Classical limit of the d-bar operators on quantum domains

0.2 MB·English

by  Slawomir Klimek

CLASSICAL LIMIT OF THE D-BAR OPERATORS ON QUANTUM
DOMAINS
SLAWOMIR KLIMEK AND MATT MCBRIDE
1
1 Abstract. We study one parameter families Dt, 0 < t < 1 of non-commutative analogs
0 of the d-bar operator D0 = ∂∂z on disks and annuli in complex plane and show that, under
2
suitable conditions, they converge in the classical limit to their commutative counterpart.

n Moreprecisely,weendowthecorrespondingfamiliesofHilbertspaceswiththestructuresof
a
continuous fields over the interval [0,1) and we show that the inverses of the operators Dt
J
subject to APS boundary conditions form morphisms of those continuous fields of Hilbert
3
spaces.
1

]
A
F
1. Introduction
.
h
t According to the broadest and the most flexible definition, a quantum space is simply a
a
m noncommutative algebra. Noncommutative geometry [4] studies what could be considered
[ “geometric properties” of such quantum spaces.

1 One of the most basic examples of a quantum space is the quantum unit disk C(Dt) of [8].
v It is defined as theuniversal unital C∗-algebra with thegenerators z andz which areadjoint
t t
5
to each other, and satisfy the following commutation relation: [z ,z ] = t(I z z )(I z z ),
4 t t t t t t
− −
6 for a continuous parameter 0 < t < 1.
2 It was proved in [8] that C(D ) has a more concrete representation as the C∗-algebra
. t
1 generated by the unilateral weighted shift with the weights given by the formula:
0
1
(k +1)t
1
w (k) = . (1.1)
: t
v s1+(k +1)t
i
X In fact, as a C∗-algebra, C(D ) is isomorphic to the Toeplitz algebra. Moreover the family
t
ar C(Dt) is a deformation, and even deformation - quantization of the algebra of continuous
functions on the disk C(D) obtained in the limit as t 0, called the classical limit.
→
The quantum unit disk is one of the simplest examples of a quantum manifold with
boundary. It is also an example of a quantum complex domain, with z playing the role of a
t
quantum complex coordinate. Additionally, biholomorphisms of the unit disk naturally lift
to automorphisms of C(D ), see [8].
t
In view of this complex analytic interpretation of the quantum unit disk, there is a natural
need to define analogs of complex partial derivatives as some kind of unbounded operators
on C(D ) and its various Hilbert space completions. Such constructions have been described
t
in several places in the literature, see for example [2], [3], [7], [9], [10],[11]. In this paper we
are primarily concerned with one such choice, the so-called balanced d and d-bar operators
of [9] which we describe below.
Date: January 14, 2011.
1
2 SLAWOMIRKLIMEK AND MATT MCBRIDE
One notices that S := [z ,z ] is an invertible trace class operator (with an unbounded
t t t
inverse) and defines
−1/2 −1/2
D a = S [a,z ]S
t t t t
and
−1/2 −1/2
D a = S [z ,a]S ,
t t t t
for appropriate a C(D ). These two operators have the following easily seen properties
t
∈
D (1) = 0, D (z ) = 0, D (z ) = 1
t t t t t
D (1) = 0, D (z ) = 1, D (z ) = 0
t t t t t
which makes themplausible candidatesforquantum complexpartialderivatives. To makean
evenbettercaseoftheirsuitability, onewouldliketoknowthatinsomekindofinterpretation
of the limit as t 0, they indeed become the classical partial derivatives. This problem was
→
posed at the end of [7] and it is the subject of the present paper.
Infactwe consider hereabroaderclassical limit problembystudying quitegeneral families
of unilateral weights w (k), and not just those given by (1.1). Like in [9] such unilateral shifts
t
are still considered coordinates of quantum disks. Additionally we also consider bilateral
shifts and the C∗-algebras they generate. They are quantum analogs of annuli and can be
analyzed very similarly to the quantum disks.
We start with giving a concrete meaning to the classical limit t 0, which involves two
→
important steps. The first step is to consider certain bounded functions of the quantum d
and d-bar operators to properly manage their unboundedness. In this paper we choose to
work with the inverses of the operators D subject to APS boundary conditions [1] since
t
they are easy to describe and we can use the results of [3], [9].
The second step of our approach to the classical limit is the choice of framework for
studying limits of objects living in different spaces. Such a natural framework is provided
by the language of continuous fields, in our case of continuous fields of Hilbert spaces, see
[5]. Following [3] and [9] we define, using operators S , weighted Hilbert space completions
t
, 0 < t < 1, of the above quantum domains, while is the classical L2 space. We
t 0
H H
then equip that family of Hilbert spaces with a natural structure of continuous field, namely
the structure generated by the polynomials in complex quantum and classical coordinates.
In this setup the study of the classical limit becomes a question of continuity, a property
embedded in the definition of the continuous field. Consequently, inverses of the operators
D subject toAPS boundaryconditions, areconsidered asmorphisms ofthecontinuous fields
t
of Hilbert spaces. The main result of this paper is that in such a sense the limit of D is
t
indeed ∂ .
∂z
The paper is organized as follows. In section 2 we review the definitions and properties
of continuous fields of Hilbert spaces and their morphisms. In Section 3 we describe the
constructions of the quantum disk, the quantum annulus, Hilbert spaces of L2 “functions”
on those quantum spaces, d-bar operators and their inverses subject to APS conditions. We
state the conditions on weights w (k) and provide example of such weights. We construct the
t
generating subspace Λ needed for the construction of the continuous field of Hilbert spaces.
The main results of this paper are also formulated at the end of that section. Finally, section
4 contains the proofs of the results.
CLASSICAL LIMIT OF THE D-BAR OPERATORS ON QUANTUM DOMAINS 3
2. Continuous Fields of Hilbert Spaces
In this section we review some aspects of the theory of continuous fields of Hilbert spaces.
The main reference here is Dixmier’s book [5].
Definition 2.1. A continuous field of Hilbert spaces is a triple, denoted (Ω, ,Γ), where Ω
H
is a locally compact topological space, = H(ω) : ω Ω is a family of Hilbert spaces,
H { ∈ }
and Γ is a linear subspace of H(ω), such that the following conditions hold
ω∈Ω
(1) for every ω Ω, the set of x(ω), x Γ, is dense in H(ω),
∈ Q ∈
(2) for every x Γ, the function ω x(ω) is continuous,
∈ 7→ k k
(3) let x H(ω); if for every ω Ω and every ε > 0, there exists x′ Γ such that
∈ ω∈Ω 0 ∈ ∈
x(ω) x′(ω) ε for every ω in some neighborhood (depending on ε) of ω , then
0
k −Q k ≤
x Γ.
∈
The point of this definition is to describe a continuous arrangement of a family of different
Hilbert spaces. If they are all the same, then the space Γ of continuous functions on Ω with
values in that Hilbert space clearly satisfies all the conditions.
Below we will use the following terminology. We say that a section x H(ω) is
∈ ω∈Ω
approximable by Γ at ω if for every ε > 0, there exists an x′ Γ and a neighborhood of ω
0 0
∈ Q
such that x(ω) x′(ω) ε for every ω in that neighborhood. In this terminology condition
k − k ≤
3 of the above definition says that if a section x is approximable by Γ at every ω Ω, then
∈
x Γ.
∈
The above definition is a little cumbersome to work with, namely, trying to describe Γ in
full detail is usually very difficult since the third condition isn’t easy to verify. The following
proposition, proved in [5], makes it easier to construct continuous fields.
Proposition 2.2. Let Ω be a locally compact topological space, and let = H(ω) : ω Ω
H { ∈ }
be a family of Hilbert spaces. If Λ is a linear subspace of H(ω) such that
ω∈Ω
(1) for every ω Ω, the set of x(ω), x Λ, is dense in H(ω),
∈ ∈ Q
(2) for every x Λ, the function ω x(ω) is continuous,
∈ 7→ k k
then Λ extends uniquely to a linear subspace Γ H(w) such that (Ω, ,Γ) is a con-
⊂ ω∈Ω H
tinuous field of Hilbert spaces.
Q
Here one says that if a linear subspace Λ of H(ω) satisfies the two conditions above
ω∈Ω
then Λ generates the continuous field of Hilbert spaces (Ω, ,Γ). In fact, Γ is simply con-
Q H
structed as a local completion of Λ, i.e. Γ consists of all those sections x H(ω) which
∈ ω∈Ω
are approximable by Λ at every ω Ω.
∈ Q
Next we consider morphisms of continuous fields of Hilbert spaces. For this we have the
following definition.
Definition 2.3. Let (Ω, ,Γ) be a continuous field of Hilbert spaces and let T(ω) : H(ω)
H →
H(ω) be a collection of operators acting on the Hilbert spaces H(ω). Define T = T(ω) :
ω∈Ω
H(ω) H(ω). We say that T(ω) is a continuous family of bounded operators
ω∈Ω → ω∈Ω { } Q
in (Ω, ,Γ) if
Q H Q
(1) T(ω) is bounded for each ω,
(2) sup T(ω) < ,
k k ∞
ω∈Ω
(3) T maps Γ into Γ.
4 SLAWOMIRKLIMEK AND MATT MCBRIDE
The proposition below contains an alternative description of the third condition above, so
it is more manageable.
Proposition 2.4. With the notation of the above definition, the following three conditions
are equivalent:
(1) T maps Γ into Γ,
(2) T maps Λ into Γ,
(3) for every x Λ and for every ω Ω, T(ω)x(ω) is approximable by Λ at ω.
∈ ∈
Proof. The items above are arranged from stronger to weaker. The proof that (2) is equiva-
lent to(3)isa simple consequence oftheway thatΓ isobtainedfromΛdescribed inthepara-
graph following Proposition 2.2. Condition (2) implies condition (1) because sup T(ω) <
k k
ω∈Ω
andso, ifx(ω)andy(ω)arelocallyclosetoeachother, so areT(ω)x(ω)andT(ω)y(ω). (cid:3)
∞
3. D-Bar operators on non-commutative domains
In this section we review a variety of constructions needed to formulate and prove the
results of this paper. Those constructions include the definitions of the quantum disk, the
quantum annulus, Hilbert spaces of L2 “functions” on those quantum spaces, and d-bar
operators that were discussed in [9]. Other items discussed in this section are APS boundary
conditions, inverses of d-bar operators subject to APS conditions, conditions on weights, and
a construction of the generating subspace Λ of the continuous field of Hilbert spaces. The
main results are stated at the end of this section.
In the following formulas we let S be either N or Z. Let t (0,1) be a parameter. Let
e , k S be the canonical basis for ℓ2(S). Given a t-depe∈ndent, bounded sequence of
k
n{um}bers∈w (k) , called weights, the weighted shift U is an operator in ℓ2(S) defined by:
{ t } wt
U e = w (k)e . The usual shift operator U satisfies Ue = e .
wt k t k+1 k k+1
If S = N then the shift U is called a unilateral shift and it will be used to define a
wt
quantum disk. If S = Z then the shift U is called a bilateral shift and we will use it to
wt
define a quantum annulus (also called a quantum cylinder). For the choice of weights 1.1
the shifts U are the quantum complex coordinates z of the introduction.
wt t
We require the following condition on the one-parameter family of weights w (k).
t
Condition 1. The weights w (k) form a positive, bounded, strictly increasing sequence in
t
k such that the limits w := lim w (k) exist, are positive, and independent of t.
± t
k→±∞
Consider the commutator S = U∗ U U U∗ . It is a diagonal operator S e = S (k)e ,
t wt wt− wt wt t k t k
where
S (k) := w (k)2 w (k 1)2.
t t t
− −
Moreover S is a trace class operator with easily computable trace:
t
tr(S ) = S (k) = (w )2 (w )2 (3.1)
t t + −
−
k∈S
X
in the bilateral case, and tr(S ) = (w )2 in the unilateral case. Additionally S is invertible
t + t
with unbounded inverse.
CLASSICAL LIMIT OF THE D-BAR OPERATORS ON QUANTUM DOMAINS 5
Weassume further conditionsonthew (k)’sandtheS (k)’s. Thoseconditions weresimply
t t
extracted from the proofs in the next section to make the estimates work. They are possibly
not optimal, but they cover our motivating example described in the introduction.
Condition 2. The function t w (k) is continuous for every k, and for every ε > 0, w (k)
t t
7→
converges to w as k uniformly on the interval t ε.
±
→ ±∞ ≥
Condition 3. If h (t) := sup S (k) then h (t) 0 as t 0+.
1 t 1
k∈S → →
Condition 4. The supremum h (t) := sup 1 St(k+1) exists, and is a bounded function
2 k∈S − St(k)
of t, and h (t) 0 as t 0+. (cid:12) (cid:12)
2 (cid:12) (cid:12)
→ → (cid:12) (cid:12)
Condition 5. The supremum h (k) := sup 1 wt(k−1) exists for every k, and h (k) 0
3 − wt(k) 3 →
t∈[0,1)
(cid:12) (cid:12)
as k .
(cid:12) (cid:12)
→ ±∞ (cid:12) (cid:12)
Notice that the last condition implies that
w (k) constw (k 1) (3.2)
t t
≤ −
where the const above does not depend on t and k. This observation will be used in the
proofs in the next section.
Beforemovingon, weverifythattheweightsequence 1.1intheexampleintheintroduction
satisfies all of the conditions. First we compute:
t
S (k) = .
t
(1+kt)(1+(k +1)t)
Conditions 1 and 2 are all easily seen to be true with w = 1. For conditions 3, 4, and
+
5 simple computations give h (t) = t/(1 + t), h (t) = 2t/(1 + 2t) = O(t), and h (k) =
1 2 3
(k + 1 + √k2 +k)−1 = O(1/k), and so, by inspection, these weights meet all the required
conditions. Examples of bilateral shifts satisfying the above conditions are:
tk
w2(k) = α+β .
t 1+t k
| |
For this example h (t) = βt/(1 + t) and h (t) = O(t), h (k) = O(1/k), w2 = α + β,
1 2 3 +
w2 = α β. Another similar example is w2(k) = α+βtan−1(tk).
− − t
Next we proceed to the definition of the continuous field of Hilbert spaces over the interval
I = [0,1). Let C∗(U ) be the C∗-algebra generated by U . Then, in the unilateral case,
wt wt
the algebra C∗(U ) is called the non-commutative disk. There is a canonical map:
wt
C∗(U ) r C(S1)
wt −→
called the restriction to the boundary map.
In the bilateral case the algebra C∗(U ) is called the non-commutative cylinder, and we
wt
also have restriction to the boundary maps:
C∗(U ) r=r+⊕r− C(S1) C(S1).
wt −→ ⊕
We then define the Hilbert space , for t > 0, to be the completion of C∗(U ) with
Ht wt
respect to the inner product given by:
6 SLAWOMIRKLIMEK AND MATT MCBRIDE
a 2 = tr S1/2aS1/2a∗ .
k kt t t
For t = 0 we set = L2(D ) in the un(cid:16)ilateral/disk(cid:17)case and = L2(A ) in the
bilateral/annulus cHa0se where wD+ := z C : z w is theHd0isk of radwi−u,sw+w , and
A := z C : w z ww+ is{the∈annul|us| w≤ith+in}ner radius w and outer+radius
w−,w+ { ∈ − ≤ | | ≤ +} −
w . In what follows we usually skip the norm subscript as it will be clear from other terms
+
subscript whichHilbertspacenormoroperatornormisused. Alsonoticethatsettingw = 0
−
reduces most annulus formulas below to the disk case.
So far we have the space of parameters I, and for every t I we defined the Hilbert space
∈
. We also have distinguished elements of , namely quantum complex coordinates U .
Ht Ht wt
We use them to generate the continuous field of Hilbert spaces. More precisely we define the
generating linear space Λ to consists of all those x = x(t) such that there exists
⊂ t∈I Ht { }
N > 0, (depending onx), andfor every n N there arefunctions f ,g C([(w )2,(w )2]),
n n − +
Q ≤ ∈
such that for t > 0:
x (k) = Unf w (k)2 + g w (k)2 (U∗)n, (3.3)
t n t n t
n≤N n≤N
X (cid:0) (cid:1) X (cid:0) (cid:1)
and for t = 0:
x (r,ϕ) = f (r2)einϕ + g (r2)e−inϕ. (3.4)
0 n n
n≤N n≤N
X X
Now we proceed to the definitions of the quantum d-bar operators. The operator D in
t
is given by the following expression:
t
H
−1/2 −1/2
D a = S [a,U ]S
t t wt t
for t > 0, and for t = 0, D = ∂/∂z. Of course we need to specify the domain of D since it is
0 t
an unbounded operator. For reasons indicated in the introduction, in this paper we consider
the operators subject to the APS boundary conditions. Let P± be the spectral projections
in L2(S1) of the boundary operators 1 ∂ onto the interval ( ,0]. The domain of D is
±i ∂θ −∞ t
then defined to be:
dom(D ) = a : D a < , r(a) Ran P+
t t t
{ ∈ H k k ∞ ∈ }
for the disk. For the annulus we set:
dom(D ) = a : D a < , r (a) Ran P+, r (a) Ran P− .
t t t + −
{ ∈ H k k ∞ ∈ ∈ }
Here the maps r, r are the restriction to the boundary maps, that by the results of [9],
±
continue to make sense for those a for which D a < .
t t
∈ H k k ∞
If t = 0 the domain of D consists of all those first Sobolev class functions f on the disk
0
or the annulus for which the APS condition holds i.e. either r(f) Ran P+ or r (f)
+
∈ ∈
Ran P+, r (f) Ran P−, depending on the case. Here, by slight notational abuse, the
−
∈
symbols r, r are the classical restriction to the boundary maps.
±
It was verified in [9] that the above defined operators D are invertible, with bounded, and
t
even compact inverses Q . Using [9] we can write down the formulas for Q . If x Λ we
t t
∈
have the following for t > 0:
CLASSICAL LIMIT OF THE D-BAR OPERATORS ON QUANTUM DOMAINS 7
Q x (k) =
t t
N w (k +1) w (k +n) S (i)1/2S (i+n+1)1/2
Un t ··· t t t f (w (i)2)
n+1 t
− w (i+1) w (i+n) · w (k +n)
t t t !
n=0 i≥k ···
X X
N w (i) w (i+n 1) S (i)1/2S (i+n 1)1/2
+ t ··· t − t t − g (w (i)2) (U∗)n.
n−1 t
w (k) w (k +n 1) · w (i+n 1)
t t t !
n=1 i≤k ··· − −
X X
For the disk the second sum is from 0 to k, while for the annulus it is from to k.
−∞
For t = 0 we have
N ei(n+1)θ n N n e−i(n−1)θ
D x = 2rf′(r2) f (r2) + 2rg′(r2)+ g (r2) .
0 0 2 n − r n n r n 2
Xn=0 (cid:16) (cid:17) Xn=1(cid:16) (cid:17)
for both the disk and annulus. From this we can compute the inverse Q of D . A straight-
0 0
forward calculation gives the following result:
N (w+)2 rn−1 N r2 ρn−1
Q x = einθ f (ρ2) d(ρ2)+ e−inθ g (ρ2) d(ρ2),
0 0 − n+1 ρn n−1 rn
n=0 Zr2 n=1 Z(w−)2
X X
for the annulus, and the same formula with w replaced by 0 for the disk.
−
We are now in the position to state the main results of the paper. They are summarized
in the following two theorems:
Theorem 3.1. Given I = [0,1), let = : t I be the family of Hilbert spaces
t
H {H ∈ }
defined above and let Λ be the linear subspace of defined by 3.3 and 3.4. Also let the
t∈I Ht
conditions on w (k) and S (k) hold. Then Λ generates a continuous field of Hilbert spaces
t t
Q
denoted below by (I, ,Γ).
H
Theorem 3.2. Let Q : be the collection of operators for t [0,1) defined above.
t t t
H → H ∈
Then Q is a continuous family of bounded operators in the continuous field (I, ,Γ).
t
{ } H
We finish this section by shortly indicating that the above results are also valid for families
of d-bar operators studied in [3]. Let us quickly review the differences. The Hilbert space
studied in [3] is the completion of C∗(U ) with respect to the following inner product:
Ht wt
a 2 = tr(S aa∗).
k kt t
The quantum d-bar operator D of [3], acting in , is given by the following formula:
t t
H
D a = S−1[a,U ].
t t wt
It turns out that Theorems 3.1 and 3.2 are also true for the above spaces and operators. In
fact the proofs are even easier in this case and Condition 4, designed to handle expressions
like S (k +n)1/2S (k)1/2 is not even needed.
t t
The next section will contain all the analysis needed to prove the two theorems.
8 SLAWOMIRKLIMEK AND MATT MCBRIDE
4. Continuity and the classical limit
We will prove the two theorems from the above section by a series of steps that verify the
assumptions in the definitions of the continuous field of Hilbert spaces and the continuous
family of bounded operators. We concentrate mainly on the annulus case. The disk case is
in some respects simpler. Most of the formulas for the annulus are true also in the disk case
witha modification: replacing w byzero. Thesummation index intheannulus case extends
−
to and in couple of places the corresponding sums need to be estimated. This is not the
−∞
issue in the disk case where the summation starts at zero. However the major difficulty in
the disk case are the w terms in the denominator in the formula for the parametrix, since
t
they go to zero as t goes to zero. The proofs we describe below work in both cases, but much
shorter arguments are possible in the annulus case.
We first verify that Λ generates a continuous field of Hilbert spaces. To this end we need
to check two things: the density in of x(t), x Λ, and the continuity of the norm. The
t
H ∈
density is immediate, since, for example, the canonical basis elements of , see the proof of
t
H
Lemma 5.1 in [9], come from Λ.
The verification of the continuity of the norm is done in two steps: continuity at t = 0,
and at t > 0. If x Λ, i.e. x is given by formulas 3.3 and 3.4, then the norm of x in is,
t t
∈ H
for t > 0, given by
N
x 2 = S (k +n)1/2S (k)1/2 f w (k)2 2
t t t n t
k k
n=0 k∈S
XX (cid:12) (cid:0) (cid:1)(cid:12) (4.1)
N (cid:12) (cid:12)
+ S (k +n)1/2S (k)1/2 g w (k)2 2,
t t n t
n=1 k∈S
XX (cid:12) (cid:0) (cid:1)(cid:12)
while for t = 0 the norm of x is (cid:12) (cid:12)
0
N (w+)2 N (w+)2
x 2 = f r2 2d r2 + g r2 2d r2 . (4.2)
0 n n
k k
n=0Z(w−)2 n=1Z(w−)2
X (cid:12) (cid:0) (cid:1)(cid:12) (cid:0) (cid:1) X (cid:12) (cid:0) (cid:1)(cid:12) (cid:0) (cid:1)
The next lemma is needed to h(cid:12)andle t(cid:12)he product of S terms(cid:12) with d(cid:12)ifferent arguments.
Lemma 4.1. For n 1 we have
≥
S (k +n)
sup t 1 (2+h (t))n−1h (t),
2 2
k∈S (cid:12) St(k) − (cid:12) ≤
(cid:12) (cid:12)
where h (t) is the function de(cid:12)fined in Condi(cid:12)tion 4.
2
(cid:12) (cid:12)
Proof. The proof is by induction. For n = 1 we get Condition 4. The inductive step is
S (k +n+1) S (k +n+1) S (k +n) S (k +n+1)
t t t t
1 = 1 + 1
S (k) − S (k +n) S (k) − S (k +n) − ≤
(cid:12) t (cid:12) (cid:12) t (cid:18) t (cid:19) t (cid:12)
(cid:12) (cid:12) (cid:12)(1+h (t))(2+h (t))n−1h (t)+h (t) (2+h (t))nh(cid:12) (t)
(cid:12) (cid:12) (cid:12) 2 2 2 2 2 (cid:12)2
(cid:12) (cid:12) ≤ (cid:12) ≤ (cid:12)
(cid:3)
and the lemma is proved.
Now we are ready to discuss the continuity of norms 4.1, 4.2 as t 0+.
→
CLASSICAL LIMIT OF THE D-BAR OPERATORS ON QUANTUM DOMAINS 9
Proposition 4.2. If x is in Λ then
t
lim x = x
t 0
t→0+k k k k
Proof. Without loss of generality we may assume that x (k) = Unf (w (k)2) and x (r,ϕ) =
t n t 0
f (r2)einϕ , as the proof is identical for the g terms, and the elements of Λ are finite sums of
n
such x’s. We have
(w+)2
x 2 x 2 = S (k +n)1/2S (k)1/2 f (w (k)2) 2 f (r2) 2d(r2)
t 0 t t n t n
(cid:12)k k −k k (cid:12) (cid:12)(cid:12)Xk∈S (cid:12) (cid:12) −Z(w−)2 (cid:12) (cid:12) (cid:12)(cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12) (w+)2 (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) S (k) f (w (k)2) 2 f (r2) 2d(r2) + (cid:12)
t n t n
≤ −
(cid:12)(cid:12)Xk∈S (cid:12) (cid:12) Z(w−)2 (cid:12) (cid:12) (cid:12)(cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12)
+ (cid:12) S (k +n)1/2S (k)1/2 S (k) f (w (k)2) 2(cid:12) .
t t t n t
(cid:12) − (cid:12)
(cid:12)Xk∈S (cid:0) (cid:1)(cid:12) (cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
Since fn is continuous a(cid:12)nd hence bounded, we can estimate: (cid:12)
(w+)2
x 2 x 2 S (k) f (w (k)2) 2 f (r2) 2d(r2) +
t 0 t n t n
(cid:12)k k −k k (cid:12) ≤ (cid:12)(cid:12)Xk∈S (cid:12) (cid:12) −Z(w−)2 (cid:12) (cid:12) (cid:12)(cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) 1/2 (cid:12) (cid:12) (cid:12)
(cid:12) S (k +n) (cid:12)
+const S (k)(cid:12) t 1 . (cid:12)
t
S (k) −
(cid:12)(cid:12)Xk∈S "(cid:18) t (cid:19) #(cid:12)(cid:12)
(cid:12) (cid:12)
Using S (k) = w (k(cid:12))2 w (k 1)2 and Condition 3(cid:12), we see that the first term inside of
t t (cid:12) t (cid:12)
− −
the absolute value is a difference of a Riemann sum and the integral to which it converges
as t 0+. Hence this term is zero in the limit. As for the second term, since by 3.1,
→
S (k) = (w )2 (w )2 = const, Lemma 4.1 shows that it also goes to zero, because,
bykC∈Sontdition 4, h+(t)− 0−as t 0+. (cid:3)
2
P → →
We can now prove the first theorem.
Proof. (of Theorem 3.1) We have already verified that Λ satisfies some of the properties of
Proposition 2.2. What remains is the proof of the continuity of the norm for t > 0. Notice
that by Condition 2 all the terms in formula 4.1 are continuous in t, t > 0. Thus we need to
show that the series 4.1 converges uniformly in t (away from t = 0). Assuming again that
x (k) = Unf (w (k)2), and using the boundedness of f we have:
t n t n
M−1
x 2 S (k +n)1/2S (k)1/2 f (w (k)2) 2
t t t n t
k k − ≤
(cid:12) (cid:12)
(cid:12) k=XL+1 (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) const S (k +n)1/2S (k)1/2 +(cid:12)const S(cid:12) ((cid:12)k +n)1/2S (k)1/2.
(cid:12)≤ t t t(cid:12) t
k≥M k≤L
X X
10 SLAWOMIRKLIMEK AND MATT MCBRIDE
We use the Cauchy-Schwarz inequality to estimate the first term:
1/2 1/2
S (k +n)1/2S (k)1/2 S (k +n) S (k)
t t t t
≤ ≤
! !
kX≥M kX≥M kX≥M (4.3)
∞
S (k) = w2 w2(M).
≤ t + − t
k=M
X
The second term is only present in the annulus case and can be estimated in an analogous
way. By Condition 2 again, the difference w2 w2(M) is small for large M, uniformly in t
+ − t
on the intervals t ε > 0, and so, for t > 0, x is (locally) the uniform limit of continuous
t
≥ k k
functions and hence continuous. Therefore Λ generates a continuous field of Hilbert spaces
(I, ,Γ). (cid:3)
H
Our next concern is with the parametrices Q (k). To verify that they form a continuous
t
family of bounded operators in (I, ,Γ) we need to check that they are uniformly bounded
H
and that Q maps Γ into itself. We start with the former assertion.
Proposition 4.3. The norm of Q is uniformly bounded in t.
t
Proof. First we write Q x (k) in a more compact form:
t t
N N
Q x (k) = UnT(1,n)f (w (k)2)+ T(2,n)g (w (k)2)(U∗)n
t t − t n+1 t t n−1 t
n=0 n=1
X X
where
w (k +1) w (k +n) S (i)1/2S (i+n+1)1/2
(1,n) t t t t
T f(k) = ··· f(i)
t w (i+1) w (i+n) · w (k +n)
t t t
i≥k ···
X
w (i) w (i+n 1) S (i)1/2S (i+n 1)1/2
(2,n) t t t t
T g(k) = ··· − − g(i).
t w (k) w (k +n 1) · w (i+n 1)
t t t
i≤k ··· − −
X
HeretheoperatorsT(1,n) andT(2,n) areintegraloperatorsbetweenweightedl2 spaces, namely:
t t
T(1,n) : l2 l2 and T(2,n) : l2 l2 where
t n+1 7→ n t n−1 7→ n
l2 := f : S (k +n)1/2S (k)1/2 f(k) 2 <
n { t t | | ∞}
k∈S
X
The main technique used to estimate the norms will be the Schur-Young inequality: if
T : L2(Y) L2(X) is an integral operator Tf(x) = K(x,y)f(y)dy, then one has
−→
R
T 2 sup K(x,y) dy sup K(x,y) dx .
k k ≤ | | | |
(cid:18)x∈XZY (cid:19)(cid:18)y∈Y ZX (cid:19)
The details can be found in [6].
We will also use two integral estimates, with t independent right hand sides:
S (k) (w+)2 dx
t
= 2(w w (i)) 2(w w ), (4.4)
+ t + −
w (k) ≤ √x − ≤ −
i<k t Zwt(i)2
X