

Basic information on the classical Fourier–Wigner transforms, Wigner transforms and

Weyl transforms can be found in [13] among others.

In Section 2, we introduce the τ-Weyl transforms and prove results on the L2-bounded-

ness and the Hilbert–Schmidt property of the τ-Weyl transforms. The τ-Fourier–Wigner

transforms of Hermite functions are recalled in Section 3. A formula for e−uL̃τ f , u > 0, for ev-
ery function f in L2(C) and an estimate for ‖e−uL̃τ f ‖L2 (C), u > 0, in terms of ‖f ‖L p (C), 1 ≤ p ≤ 2,
are given in Section 4. This formula gives a formula for e−u∆H , u > 0, immediately using the
inverse Fourier transform as indicated by (1.1). In Section 5, we use the family L2s (H), s ∈ R,
of Sobolev spaces with respect to the center of the Heisenberg group as in [1, 2] to ob-

tain Sobolev estimates for e−u∆H f , u > 0, in terms of ‖f ‖L2 (H), and Sobolev estimates for



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The Semigroup and the Inverse of the Laplacian ... 87

‖e−u∆H f ‖L2(H), u > 0, in terms of the Sobolev norms ‖f ‖L2s (H) of f in L
2
s (H). In Section 6,

we obtain a formula for L̃−1τ and estimates for L̃
−1
τ which are then used to estimate ∆

−1
H

. In

Section 7, estimates for ‖∆−1
H

f ‖L2
s+2 (H)

in terms of ‖f ‖L2s (H) for all f in L
2
s (H) are given.

We end this section by putting in perspectives the results in this paper. While the semi-

group and the inverse can be studied in the framework of functional analysis as explained in

[3, 4, 5, 8, 9, 16], the results and methods in this paper are based on explicit formulas in hard

analysis and are related to the works in [1, 2, 6, 7, 10, 12, 14, 15].

2 τ-Weyl Transforms

Let f and g be functions in L2(R). Then for τ in R\{0}, the τ-Fourier–Wigner transform Vτ( f , g)

is defined by

Vτ( f , g)(q, p) = (2π)−1/2|τ|1/2
ˆ ∞

−∞
e

iτq y
f
(
y +

p

2

)
g
(
y −

p

2

)
d y

for all q and p in R. In fact,

Vτ( f , g)(q, p) = |τ|1/2V ( f , g)(τq, p), q, p ∈ R,

where V ( f , g) is the classical Fourier–Wigner transform of f and g. A proof can be found in

[1].

It can be proved that Vτ( f , g) is a function in L
2(C) and we have the Moyal identity

stating that

‖Vτ( f , g)‖L2 (C) = ‖f ‖L2 (R)‖g‖L2 (R), τ ∈ R \ {0}. (2.1)

We define the τ-Wigner transform Wτ( f , g) of f and g by

Wτ( f , g) = Vτ( f , g)∧. (2.2)

Then we have the following connection of the τ-Wigner transform with the usual Wigner

transform.

Theorem 2.1. Let τ ∈ R \ {0}. Then for all functions f and g in L2(R),

Wτ( f , g)(x,ξ) = |τ|−1/2W ( f , g)(x/τ,ξ), x,ξ ∈ R,

where W ( f , g) is the classical Wigner transform of f and g.

It is obvious that

Wτ( f , g) = Wτ( g, f ), f , g ∈ L2(R). (2.3)



88 Aparajita Dasgupta & M.W. Wong
CUBO

12, 3 (2010)

Let σ ∈ L p(C), 1 ≤ p ≤ ∞. Then for all τ in R\{0} and all functions f in the Schwartz space
S (R) on R, we define Wτσ f to be the tempered distribution on R by

(W
τ
σ f , g) = (2π)

−1/2
ˆ ∞

−∞

ˆ ∞

−∞
σ(x,ξ)Wτ( f , g)(x,ξ) dx dξ (2.4)

for all g in S (R), where (F, G) is defined by

(F, G) =
ˆ

Rn

F(z)G(z) d z

for all measurable functions F and G on Rn, provided that the integral exists. We call Wτσ

the τ-Weyl transform associated to the symbol σ. It is easy to see that if σ is a symbol in the

Schwartz space S (C) on C, then Wτσ f is a function in S (R) for all f in S (R).

We have the following estimate for the norm of the Weyl transform Wτ
σ̂

in terms of the

L p norm of the symbol σ when σ ∈ L p(C), 1 ≤ p ≤ 2.

Theorem 2.2. Let σ ∈ L p(C), 1 ≤ p ≤ 2. Then Wτ
σ̂

: L2(R) → L2(R) is a bounded linear operator
and

‖Wτσ̂‖∗ ≤ (2π)
−1/p|τ|−(1/2)+(1/p)‖σ‖L p (C),

where ‖Wτ
σ̂
‖∗ is the operator norm of Wτσ̂ : L

2(R) → L2(R).

Proof Let f and g be functions in S (R). Then

(W
τ
σ̂ f , g) = (2π)

−1/2
ˆ ∞

−∞

ˆ ∞

−∞
σ̂(x,ξ)Wτ( f , g)(x,ξ) dx dξ

= (2π)−1|τ|−1/2
ˆ ∞

−∞

ˆ ∞

−∞
σ̂(x,ξ)W ( f , g)(x/τ,ξ) dx dξ

= (2π)−1|τ|1/2
ˆ ∞

−∞

ˆ ∞

∞
σ̂(τx,ξ)W ( f , g)(x,ξ) dx dξ.

But

σ̂(τx,ξ) = |τ|−1σ̂1/τ(x,ξ), x,ξ ∈ R,

where σ1/τ is the dilation of σ with respect to the first variable by the amount 1/τ. More

precisely,

σ1/τ(q, p) = σ(q/τ, p), q, p ∈ R.

So,

(W
τ
σ̂ f , g) = (2π)

−1/2|τ|−1/2
ˆ ∞

−∞

ˆ ∞

−∞
σ̂1/τ(x,ξ)W ( f , g)(x,ξ) dx dξ

= |τ|−1/2(Wσ̂1/τ f , g),



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The Semigroup and the Inverse of the Laplacian ... 89

where Wσ̂1/τ is the classical Weyl transform with symbol σ̂1/τ. Thus, it follows from Theorem

21.1 in [14] that Wτ
σ̂

: L2(R) → L2(R) is a bounded linear operator and

‖Wτσ̂‖∗ ≤ |τ|
−1/2

(2π)
−1/p‖σ1/τ‖L p (C) = (2π)−1/p|τ|−(1/2)+(1/p)‖σ‖L p (C).

�

We have the following result for the Hilbert–Schmidt norm of the Weyl transform Wτ
σ̂

in

terms of the L2 norm of the symbol σ when σ ∈ L2(C).

Theorem 2.3. Let σ ∈ L2(C). Then Wτ
σ̂

: L2(R) → L2(R) is a Hilbert–Schmidt operator and

‖Wτσ̂‖HS = (2π)
−1/2‖σ‖L2 (C),

where ‖Wτ
σ̂
‖HS is the Hilbert–Schmidt norm of Wτσ̂ : L

2(R) → L2(R).

Proof Let f and g be functions in S (R). Then

(W
τ
σ̂ f , g) = (2π)

−1/2
ˆ ∞

−∞

ˆ ∞

−∞
σ̂(x,ξ)Wτ( f , g)(x,ξ) dx dξ

= (2π)−1/2|τ|−1/2
ˆ ∞

−∞

ˆ ∞

−∞
σ̂(x,ξ)W ( f , g)(x/τ,ξ) dx dξ

= (2π)−1/2|τ|1/2
ˆ ∞

−∞

ˆ ∞

∞
σ̂(τx,ξ)W ( f , g)(x,ξ) dx dξ.

But

σ̂(τx,ξ) = |τ|−1/2σ̂1/τ(x,ξ), x,ξ ∈ R,

where σ1/τ is the dilation of σ with respect to the first variable by the amount 1/τ, i.e.,

σ1/τ(q, p) = σ(q/τ, p), q, p ∈ R.

So,

(W
τ
σ̂ f , g) = (2π)

−1|τ|−1/2
ˆ ∞

−∞

ˆ ∞

−∞
σ̂1/τ(x,ξ)W ( f , g)(x,ξ) dx dξ

= |τ|−1/2(Wσ̂1/τ f , g),

where Wσ̂1/τ is the classical Weyl transform with symbol σ̂1/τ. Thus, it follows from Theorem

7.5 in [13] that Wτ
σ̂

: L2(R) → L2(R) is a Hilbert–Schmidt operator and

‖Wτσ̂‖HS = |τ|
−1/2‖Wσ̂1/τ ‖HS

= (2π)−1/2|τ|−1/2‖σ1/τ‖L2 (C)
= (2π)−1/2‖σ‖L2 (C).

�



90 Aparajita Dasgupta & M.W. Wong
CUBO

12, 3 (2010)

3 Fourier–Wigner Transforms of Hermite Functions

For τ ∈ R \ {0} and for k = 0, 1, 2, . . . , we define eτ
k

to be the function on R by

e
τ
k
(x) = |τ|1/4 e k (

√
|τ|x), x ∈ R.

Here, e k is the Hermite function of order k defined by

e k(x) =
1

(2k k!
p
π)1/2

e
−x2/2

Hk (x), x ∈ R,

where Hk is the Hermite polynomial of degree k given by

Hk(x) = (−1)k ex
2/2

(
d

dx

)k
(e

−x2
), x ∈ R.

For j, k = 0, 1, 2, . . . , we define eτ
j,k

on R2 by

e
τ
j,k

= Vτ(eτj , e
τ
k
).

The following theorem gives the connection of {eτ
j,k

: j, k = 0, 1, 2, . . . } with {e j,k : j, k =
0, 1, 2, . . . }, where

e j,k = V (e j , e k ), j, k = 0, 1, 2, . . . .

A proof can be found in [1].

Theorem 3.1. For τ ∈ R \ {0} and for j, k = 0, 1, 2, . . . ,

e
τ
j,k

(q, p) = |τ|1/2 e j,k
(

τ
p
|τ|

q,
√

|τ|p
)
, q, p ∈ R.

Theorem 3.2. {eτ
j,k

: j, k = 0, 1, 2, . . . } forms an orthonormal basis for L2(R2).

Theorem 3.2 follows from Theorem 3.1 and Theorem 21.2 in [13] to the effect that {e j,k :

j, k = 0, 1, 2, . . . } is an orthonormal basis for L2(R2).

Theorem 3.3. For j, k = 0, 1, 2, . . . ,

L̃τ e
τ
j,k

= (2k + 1 +|τ|)|τ|eτ
j,k

.

Theorem 3.3 can be proved using Theorem 3.1, Theorem 3.3 in [2] and Theorem 22.2

in [13] telling us that for j, k = 0, 1, 2, . . . , e j,k is an eigenfunction of L1 corresponding to the
eigenvalue 2k + 1 and the fact that, L̃τ = Lτ +τ2.



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The Semigroup and the Inverse of the Laplacian ... 91

4 A Formula and an Estimate for e−uL̃τ , u > 0

Let τ ∈ R \ {0}. Then a formula for e−uL̃τ , u > 0, is given by the following theorem.

Theorem 4.1. Let f ∈ L2(C). Then for u > 0,

e
−uL̃τ f = (2π)1/2

∞∑

k=0
e
−(2k+1+|τ|)|τ|u

Vτ(W
τ

f̂
e
τ
k
, e

τ
k
),

where the convergence of the series is understood to be in L2(C).

Proof Let f ∈ L2(C). Then from Theorem 3.3 we have for u > 0

e
−uL̃τ f =

∞∑

k=0

∞∑

j=0
e
−(2k+1+|τ|)|τ|u

( f , e
τ
j,k

)e
τ
j,k

= e−|τ|
2 u

e
−uLτ f , (4.1)

where the series is convergent in L2(C). Now, using the formula for e−uLτ f in [2] and (4.1), we

get

e
−uL̃τ f = (2π)1/2

∞∑

k=0
e
−(2k+1+|τ|)|τ|u

Vτ(W
τ

f̂
e
τ
k
, e

τ
k
)

for all f in L2(C) and u > 0. �

For all τ in R \ {0}, we have the following estimate for the L2 norm of e−uL̃τ f , u > 0, in
terms of the L p norm of f .

Theorem 4.2. Let τ ∈ R \ {0}. Then for all functions f in L p(C), 1 ≤ p ≤ 2,

‖e−uL̃τ f ‖L2(C) ≤ (2π)
−(1/p)+(1/2)|τ|−(1/2)+(1/p) e−τ

2 u 1

2 sinh(|τ|u)
‖f ‖L p (C).

Proof By Theorem 4.1, the Moyal identity (2.1) and the fact that

‖eτ
k
‖L2 (R) = 1, k = 0, 1, 2, . . . ,

we get

‖e−uL̃τ f ‖L2(C) ≤ (2π)
1/2

e
−(|τ|+|τ|2 )u

∞∑

k=0
e
−2k|τ|u‖Wτ

f̂
e
τ
k
‖L2 (R), u > 0. (4.2)

Applying Theorem 2.2 to (4.2), we get

‖e−uL̃τ f ‖L2 (C)

≤ (2π)−(1/p)+(1/2)|τ|−(1/2)+(1/p) e−(|τ|+|τ|
2 )u

(
∞∑

k=0
e
−2k|τ|u

)
‖f ‖L p (C)

= (2π)−(1/p)+(1/2)|τ|−(1/2)+(1/p) e−|τ|
2 u 1

2 sinh(|τ|u)
‖f ‖L p (C),

as asserted.

�



92 Aparajita Dasgupta & M.W. Wong
CUBO

12, 3 (2010)

5 Sobolev Estimates for e−∆H, u > 0

Let s ∈ R. Then we define L2s (H) to be the set of all tempered distributions f in S
′
(H) such

that f τ(z) is a measurable function and

ˆ

C

ˆ ∞

−∞
|τ|2s|f τ(z)|2dτ d z < ∞.

For every f in L2s (H), we define the norm ‖f ‖L2s (H) by

‖f ‖2
L2s (H)

=
ˆ

C

ˆ ∞

−∞
|τ|2s|f τ(z)|2 dτ d z.

Then it can be shown easily that L2s (H) is an inner product space in which the inner product

( , )
L2s (H)

is given by

( f , g)
L2s (H)

=
ˆ

C

ˆ ∞

−∞
|τ|2s f τ(z) gτ(z) dτ d z

for all f and g in L2s (H).

Theorem 5.1. Let s ≥ 1. Then for u > 0, e−u∆H : L2(H) → L2s (H) is a bounded linear operator
and

‖e−u∆H f ‖
L2s (H)

≤
cs

2us
‖f ‖L2 (H), f ∈ L

2
(H),

where

cs = sup
τ∈R\{0}

(|τ|s/sinh|τ|).

Proof Let u > 0 and f ∈ L2(H). Then by (1.1), Fubini’s theorem, Plancherel’s theorem and
Theorem 4.2 with p = 2,

‖e−u∆H f ‖2
L2s (H)

=
ˆ

C

ˆ ∞

−∞
|τ|2s|(e−u∆H f )τ(z)|2dτ d z

=
ˆ ∞

−∞
|τ|2s

(ˆ

C

|(e−u∆H f )τ(z)|2d z
)

dτ

=
ˆ ∞

−∞
|τ|2s

(ˆ

C

|(e−uL̃τ f τ)(z)|2d z
)

dτ

=
ˆ ∞

−∞
|τ|2s‖e−uL̃τ f τ‖2

L2(C)
dτ

≤
1

4

(
ˆ ∞

−∞

e−2τ
2 u|τ|2s

sinh2(|τ|u)
‖f τ‖2

L2 (C)
dτ

)

≤
1

4

ˆ ∞

−∞

|τ|2s

sinh2(|τ|u)

(ˆ

C

|f τ(z)|2d z
)

dτ



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The Semigroup and the Inverse of the Laplacian ... 93

=
1

4 u2s+1

ˆ ∞

−∞

|τ|2s

sinh2(|τ|u)

(ˆ

C

| f̌ (z,τ/u)|2 d z
)

dτ,

where f̌ is the inverse Fourier transform of f with respect to t. So, using a simple change of

variable and letting

Cs = sup
τ∈R\{0}

(|τ|2s/sinh2|τ|),

we get

‖e−u∆H f ‖2
L2s (H)

≤
Cs

4u2s

ˆ ∞

−∞

(ˆ

C

| f̌ (z,τ)|2 d z
)

dτ =
Cs

4u2s
‖f ‖2

L2 (H)

and this completes the proof. �

The following result complements Theorem 5.1.

Theorem 5.2. Let s ≤ −1. Then for u > 0, e−u∆H : L2s (H) → L
2(H) is a bounded linear operator

and

‖e−u∆H f ‖L2(H) ≤
c−s

2u−s
‖f ‖

L2s (H)
, f ∈ L2s (H),

where

c−s = sup
τ∈{0}

(|τ|−ssinh|τ|).

The proof of Theorem 5.2 is very similar to that of Theorem 5.1 and is hence omitted.

6 Two Formulas and an Estimate for L̃−1τ

Let τ ∈ R \ {0}. Then a formula for L−1τ is given by the following theorem.

Theorem 6.1. Let f ∈ L2(C). Then

L̃
−1
τ f = (2π)

1/2
∞∑

k=0

1

(2k + 1 +|τ|)|τ|
Vτ(W

τ

f̂
e
τ
k
, e

τ
k
),

where the convergence of the series is understood to be in L2(C).

Proof Let f ∈ L2(C). Then

L̃
−1
τ f =

∞∑

k=0

∞∑

j=0

1

(2k + 1 +|τ|)|τ|
( f , e

τ
j,k

)e
τ
j,k

, (6.1)

where the series is convergent in L2(C). Now, by Plancherel’s theorem and (2.2)–(2.4),

( f , e
τ
j,k

) =
ˆ

C

f (z)Vτ(e
τ
j
, eτ

k
)(z) d z =

ˆ

C

f̂ (ζ)Vτ(e
τ
j
, eτ

k
)∧(ζ) dζ



94 Aparajita Dasgupta & M.W. Wong
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12, 3 (2010)

=
ˆ

C

f̂ (ζ)Wτ(e
τ
j
, eτ

k
)(ζ) dζ = (2π)1/2(W

f̂
e
τ
k
, e

τ
j
) (6.2)

for j, k = 0, 1, 2, . . . . Similarly, for j, k = 0, 1, 2, . . . , and g in L2(C), we get

(e
τ
j,k

, g) = ( g, eτ
j,k

) = (2π)1/2(Wτ
ĝ

eτ
k
, eτ

j
) = (2π)1/2(eτ

j
, W

τ
ĝ

e
τ
k
). (6.3)

So, by (6.1)–(6.3), Fubini’s theorem and Parseval’s identity,

(L̃
−1
τ f , g) = 2π

∞∑

k=0

1

(2k + 1 +|τ|)|τ|

∞∑

j=0
(W

τ

f̂
e
τ
k
, e

τ
j
)(e

τ
j
, W

τ
ĝ

e
τ
k
)

= 2π
∞∑

k=0

1

(2k + 1 +|τ|)|τ|
(W

τ

f̂
e
τ
k
, W

τ
ĝ

e
τ
k
). (6.4)

By Plancherel’s theorem and (2.2)–(2.4),

(W
τ

f̂
e
τ
k
, W

τ
ĝ

e
τ
k
) = (2π)−1/2

ˆ

C

ĝ(z)Wτ(e
τ
k
, Wτ

f̂
eτ

k
)(z) d z

= (2π)−1/2
ˆ

C

Wτ(W
τ

f̂
e
τ
k
, e

τ
k
)(z) ĝ(z) d z

= (2π)−1/2
ˆ

C

Vτ(W
τ

f̂
e
τ
k
, e

τ
k
)(z) g(z) d z (6.5)

for k = 0, 1, 2, . . . . Thus, by (6.4), (6.5) and Fubini’s theorem,

(L̃
−1
τ f , g) = (2π)

1/2
∞∑

k=0

1

(2k + 1 +|τ|)|τ|
(Vτ(W

τ

f̂
e
τ
k
, e

τ
k
), g)

= (2π)1/2
(

∞∑

k=0

1

(2k + 1 +|τ|)|τ|
Vτ(W

τ

f̂
e
τ
k
, e

τ
k
), g

)
(6.6)

for all f and g in L2(C). Thus, by (6.6),

L̃
−1
τ f = (2π)

1/2
∞∑

k=0

1

(2k + 1 +|τ|)|τ|
Vτ(W

τ

f̂
e
τ
k
, e

τ
k
)

for all f in L2(C). �

The formula (6.4) is an important formula in its own right and we upgrade it to the status

of a theorem.

Theorem 6.2. For all τ ∈ R \ {0}, the inverse L̃−1τ of the parametrized partial differential oper-
ators L̃τ is given by

(L̃
−1
τ f , g) = 2π

∞∑

k=0

1

(2k + 1 +|τ|)|τ|
(W

τ

f̂
e
τ
k
, W

τ
ĝ

e
τ
k
), f , g ∈ L2(C).



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The Semigroup and the Inverse of the Laplacian ... 95

For all τ in R\ {0}, we have the following estimate for the L2 norm of L̃−1τ f in terms of the

L2 norm of f .

Theorem 6.3. Let τ ∈ R \ {0}. Then for all functions f in L2(C),

‖L̃−1τ f ‖L2 (C) ≤ |τ|
−2‖f ‖L2 (C).

Proof Let f and g be functions in L2(R). Then by Theorems 2.3 and 6.2,

|(L̃−1τ f , g)| ≤ 2π
1

|τ|2
∞∑

k=0
|(Wτ

f̂
e
τ
k
, W

τ
ĝ

e
τ
k
)|

≤ 2π
1

|τ|2
‖Wτ

f̂
‖HS‖Wτĝ ‖HS

=
1

|τ|2
‖f ‖L2 (C)‖g‖L2 (C)

and this completes the proof. �

7 Sobolev Estimates for ∆−1
H

We have the following simple result giving the connection of ∆−1
H

with L̃−1τ , τ ∈ R \ {0}, which
can be proved easily using the elementary properties of the Fourier transform and the Fourier

inversion formula.

Theorem 7.1. Let f ∈ L2(H). Then

(∆
−1
H

f )(z, t) = (2π)−1/2
ˆ ∞

−∞
e
−itτ

(L̃
−1
τ f

τ
)(z) dτ, (z, t)∈ H.

We can now give the following theorem, which can be seen as another manifestation of

the ellipticity of ∆H.

Theorem 7.2. Let s ∈ R. Then ∆−1
H

: L2s (H) → L
2
s+2(H) and

‖∆−1
H

f ‖L2
s+2 (H)

≤ ‖f ‖L2s (H), f ∈ L
2
s (H).

Proof By Fubini’s theorem, Plancherel’s theorem, Theorems 6.3 and 7.1,

‖∆−1
H

f ‖2
L2

s+2 (H)
=
ˆ

C

ˆ ∞

−∞
|τ|2(s+2)|(∆−1

H
f )

τ
(z)|2dτ d z

=
ˆ ∞

−∞
|τ|2(s+2)

(ˆ

C

|(∆−1
H

f )
τ
(z)|2d z

)
dτ



96 Aparajita Dasgupta & M.W. Wong
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12, 3 (2010)

=
ˆ ∞

−∞
|τ|2(s+2)

(ˆ

C

|(L̃−1τ f
τ
)(z)|2d z

)
dτ

=
ˆ ∞

−∞
|τ|2(s+2)‖L̃−1τ f

τ‖2
L2 (C)

dτ

≤
ˆ ∞

−∞
|τ|2s‖f τ‖2

L2 (C)
dτ

=
ˆ ∞

−∞
|τ|2s

(ˆ

C

|f τ(z)|2d z
)

dτ

=
ˆ

C

ˆ ∞

−∞
|τ|2s|f τ(z)|2dτ d z

= ‖f ‖2
L2s (H)

,

as asserted. �

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