Pseudodifferential operators and weighted normed symbol spaces
Pseudodifferential operators and weighted normed
symbol spaces
J. Sjöstrand
Ecole Polytechnique
FR 91120 Palaiseau cédex, France
johannes@math.polytechnique.fr
UMR7640–CNRS
Abstract
In this work we study some general classes of pseudodifferential operators
where the classes of symbols are defined in terms of phase space estimates.
Résumé
On étudie des classes générales d’opérateurs pseudodifférentiels dont les
classes de symboles sont définis en termes d’éstimations dans l’espace de phase.
Keywords and Phrases: Pseudodifferential operator, symbol, modulation space.
Mathematics Subject Classification 2000: 35S05
Contents
1 Introduction 2
2 Symbol spaces 5
3 Effective kernels and L2-boundedness 7
4 Composition 12
5 More direct approach using Bargmann transforms 18
http://arxiv.org/abs/0704.1230v1
6 Cp classes 21
7 Further generalizations 24
1 Introduction
This paper is devoted to pseudodifferential operators with symbols of limited reg-
ularity. The author [28] introduced the space of symbols a(x) on the phase space
E = Rn × (Rn)∗ with the property that
|χ̂γa(x
∗)| ≤ F (x∗), ∀γ ∈ Γ (1.1)
for some L1 function F on E∗. Here the hat indicates that we take the Fourier
transform, Γ ⊂ E is a lattice and χγ(x) = χ0(x − γ) form a partition of unity,
γ∈Γ χγ, χ0 ∈ S(E). A. Boulkhemair [4] noticed that this space is identical to
a space that he had defined differently in [3].
It was shown among other things that this space of symbols is an algebra for
the ordinary multiplication and that this fact persists after quantization, namely
the corresponding pseudodifferential operators (say under Weyl quantization) form
a non-commutative algebra: If a1, a2 belong to the class above with corresponding
L1 functions F1 and F2 then a
1 ◦ a
2 = a
3 where a3 belongs to the same class and
as a correponding function we may take F3 = CNF1 ∗ F2 ∗ 〈·〉
−N for any N > 2n.
Here ∗ indicates convolution and aw : S(Rn) → S ′(Rn) is the Weyl quantization of
the symbol a, given by
awu(x) =
(2π)n
ei(x−y)·θa(
, θ)u(y)dydθ. (1.2)
The definition (1.1) is independent of the choice of lattice and the corresponding
function χ0. When passing to a different choice, we may have to change the function
F to m(x∗) = F ∗ 〈·〉−N0 for any fixed N0 > 2n. We then gain the fact that the
weight m is an order function in the sense that
m(x∗) ≤ C0〈x
∗ − y∗〉N0m(y∗), x∗, y∗ ∈ E∗. (1.3)
(See [11] where this notion is used for developing a fairly simple calculus of semi-
classical pseudodifferential operators, basically a special case of Hörmander’s Weyl
calculus [26].)
The space of functions in (1.1) is a special case of the modulation spaces of
H.G. Feichtinger (see [12, 14]), and the relations between these spaces and pseudod-
ifferential operators have been developed by many authors; K. Gröchenig [18, 19],
Gröchenig, T. Strohmer [22], K. Tachigawa [32], J. Toft [33], A. Holst, J. Toft, P.
Wahlberg [25]. Here we could mention that Boulkhemair [5] proved L2-continuity
for Fourier integral operators with symbols and phases in the original spaces of the
type (1.1), that T. Strohmer [31] has applied the theory to problems in mobile com-
munications and that Y. Morimoto and N. Lerner [27] have used the original space
to prove a version of the Fefferman-Phong inequality for pseudodifferential operators
with symbols of low regularity. This result was recently improved by Boulkhemair
Closely related works on pseudodifferential - and Fourier - integral operators with
symbols of limited regularity include the works of Boulkhemair [6, 7], and many
others also contain a study of when such operators or related Gabor localization
operators belong to to Schatten-von Neumann classes: E. Cordero, Gröchenig [9, 10],
C. Heil, J. Ramanathan, P. Topiwala [24], Heil [23], J. Toft [34], and M.W. Wong
[37].
The present work has been stimulated by these developments and the prospect
of using “modulation type weights” to get more flexibility in the calculus of pseu-
dodifferential operators with limited regularity. In the back of our head there were
also some very stimulating discussions with J.M. Bony and N. Lerner from the time
of the writing of [28, 29] and at that time Bony explained to the author a nice
very general point of view of A. Unterberger [36] for a direct microlocal analysis of
very general classes of operators. Bony used it in his work [1] and showed how his
approach could be applied to recover and generalize the space in [28]. However, the
aim of the work [1] was to develop a very general theory of Fourier integral operators
related to symplectic metrics of Hörmander’s Weyl calculus of pseudodifferential op-
erators, and the relation with [28] was explained very briefly. See [2] for even more
general classes of Fourier integral operators.
In the present paper we make a direct generalization of the spaces of [28]. Instead
of using order functions only depending on x∗ we can now allow arbitrary order
functions m(x, x∗). See Definition 2.1 below. In Proposition 2.4 we show that this
definition gives back the spaces above when the weight m(x∗) is an order function
of x∗ only.
In Section 3 we consider the quantization of our symbols and show how to define
an associated effective kernel on E ×E, E = T ∗Rn, which is O(1)m(γ(x, y)) where
γ(x, y) = (x+y
, J−1(y − x)) and J : E∗ → E is the natural Hamilton map induced
by the symplectic structure. We show that if the effective kernel is the kernel
of a bounded operator : L2(E) → L2(E) then our pseudodifferential operator is
bounded in L2(Rn). In particular if m = m(x∗) only depends on x∗, we recover the
L2-boundedness when m is integrable. This result was obtained previously by Bony
[1], but our approach is rather different.
In Section 4 we study the composition of pseudodifferential operators in our
classes. If aj are symbols associated to the order functions mj, j = 1, 2, then the
Weyl composition is a well defined symbol associated to the order function m3(z, z
given in (4.11), provided that the integral there converges for at least one value
of (z, z∗) (and then automatically for all other values by Proposition 4.1). This
statement is equivalent to the corresponding natural one for the effective kernels,
namely the composition is well defined if the composition of the majorant kernels
, J−1(y − x)) and m2(
, J−1(y − x)) is well-defined, see (4.16), (4.17).
In Section 5 we simplify the results further (for those readers who are familiar
with Bargmann transforms from the FBI - complex Fourier integral operator point
of view).
In Section 6 we use the same point of view to give a simple sufficient condition
on the order function m and the index p ∈ [1,∞], for the quantization aw to belong
to the Schatten–von Neumann class Cp for every symbol a belonging to the symbol
class with weight m. See [34, 35, 25, 20, 21] for related results and ideas.
In Section 7 we finally generalize our results by replacing the underlying space
ℓ∞ on certain lattices by more general translation invariant Banach spaces. We
believe that this generalization allows to include modulation spaces, but we have
contented ourselves by establishing results allowing to go from properties on the level
of lattices to the level of pseudodifferential operators. The results could undoubtedly
be even further generalized. In this section and the preceding one, we have been
inspired by the use of lattices and amalgan spaces in time frequency analysis, in
particular by the work of Gröchenig and Strohmer [22] that uses previous results by
Fournier–Stewart [15] and Feichtinger [13].
We have chosen to work with the Weyl quantization, but it is clear that the
results carry over with the obvious modifications to other quantizations like the
Kohn-Nirenberg one, actually for the general symbol-spaces under consideration the
results could also have been formulatated directly for classes of integral operators.
Similar ideas and results have been obtained in many other works, out of which
some are cited above and later in the text.
Acknowledgements. We thank J.M. Bony for a very stimulating and helpful re-
cent discussion. The author also thanks K. Gröchenig, T. Strohmer, A. Boulkhemair
and J. Toft for several helpful comments and references.
2 Symbol spaces
Let E be a d-dimensional real vector space. We say that m : E →]0,∞[ is an order
function on E if there exist constants C0 > 0, N0 ≥ 1, such that
m(ρ) ≤ C0〈ρ− µ〉
N0m(µ), ∀ρ, µ ∈ E. (2.1)
Here 〈ρ− µ〉 = (1 + |ρ− µ|2)1/2 and | | is a norm on E.
Let E be as above, let E∗ be the dual space and let Γ be a lattice in E ×E∗, so
that Γ = Ze1+Ze2+...+Ze2d where e1, ..., e2d is a basis in E×E
∗. Let χ ∈ S(E×E∗)
have the property that
τγχ = 1, τγχ(ρ) = χ(ρ− γ). (2.2)
Let m be an order function on E ×E∗, a ∈ S ′(E).
Definition 2.1 We say that a ∈ S̃(m) if there is a constant C > 0 such that
‖χwγ a‖ ≤ Cm(γ), γ ∈ Γ, (2.3)
where χγ = τγχ and χ
γ denotes the Weyl quantization of χγ . The norm will always
be the the one in L2 if nothing else is indicated.
To define the L2-norm we need to choose a Lebesgue measure on E, but clearly
that can only affect the choice of the constant in (2.3).
Proposition 2.2 S̃(m) is a Banach space with ‖a‖eS(m) equal to the smallest possible
constant in (2.3). Changing Γ, χ and replacing the L2 norm by the Lp-norm for any
p ∈ [1,∞] in the above definition, gives rise to the same space with an equivalent
norm.
Proof The Banach space property will follow from the other arguments so we do
not treat it explicitly. Let m,Γ, a be as in Definition 2.1.
Let Γ̃ be another lattice and let χ̃ be another function with the same properties
as χ. We have to show that
eγ a‖Lp ≤ C̃m(γ̃), γ̃ ∈ Γ̃.
Lemma 2.3 ∃ψ ∈ S(E × E∗) such that
γ∈Γ ψ
γ = 1, where ψγ = τγψ.
Proof Let χ̃ ∈ S(E × E∗) be equal to 1 near (0, 0), and put χ̃ǫ(x, ξ) = χ̃(ǫ(x, ξ)).
γ∈Γ(1− χ̃
γ)#χγ → 0 in S
0(E×E∗), when ǫ→ 0, so for ǫ > 0 small enough,
(χ̃ǫγ)
wχwγ = 1−
(1− χ̃ǫγ)
has a bounded inverse in L(L2, L2). Here S0 is the space of all a ∈ C∞(E × E∗)
that are bounded with all their derivatives. By a version of the Beals lemma (see
for instance [11]), we then know that the inverse is of the form Ψw where Ψ ∈ S0.
Also τγΨ = Ψ, γ ∈ Γ. Put ψ
γ = Ψ
w ◦ (χ̃ǫγ)
w for ǫ small enough and fixed, so that
ψγ = τγψ0, ψ0 ∈ S (using for instance the simple pseudodifferential calculus in [11]).
γ = 1. ✷
Now, write
eγ a =
Here (using for instance [11])
‖χ̃eγψ
γ ‖L(L2,Lp) ≤ Cp,N〈γ̃ − γ〉
−N , 1 ≤ p ≤ ∞, N ≥ 0.
Hence, if N is large enough,
eγ a‖Lp ≤ Cp,N
〈γ̃ − γ〉−N‖χwγ a‖L2 (2.4)
≤ C̃p,N,a
〈γ̃ − γ〉−Nm(γ)
≤ Ĉp,N,a,m(
〈γ̃ − γ〉−N+N0)m(γ̃)
≤ Čm(γ̃).
Conversely, if ‖χ̃w
eγ a‖Lp ≤ Constm(γ̃), γ̃ ∈ Γ̃, we see that by the same arguments
that ‖χwγ a‖L2 ≤ O(1)m(γ), γ ∈ Γ. ✷
Next, we check that this is essentially a generalization of a space introduced by
Sjöstrand [28] and independently and in a different way by Boukhemair [3]. It is a
special case of more general modulation spaces (see [12, 14]). That follows from the
next result if we take an order function m(x, x∗) independent of x.
Proposition 2.4 Let m = m(x, x∗) be an order function on E × E∗ and let χ ∈
S(E),
j∈J χj = 1, where J ⊂ E is a lattice and χj = τjχ. Then
S̃(m) = {a ∈ S ′(E); ∃C > 0, |χ̂ju(x
∗)| ≤ Cm(j, x∗)}. (2.5)
Proof Let K ⊂ E∗ be a lattice and choose χ∗ ∈ S(E∗), such that
k∈K χ
k = 1,
where χk = τkχ. If a belongs to the set in the right hand side of (2.5), then by
Parseval’s relation,
‖χ∗k(D)(χj(x)u(x))‖L2 ≤ C̃m(j, k). (2.6)
Now χ∗k(D) ◦ χj(x) = χ
j,k, where χj,k = τj,kχ0,0, χ0,0 ∈ S, (j, k) ∈ J × J
∗, so
a ∈ S̃(m). Conversely, if a ∈ S̃(m), we get (2.6). According to Proposition 2.2, we
can replace the L2 norm by any Lp norm, and the proof shows that we can equally
well replace the L2 norm that of FLp. Taking FL∞, we get
‖χ∗k(x
∗)χ̂ju(x
∗)‖L∞ ≤ Ĉm(j, k),
and since m is an order function, we deduce that a belongs to the set in the right
hand side of (2.5). ✷
3 Effective kernels and L2-boundedness
A closely related notion for effective kernels in terms of short time Fourier transforms
has been introduced by Gröchenig and Heil [20].
We now take E = R2n ≃ T ∗Rn. If a, b ∈ S(E), we let
a#b = (e
σ(Dx,Dy)a(x)b(y))y=x (3.1)
denote the Weyl composition so that (a#b)w = aw ◦ bw. Here σ(Dx,ξ, Dy,η) =
Dξ ·Dy −Dx ·Dη where we write (x, ξ), (y, η) instead of x, y whenever convenient.
We know that the Weyl composition is still well-defined when a, b belong to
various symbol spaces like
S(m) = {a ∈ C∞(E); |Dαxa(x)| ≤ Cαm(x)}, (3.2)
when m is an order function on E. (See Example 4.3 below for a straight forward
generalization.)
Let ℓ(x) = x · x∗ be a linear form on E and let a be a symbol. Then,
eiℓ#a = e
σ(Dx,Dy)(eiℓ(x)a(y))y=x (3.3)
= (eiℓ(x)e
σ(ℓ′(x),Dy)a(y))y=x
= eiℓ(x)(e
where Hℓ = ℓ
− ℓ′x ·
(with “x = (x, ξ)”) is the Hamilton field of ℓ. Similarly,
a#eiℓ = eiℓ(x)(e−
Hℓa). (3.4)
From (3.3), (3.4), we get
eiℓ#a#e−iℓ = eHℓa, (3.5)
where we notice that (eHℓa)(x) = a(x+Hℓ), and
2 #a#ei
2 = eima, (3.6)
if m is a second linear form on E.
If a ∈ S(E) is fixed, we may consider that a is concentrated near (0, 0) ∈ E×E∗.
Then we say that e−Hℓeima is concentrated near (Hℓ, m) ∈ E ×E
∗. Conversely, if b
is concentrated near a point (x0, x
0) ∈ E ×E
∗, we let y∗0 ∈ E
∗ be the unique vector
with x0 = Hy∗0 and write
b = e
0 eix
0a = e−iy
2 #a#ei
2 #eiy
0 , (3.7)
where a is concentrated near (0, 0) ∈ E × E∗.
To make this more precise, let (as in [30])
Tu = C
eiφ(x,y)u(y)dy, C > 0, (3.8)
be a generalized Bargmann transform where φ(x, y) is a quadratic form on Cn ×
Cn with detφ′′xy 6= 0, Imφ
yy > 0, and with C > 0 suitably chosen, so that T is
unitary L2(Rn) → HΦ(C
n) = Hol (Cn) ∩ L2(e−2Φ(x)L(dx)), where L(dx) denotes
the Lebesgue measure on Cn and Φ is the strictly plurisubharmonic quadratic form
given by
Φ(x) = sup
−Imφ(x, y). (3.9)
We know ([30]) that if ΛΦ = {(x,
); x ∈ Cn}, then
ΛΦ = κT (E), (3.10)
where
κT : C
2n ≃ EC ∋ (y,−φ′y(x, y)) → (x, φ
x(x, y)) ∈ C
2n (3.11)
is the linear canonical transformation associated to T . Here ∂
∂Re x
following standard conventions in complex analysis.
If a ∈ S0(E) we have an exact version of Egorov’s theorem, saying that
TawT−1 = ãw, (3.12)
where ã ∈ S0(ΛΦ) is given by ã ◦ κT = a. In [30] it is dicussed how to define and
estimate the Weyl quantization of symbols on the Bargmann transform side, by
means of almost holomorphic extensions and contour deformations. We retain from
the proof of Proposition 1.2 in that paper that
ãwu(x) =
eΦ(x)Keff
ea (x, y)u(y)e
−Φ(y)L(dy), u ∈ HΦ(C
n), (3.13)
where the kernel is non-unique but can be chosen to satisfy
ea (x, y) = ON (1)〈x− y〉
−N , (3.14)
for every N ≥ 0. (This immediately implies the Calderón-Vaillancourt theorem for
the class Op (S0(E)).)
If a ∈ S(E), then for every N ∈ N
|KeffTawT−1(x, y)| ≤ CN(a)〈x〉
−N〈y〉−N , x, y ∈ Cn, (3.15)
where CN(a) are seminorms in S.
Identifying x ∈ Cn with κ−1T (x,
) ∈ E, we can view Keff
TawT−1
as a function
Keffaw(x, y) on E × E and (3.15) becomes
|Keffaw(x, y)| ≤ CN(a)〈x〉
−N〈y〉−N , x, y ∈ E. (3.16)
Now, let b in (3.7) be concentrated near (x0, x
0) = (Jy
0) ∈ E × E
∗ with
a ∈ S(E), where we let J : E∗ → E be the map y∗ 7→ Hy∗ (and we shall prefer
to write Jy∗ when we do not think of this quantity as a constant coefficient vector
field). Then by (3.5)–(3.7), we have
b = e−iy
0#eix
0/2#a#eix
0/2#eiy
0 , (3.17)
bw = e−i(y
◦ ei(x
w/2 ◦ aw ◦ ei(x
w/2 ◦ ei(y
. (3.18)
Now it is wellknown that if z∗ ∈ E∗ then e−i(z
∗)w = (e−iz
)w is a unitary oper-
ator that can be viewed as a quantization of the phase space translation E ∋ x 7→
x+Hz∗ ∈ E. On the Bargmann transform side these quantizations can be explicitly
represented as magnetic translations, i.e. translations made unitary by multiplica-
tion by certain weights. In fact, let ℓ(x, ξ) = x∗0 · x+ x0 · ξ be a linear form on C
which is real on ΛΦ, so that
x∗0 · x+ x0 ·
(x) ∈ R, ∀x ∈ Cn. (3.19)
By essentially the same calculation as in the real setting, we see that
(eiℓ)wu(x) = eix
0·(x+
x0)u(x+ x0), u ∈ HΦ,
and here we recall from the unitary and metaplectic equivalence with L2(Rn) (via
T ) that (eiℓ)w : HΦ → HΦ is unitary, or equivalently that
− Φ(x) + Φ(x+ x0) + Re
ix∗0 · (x+
= 0, ∀x ∈ Cn. (3.20)
(A simple calculation shows more directly the equivalence of (3.19) and (3.20).)
Notice also that if we identify u with a function ũ(ρ) on ΛΦ via the natural projection
(x, ξ) 7→ x, then u(x+ x0) is identified with ũ(ρ+Hℓ), where the Hamilton field Hℓ
is viewed as a real constant vector field on ΛΦ.
It follows that bw has a kernel satisfying
|Keffbw (x, y)| = |K
aw(x+
Jx∗0 − x0, y −
Jx∗0 − x0)|
and from (3.16) we get
|Keffbw (x, y)| ≤ CN(a)〈x− (x0 −
Jx∗0)〉
−N〈y − (x0 +
Jx∗0)〉
−N , (3.21)
so the kernel of bw is concentrated near (x0 −
Jx∗0, x0 +
Jx∗0).
Now, let m be an order function on E × E∗ and let a ∈ S̃(m). Choose a lattice
Γ ⊂ E×E∗ and a partition of unity as in (2.2) as well as a function ψ ∈ S(E×E∗)
as in Lemma 2.3. Write
aγ, aγ = ψ
γ ãγ, ãγ = χ
γ a, (3.22)
where ‖ãγ‖ ≤ Cm(γ). Then, using that ψ
0 is continuous: L
2(E) → S(E), we see
that aγ is concentrated near γ in the above sense and more precisely,
|Keffaw(x, y)| ≤ CNm(γ)〈x− (γx−
Jγx∗)〉
−N〈y− (γx+
Jγx∗)〉
−N , x, y ∈ E, (3.23)
where we write γ = (γx, γx∗) ∈ E ×E
Let q(x, y) = (x+y
, J−1(y − x)) = (qx(x, y), qx∗(x, y)), so that
q−1(γ) = (γx −
Jγx, γx +
Jγx),
and hence
〈q(x, y)− γ〉 ≤ O(1)〈x− (γx −
Jγx∗)〉〈y − (γx +
Jγx∗)〉,
so (3.23) implies
|Keffawγ (x, y)| ≤ CN(a)m(γ)〈q(x, y)− γ〉
−N (3.24)
≤ C̃N(a)m(q(x, y))〈q(x, y)− γ〉
N0−N ,
where we used that m is an order function in the last inequality. Choose N with
N0 −N < −4n, sum over γ and use (3.22) to get
|Keffaw(x, y)| ≤ C(a)m(q(x, y)) = C(a)m(
, J−1(y − x)), x, y ∈ E. (3.25)
We get
Theorem 3.1 Let a ∈ S̃(m), where m is an order function on E × E∗, E =
T ∗Rn. Then aw has an effective kernel (rigorously defined after applying a Bargmann
transform as above) satisfying (3.25), where C(a) is a S̃(m) norm of a. In particular,
if M(x, y) = m(x+y
, J−1(y − x)) is the kernel of an L2(E)-bounded operator, then
aw is bounded: L2(Rn) → L2(Rn).
As mentioned in the introduction, the statement on L2-boundedness here is due
to Bony [1], who obtained it in a rather different way. A calculation, similar to the
one leading to (3.25), has been given by Gröchenig [18].
Corollary 3.2 If M is the kernel of a Shur class operator i.e. if
, J−1(y − x))dy, sup
, J−1(y − x))dx <∞,
then aw is bounded: L2(Rn) → L2(Rn).
Corollary 3.3 Assume m(x, x∗) = m(x∗) is independent of x, for (x, x∗) ∈ E×E∗
and m(x∗) ∈ L1(E∗), then aw is bounded: L2(Rn) → L2(Rn).
4 Composition
Let a, b ∈ S(E), E = Rn× (Rn)∗, (x0, x
0), (y0, y
0) ∈ E ×E
∗ and consider the Weyl
composition of the two symbols ex·x
0a(x − x0), e
x·y∗0b(x − y0) , concentrated near
(x0, x
0) and (y0, y
0) respectively:
σ(Dx,Dy)(ex·x
0a(x− x0)e
y·y∗0b(y − y0))(z, z). (4.1)
We work in canonical coordinates x ≃ (x, ξ) and identify E and E∗. Then
σ(x∗, y∗) = Jx∗ · y∗, J =
, tJ = −J, J2 = −1,
and e
σ(Dx,Dy) is convolution with k, given by
k(x, y) =
(2π)2n
ei(x·x
∗+y·y∗+ 1
Jx∗·y∗)dx∗dy∗.
The phase Φ = x · x∗ + y · y∗ + 1
Jx∗ · y∗ has a unique nondegenerate critical point
(x∗, y∗) = (2Jy,−2Jx) and the corresponding critical value is equal to −2σ(x, y) =
−2Jx · y. Hence k = Ce−2iσ(x,y) = Ce−2iJx·y for some (known) constant C.
The composition (4.1) becomes
ei(−2J(z−x)·(z−y)+x·x
0+y·y
0)a(x− x0)b(y − y0)dxdy = (4.2)
Ceiz·(x
ei(−2Jx·y+x·x
0+y·y
0)a(x+ z − x0)b(y + z − y0)dxdy.
The exponent in the last integral can be rewritten as
−2Jx · y + x · x∗0 + y · y
0 = −2J(x−
J−1y∗0) · (y +
J−1x∗0) +
Jx∗0 · y
and the composition (4.1) takes the form eiz·(x
0)d(z), where
d(z) = Ce
σ(x∗0 ,y
e−2iσ(x,y)a(x+ z − (x0 +
Jy∗0))b(y + z − (y0 −
Jx∗0))dxdy.
Since σ(x, y) is a nondegenerate quadratic form, we have for every N ≥ 0 by inte-
gration by parts,
|d(z)| ≤ CN
〈(x, y)〉−N〈x+ z − (x0 +
Jy∗0)〉
−N〈y + z − (y0 −
Jx∗0)〉
−Ndxdy.
Hence for every N ≥ 0,
|d(z)| ≤ CN〈z − (x0 +
Jy∗0)〉
−N〈z − (y0 −
Jx∗0)〉
Using the triangle inequality, we get
(1 + |z − a|)(1 + |z − b|) ≥ 1 + |z − a|+ |z − b| ≥ 1 +
|a− b|+ |z −
(1 + |z − a|)(1 + |z − b|) ≥
(1 + |a− b|)1/2(1 + |z −
|)1/2
and hence for every N ≥ 0,
|d(z)| ≤ CN〈(x0+
Jx∗0)− (y0−
Jy∗0)〉
−N〈z−
Jx∗0+y0+
Jy∗0)〉
−N . (4.3)
Clearly, we have the same estimates for the derivatives of d(z). It follows that the
composition (4.1) is equal to eiz·z
0c(z − z0), where
z∗0 = x
0 + y
0, z0 =
(x0 −
Jx∗0 + y0 +
Jy∗0), (4.4)
and where c ∈ S and for every seminorm p on S and every N , there is a seminorm
q on S such that
p(c) ≤ 〈(x0 +
Jx∗0)− (y0 −
Jy∗0)〉
−Nq(a)q(b). (4.5)
It follows that :
eiz·z
0c(z − z0) ∈ S̃(〈· − (z0, z
with corresponding norm bounded by
qN,M(a)qN,M(b)〈(x0 +
Jx∗0)− (y0 −
Jy∗0)〉
for all N,M ≥ 0 where qN,M are suitable seminorms on S.
If a1 ∈ S̃(m1), a2 ∈ S̃(m2) then c = a1#a2 is well-defined and belongs to S̃(m
provided that the integrals defining m
3 and m3 below converge. Here (replacing
summation over lattices by integration)
3 (z, z
〈z∗ − (x∗ + y∗)〉−N〈z −
Jx∗ + y +
Jy∗)〉−N(4.6)
×〈(x+
Jx∗)− (y −
Jy∗)〉−Nm1(x, x
∗)m2(y, y
∗)dxdydx∗dy∗
In order to understand the integral (4.6), we put x̃ = 1
Jx∗, ỹ = 1
Jy∗, z̃ = 1
and study the set Σ(z, z∗) where the arguments inside the three brackets vanish
simultaneously: 
x̃+ ỹ = z̃,
x+ y − x̃+ ỹ = 2z,
x− y + x̃+ ỹ = 0,
which can be transformed to
Σ(z, z∗) :
x̃− x = z̃ − z,
ỹ + y = z̃ + z,
x̃+ ỹ = z̃.
(4.7)
Now it is clear that for every M > 0 there is an N > 0 such that
3 (z, z
∗) ≤ O(1)
dist (x, x∗, y, y∗; Σ(z, z∗))−Mm1(x, x
∗)m2(y, y
∗)dxdydx∗dy∗.
(4.8)
Since m1, m2 are order functions, we have
m1(x, x
∗) ≤ O(1)dist (x, x∗, y, y∗; Σ(z, z∗))N0m1(Π
Σ (x, x
∗, y, y∗))
m2(y, y
∗) ≤ O(1)dist (x, x∗, y, y∗; Σ(z, z∗))N0m2(Π
Σ (x, x
∗, y, y∗)),
where ΠΣ : (E × E
∗)2 → Σ(z, z∗) is the affine orthogonal projection and we write
ΠΣ(x, x
∗; y, y∗) = (Π
Σ (x, x
∗; y, y∗),Π
Σ (x, x
∗; y, y∗)). We conclude that for N large
enough,
3 (z, z
∗) ≤ O(1)m3(z, z
∗), (4.9)
where
m3(z, z
Σ(z,z∗)
m1(x, x
∗)m2(y, y
∗)dΣ (4.10)
or more explicitly,
m3(z, z
Jx∗−x=1
Jz∗−z
Jy∗+y=1
Jz∗+z
x∗+y∗=z∗
m1(x, x
∗)m2(y, y
∗))dx. (4.11)
Reversing the above estimates, we see that m3(z, z
∗) ≤ O(1)m
3 (z, z
∗), if N > 0 is
large enough.
Proposition 4.1 If the integral in (4.10) converges for one value of (z, z∗), then it
converges for all values and defines an order function m3.
Proof Suppose the integral converges for the value (z, z∗) and consider any other
value (z + t, z∗ + t∗). We have the measure preserving map
Σ(z, z∗) ∋ (x, x∗, y, y∗) 7→ (x+ t, x∗ + t∗, y +
Jt∗ + t, y∗) ∈ Σ(z + t, z∗ + t∗),
m3(z + t, z
∗ + t∗) =
Σ(z,z∗)
m1(x+ t, x
∗ + t∗)m2(y +
Jt∗ + t, y∗)dx
≤ C〈(t, t∗)〉N0〈t+
t∗〉N0m3(z, z
≤ C̃〈(t, t∗)〉2N0m3(z, z
The proposition follows. ✷
From the above discussion, we get
Theorem 4.2 Let m1, m2 be order functions on E × E
∗ and define m3 by (4.11).
Assume that m3(z, z
∗) is finite for at least one (z, z∗) so that m3 is a well-defined
order function by Proposition 4.1. Then the composition map
S(E)× S(E) ∋ (a1, a2) 7→ a1#a2 ∈ S(E) (4.12)
has a bilinear extension
S̃(m1)× S̃(m2) ∋ (a1, a2) 7→ a1#a2 ∈ S̃(m3), (4.13)
Moreover,
‖a1#a2‖eS(m3) ≤ O(1)‖a1‖eS(m1)‖a2‖eS(m2). (4.14)
We end this section by establishing a connection with the effective kernels of
Section 3. Let aj be as in the theorem with a3 = a1#a2. According to Theorem 3.1,
we then know that awj has an effective kernel Kj = K
(x, y) satisfying
Kj(x, y) = O(1)mj(q(x, y)), where q(x, y) = (
, J−1(y − x)). (4.15)
Since the composition of the effective kernels of aw1 and a
2 is an effective kernel for
aw3 = a
1 ◦ a
2 we expect that
m3(q(x̃, ỹ)) = C
m1(q(x̃, z̃))m2(q(z̃, ỹ))dz̃, (4.16)
or more explicitly,
x̃+ ỹ
, J−1(ỹ − x̃)) = C
x̃+ z̃
, J−1(z̃ − x̃))m2(
z̃ + ỹ
, J−1(ỹ − z̃))dz̃,
(4.17)
Writing
x̃+ ỹ
z∗ = J−1(ỹ − x̃),
x̃+ z̃
x∗ = J−1(z̃ − x̃),
z̃ + ỹ
y∗ = J−1(ỹ − z̃),
we check that the integral in (4.17) coincides with the one in (4.11) up to a constant
Jacobian factor, so the results of this section fit with the ones of Section 3.
Example 4.3 Let aj ∈ S̃(mj), j = 1, 2, where mj are order functions on E×E
the form
mj(x, x
∗) = m̃j(x)〈x
∗〉−Nj , Nj ∈ R,
m̃j(x) ≤ C〈x− y〉
Mjm̃j(y), x, y ∈ E, Mj ≥ 0.
Then, the effective kernels K1, K2 of a
1 , a
2 satisfy
Kj(x, y) = O(1)mj(
, J−1(y − x)) = O(1)m̃j(
)〈x− y〉−Nj .
Then a1#a2 is well-defined and belongs to S̃(m3), where
, J−1(y − x)) =
)〈x− z〉−N1〈z − y〉−N2m̃2(
z + y
provided that the last integral converges for at least one (and then all) value(s) of
((x+ y)/2, J−1(y − x)). If we use that
) ≤ O(1)m̃1(
)〈z − y〉M1
z + y
) ≤ O(1)m̃2(
)〈x− z〉M2 ,
we get
, J−1(y−x)) ≤ O(1)m̃1(
)m̃2(
〈x−z〉−N1+M2〈z−y〉−N2+M1dz.
(4.18)
Thus m3 and a1#a2 ∈ S̃(m3) are well-defined if
− (N1 +N2) +M1 +M2 < −2n. (4.19)
The integral I in (4.18) is O(1) in any region where x− y = O(1). For |x− y| ≥ 2,
we write I ≤ I1 + I2 + I3, where
• I1 is the integral over |x− z| ≤
|x− y|. Here 〈z − y〉 ∽ 〈x− y〉.
• I2 is the integral over |z − y| ≤
|x− y|. Here 〈x− z〉 ∽ 〈x− y〉.
• I3 is the integral over |x−z|, |z−y| ≥
|x−y|. Here 〈x−z〉 ∽ 〈y−z〉 ≥ 1
〈x−y〉.
We get
I1 ∽ 〈x− y〉
−N2+M1
∫ 〈x−y〉
〈r〉−N1+M2+2n−1dr ∽ 〈x− y〉−N2+M1+(−N1+M2+2n)+ ,
with the convention that we tacitly add a factor ln〈x−y〉 when the expression inside
(..)+ is equal to 0. Similarly (with the same convention),
I2 ∽ 〈x− y〉
−N1+M2+(−N2+M1+2n)+ .
In view of (4.19), we have
〈x−y〉
r−(N1+N2)+M1+M2+2n−1dr ∽ 〈x− y〉−(N1+N2)+M1+M2+2n.
it follows that
I ∽ 〈x− y〉max(−N2+M1+(−N1+M2+2n)+,−N1+M2+(−N2+M1+2n)+), (4.20)
so with the same convention, we have
m3(x, x
∗) ≤ O(1)m̃1(x)m̃2(x)〈x
∗〉max(−N2+M1+(−N1+M2+2n)+,−N1+M2+(−N2+M1+2n)+).
(4.21)
This simplifies to
m3(x, x
∗) ≤ O(1)m̃1(x)m̃2(x)〈x
∗〉max(−N2+M1,−N1+M2) (4.22)
if we strengthen the assumption (4.19) to:
−N1 +M2, −N2 +M1 < −2n. (4.23)
5 More direct approach using Bargmann trans-
forms
By using Bargmann transforms more systematically (from the point of view of
Fourier integral operators with complex phase) the results of Section 3, 4 can be
obtained more directly. The price to pay however, is the loss of some aspects that
might be helpful in other situations like the ones with variable metrics.
Let F be real d-dimensional space as in Section 2 and define T : L2(F ) →
C) as in (3.8)–(3.11). Then we have
Proposition 5.1 If m is an order function on F × F ∗, then
S̃(m) = {u ∈ S ′(F ); e−Φ(x)|Tu(x)| ≤ Cm(κ−1T (x,
(x)))}, (5.1)
where the best constant C = C(m) is a norm on S̃(m).
Proof Assume first that u belongs to S̃(m) and write u =
γ∈Γ ψ
γ u as in
Lemma 2.3. The effective kernel of ψwγ satisfies
|Keffψwγ (x, y)| ≤ CN〈x− γ〉
−N〈y − γ〉−N , (5.2)
for every N > 0, where throughout the proof we identify FC with F ×F ∗ by means
of π ◦κT and work on the latter space. Here π : ΛΦ → F
C is the natural projection.
Then we see that
|e−Φ/hTu(x)| ≤ CN(u)
m(γ)〈x− γ〉−N = O(m(x)).
Conversely, if e−Φ/hTu = O(m(x)), then since the effective kernel of χwγ also satis-
fies (5.2), we see that e−Φ/hTχwγ u = ON(〈x−γ〉
−Nm(γ)), implying ‖e−Φ/hTχwγ u‖L2 =
O(m(γ)), and hence ‖χwγ u‖ = O(m(γ)). ✷
With this in mind, we now take a ∈ S̃(Rn × (Rn)∗;m) and look for an explicit
choice of effective kernel for aw. Let T : L2(Rn) → HΦ(C
n) be a Bargmann trans-
form as above. Consider first the map a 7→ Kaw(x, y) ∈ S
′(Rn ×Rn) from a to the
distribution kernel of aw, given by
Kaw(x, y) =
(2π)n
ei(x−y)·τa(
, τ)dτ (5.3)
(2π)2n
ei(x−y)·τ+i(
−t)·sa(t, τ)dtdsdτ.
We view this as a Fourier integral operator B : a 7→ Kaw(x, y) with quadratic phase.
The associated linear canonical transformation is given by:
κB : (t, τ ; t
∗, τ ∗) = (
, τ ; s, y − x) 7→ (x, τ +
; y,−τ +
) = (x, x∗; y, y∗),
which we can write as
κB : (t, τ ; t
∗, τ ∗) 7→ (t−
, τ +
,−τ +
). (5.4)
From the unitarity of T , we know that T ∗T = 1, where
T ∗v(y) = C
e−iφ(x,y)v(x)e−2Φ(x)L(dx). (5.5)
We can therefore define the effective kernel of aw to be
Keff(x, y) = e−Φ(x)K(x, y)e−Φ(y), (5.6)
where
TawT ∗v(x) =
K(x, y)v(y)e−2Φ(y)L(dy), v ∈ HΦ(C
n), (5.7)
K(x, y) = C2
ei(φ(x,t)−φ(y,s))Kaw(t, s)dtds.
We write this as
K(x, y) = C2
ei(φ(x,t)−φ
∗(y,s))Kaw(t, s)dtds,
with φ∗(y, s) = φ(y, s), so
K(x, y) = (T ⊗ T̃ )(Kaw)(x, y), (5.8)
where
(T̃ u)(y) = C
∗(y,s)u(s)ds = (Tu)(y). (5.9)
We see that T̃ : L2(Rn) → HΦ∗(C
n) is a unitary Bargmann transform, where
Φ∗(y) = sup
Imφ∗(y, s) = sup
Imφ(y, s) = Φ(y). (5.10)
The canonical transformation associated to T̃ is
κ eT : (s,
(y, s)) 7→ (y,−
(y, s)). (5.11)
ι(s, σ) = (s,−σ), (5.12)
we check that
κ eT = ικT ι, ι : (x,
(x)) 7→ (x,
(x)). (5.13)
Clearly T⊗ T̃ is a Bargmann transform with associated canonical transformation
κT × (ικT ι), so in view of (5.4) the map a 7→ K is also a Bargmann transform with
associated canonical transformation
(E×E∗)C ∋ (t, τ ; t∗, τ ∗) 7→ (κT ((t, τ)−
J(t∗, τ ∗)), ικT (((t, τ)+
J(t∗, τ ∗))), (5.14)
where E = Rn × (Rn)∗. The restriction to the real phase space is
E ×E∗ ∋ (t, τ ; t∗, τ ∗) 7→ (5.15)
(κT ((t, τ)−
J(t∗, τ ∗)), ικT (((t, τ) +
(t∗, τ ∗))) ∈ ΛΦ × ιΛΦ = ΛΦ × ΛΦ∗ ,
and this restriction determines our complex linear canonical transformation uniquely.
As in Section 3 we may view the effective kernel Keff(x, y) in (5.6) as a function
on E×E, by identifying x, y ∈ Cn with κ−1T (x,
(x)), κ−1T (y,
(y)) ∈ E respec-
tively. With this identification and using also the general characterization in (5.1)
(with T replaced by T ⊗ T̃ )), we see that if a ∈ S ′(E), then a ∈ S̃(m) iff
Keff(t−
Jt∗, t+
Jt∗) = O(1)m(t, t∗), (t, t∗) ∈ E × E∗, (5.16)
where we shortened the notation by writing t instead of (t, τ) and t∗ instead of
(t∗, τ ∗).
Theorem 3.1 now follows from (5.16), (5.6), (5.7).
Theorem 4.2 also follows from (5.16), (5.6), (5.7) together with the remark that
the kernel K(x, y) = Ka(x, y) is the unique kernel which is holomorphic on C
n×Cn,
such that the corresponding Keffaw given in (5.6) is of temperate growth at infinity
and (5.7) is fulfilled. Indeed, then it is clear that
Keff(a1#2)w(x, y) =
Keffaw1
(x, z)Keffaw2
(z, y)L(dz) (5.17)
and the bound (5.16) for a1#a2 withm = m3 follows directly from the corresponding
bounds for aj with m = mj.
6 Cp classes
In this section we give a simple condition on an order function m on E × E∗ (E =
T ∗Rn) and a number p ∈ [1,∞] that implies the property:
∃C > 0 such that: a ∈ S̃(m) ⇒ aw ∈ Cp(L
2, L2) and ‖aw‖Cp ≤ C‖a‖eS(m). (6.1)
Here Cp(L
2, L2) is the Schatten–von Neumann class of operators: L2(Rn) → L2(Rn),
see for instance [16].
Letm be an order function on E×E∗ and let p ∈ [1,+∞]. Consider the following
property, where q is given in (4.15) and Γ ⊂ E is a lattice,
∃C > 0 such that if |aα,β| ≤ m(q(α, β)), α, β ∈ Γ, (6.2)
then (aα,β)α,β∈Γ ∈ Cp(ℓ
2(Γ), ℓ2(Γ)) and ‖(aα,β)‖Cp ≤ C.
Notice that if (6.2) holds and if we fix some number N0 ∈ N
∗, then if (Aα,β)α,β∈Γ is
a block matrix where every Aα,β is an N0 ×N0 matrix then
same as (6.2) with aα,β replaced by Aα,β and | · | by ‖ · ‖L(CN0 ,CN0 ). (6.3)
Proposition 6.1 The property (6.2) only depends on m, p but not on the choice of
Proof Let m, p,Γ satisfy (6.2) and let Γ̃ be a second lattice in E. Let (a
eα,eβ
) be a
Γ̃ × Γ̃ matrix satisfying |a
eα,eβ
| ≤ m(q(α̃, β̃)). Let π(α̃) ∈ Γ be a point that realizes
the distance from α̃ to Γ, so that |π(α̃) − α̃| ≤ C0 for some constant C0 > 0. Let
N0 = max#π
−1(α) and choose an enumeration π−1(α) = {α̃1, ..., α̃N(α)}, N(α) ≤
N0, for every α ∈ Γ. Then we can identify (aeα,eβ)eΓ×eΓ with the matrix (Aα,β)α,β∈Γ×Γ
where Aα,β is the N0 ×N0 matrix with the entries
(Aα,β)j,k =
eαj ,eβk
, if 1 ≤ j ≤ N(α), 1 ≤ k ≤ N(β),
0, otherwise.
Then ‖Aα,β‖ ≤ Cm(q(α, β)) and we can apply (6.3) to conclude. ✷
Theorem 6.2 Let m be an order function and p ∈ [1,∞]. If (6.2) holds, then we
have (6.1).
Proof Assume that (6.2) holds and let a ∈ S̃(m). Define K(x, y) as in (5.7). It suf-
fices to estimate the Cp norm of the operator A : L
2(e−2ΦL(dx)) → L2(e−2ΦL(dx)),
given by
Au(x) =
K(x, y)u(y)e−2Φ(y)L(dy),
or equivalently the one of Aeff : L
2(Cn) → L2(Cn), given by
Aeffu(x) =
Keff(x, y)u(y)L(dy), (6.4)
with Keff given in (5.6). Recall that Keff(x, y) = O(1)m(q(x, y)) (identifying Cn
with T ∗Rn via πx ◦ κT ), so K(x, y) = O(1)m(q(x, y))e
Φ(x)+Φ(y).
For α, β ∈ Γ we have (identifying Γ with a lattice in Cn)
K(x, y) = eFα(x−α)K̃α,β(x, y)e
Fβ(y−β), (6.5)
where
Fα(x− α) = Φ(α) + 2
(α) · (x− α) (6.6)
is holomorphic with
Re Fα(x− α) = Φ(x) +Rα(x− α), Rα(x− α) = O(|x− α|
2), (6.7)
|∇kx∇
yK̃α,β(x, y)| ≤ C̃k,ℓm(q(α, β)), |x− α|, |y − β| ≤ C0. (6.8)
Here we identify α, β ∈ E with their images πxκT (α), πxκT (β) ∈ C
n respectively. In
fact, the case k = ℓ = 0 is clear and we get the extension to arbitrary k, ℓ from the
Cauchy inequalities, since K̃α,β is holomorphic.
We can also write
Keff(x, y) = eiGα(x−α)Kα,β(x, y)e
−iGβ(y−β), (6.9)
where
Gα(x− α) = ImFα(x− α), Kα,β = e
Rα(x−α)K̃α,β(x, y)e
Rβ(y−β),
|∇kx∇
yKα,β(x, y)| ≤ Ck,ℓm(q(α, β)), |x− α|, |y − β| ≤ C0. (6.10)
Consider a partition of unity
χα(x), χα(x) = χ0(x− α), χ0 ∈ C
0 (Ω0;R), (6.11)
where Ω0 is open with smooth boundary. Let Ωα = Ω0 + α, so that (6.10) holds for
(x, y) ∈ Ωα × Ωβ.
Let W : L2(Cn) →
β∈Γ L
2(Ωβ) be defined by
(e−iGβ(x−β)u(x))|Ωβ
so that the adjoint of W is given by
W ∗v =
eiGα(x−α)vα(x)1Ωα(x), v = (vα)α∈Γ ∈
L2(Ωα).
Then W and its adjoint are bounded operators and
Aeff =W
∗AW, (6.12)
where A = (Aα,β)α,β∈Γ and Aeff : L
2(Cn) → L2(Cn), Aα,β : L
2(Ωβ) → L
2(Ωα) are
given by the kernels Keff(x, y) and χα(x)Kα,β(x, y)χβ(y) respectively. It now suffices
to show that
L2(Ωβ) →
L2(Ωβ)
belongs to Cp with a norm that is bounded by a constant times the S̃(m)-norm of
Let e0, e1, .. ∈ L
2(Ω0) be an orthonormal basis of eigenfunctions of minus the
Dirichlet Laplacian in Ω0, arranged so that the corresponding eigenvalues form an
increasing sequence. Then eα,j := ταej , j = 0, 1, ... form an orthonormal basis of
eigenfunctions of the corresponding operator in L2(Ωα). From (6.10) it follows that
the matrix elements Kα,j;β,k of Aα,β with respect to the bases (eα,·) and (eβ,·) satisfy
|Kα,j;β,k| ≤ CNm(q(α, β))〈j〉
−N〈k〉−N , (6.13)
for every N ∈ N. We notice that (Kα,j;β,k)(α,j),(β,k)∈Γ×N is the matrix of A with
respect to the orthonormal basis (eα,j)(α,j)∈Γ×N. We can represent this matrix as a
block matrix (Kj,k)j,k∈N, where K
j,k : ℓ2(Γ) → ℓ2(Γ) has the matrix (Kα,j;β,k)α,β∈Γ.
Since (6.2) holds and a ∈ S̃(m), we deduce from (6.13) that
‖Kj,k‖Cp ≤ C̃N〈j〉
−N〈k〉−N . (6.14)
Choosing N > 2n, we get
‖A‖Cp ≤
‖Kj,k‖Cp <∞. (6.15)
Hence aw ∈ Cp and the uniform bound ‖a
w‖Cp ≤ ‖a‖eS(m) also follows from the
proof. ✷
Example 6.3 Assume that
‖m(·, x∗)‖Lp(E)dx
∗ <∞. (6.16)
Then (
m(q(α, β))
α,β∈Γ
α + β
, J−1(β − α))
α,β∈Γ
(6.17)
is a matrix where each translated diagonal {(α, β) ∈ Γ × Γ; α − β = δ} has an ℓp
norm which is summable with respect to δ ∈ Γ. Now a matrix with non-vanishing
elements in only one translated diagonal has a Cp norm equal to the ℓ
p norm of that
diagonal, so we conclude that the Cp norm of the matrix in (6.17) is bounded by
, δ)‖ℓp <∞.
We clearly have the same conclusion for every matrix (aα,β)α,β∈Γ satisfying |aα,β| ≤
m(q(α, β)), so (6.2) holds and hence by Theorem 6.2 we have the property (6.1).
7 Further generalizations
Let E be a d-dimensional real vector space and let Γ ⊂ E be a lattice. We shall
extend the preceding results by replacing the ℓ∞(Γ)-norm in the definition of the
symbol spaces by a more general Banach space norm. Let B be a Banach space of
functions u : Γ → C with the following properties:
If u ∈ B, γ ∈ Γ, then τγu ∈ B, and ‖τγu‖B = ‖u‖B. (7.1)
δγ ∈ B, ∀γ ∈ Γ, (7.2)
where τγu(α) = u(α − γ), δγ(α) = δγ,α, α ∈ Γ. (The last assumption will soon be
replaced by a stronger one.)
If u =
γ∈Γ u(γ)δγ ∈ B, we get
‖u‖B ≤
|u(γ)|‖δγ‖B = C‖u‖ℓ1,
where C = ‖δγ‖B (is independent of γ). Thus
ℓ1(Γ) ⊂ B. (7.3)
We need to strengthen (7.2) to the following assumption:
If u ∈ B and v : Γ → C satisfies |v(γ)| ≤ |u(γ)|, ∀γ ∈ Γ, (7.4)
then v ∈ B and ‖v‖B ≤ C‖u‖B, where C is independent of u, v.
It follows that ‖u(γ)δγ‖B ≤ C‖u‖B, for all u ∈ B, γ ∈ Γ, or equivalently that
|u(γ)| ≤
‖δγ‖B
‖u‖B = C̃‖u‖B,
B ⊂ ℓ∞(Γ), and ‖u‖ℓ∞ ≤ C̃‖u‖B, ∀u ∈ B. (7.5)
If f ∈ ℓ1(Γ) then using only the translation invariance (7.1), we get
u ∈ B ⇒
f ∗ u ∈ B,
‖f ∗ u‖B ≤ ‖f‖ℓ1‖u‖B.
(7.6)
Using also (7.4) we get the following partial strengthening: Let k : Γ × Γ → Γ
satisfy |k(α, β)| ≤ f(α− β) where f ∈ ℓ1(Γ). Then
u ∈ B ⇒ v(α) :=
k(α, β)u(β) ∈ B and ‖v‖B ≤ C‖f‖ℓ1‖u‖B, (7.7)
where C is independent of k, u. In fact,
u ∈ B ⇒ |u| ∈ B ⇒ f ∗ |u| ∈ B,
and v in (7.7) satisfies |v| ≤ f ∗ |u| pointwise.
Let Γ̃ ⊂ E be a second lattice and let B̃ ⊂ ℓ∞(Γ̃) satisfy (7.1), (7.4). We say
that B ≺ B̃ if the following property holds for some N > d:
If u ∈ B and ũ : Γ̃ → C satisfies |ũ(γ̃)| ≤
〈γ̃ − γ〉−N |u(γ)|, γ̃ ∈ Γ̃, (7.8)
then ũ ∈ B̃ and ‖ũ‖ eB ≤ C‖u‖B, where C is independent of u, ũ.
If (7.8) holds for one N > d and M > d then it also holds with N replaced by
M . This is obvious whenM ≥ N and if d < M < N , it follows from the observation
〈γ̃ − γ〉−M ≤ CN,M
eβ∈eΓ
〈γ̃ − β̃〉−M〈β̃ − γ〉−N
(cf. (4.20), where I is the integral in (4.18), 2n is replaced by d, and we take M1 =
M2 = 0), which allows us to write
〈γ̃ − γ〉−M |u(γ)| ≤ CN,M〈·〉
−M ∗ v,
where v(β) :=
γ〈β̃ − γ〉
−N |u(γ)| and v belongs to B̃ since (7.8) holds.
Definition 7.1 Let Γ, Γ̃ be two lattices in E and let B, B̃ be Banach spaces of
functions on Γ and Γ̃ respectively, satisfying (7.1), (7.4). Then we say that B ≡ B̃,
if B ≺ B̃ and B̃ ≺ B. Notice that this is an equivalence relation.
We can now introduce our generalized symbol spaces. With E ≃ Rd as above,
let Γ ⊂ E ×E∗ be a lattice and B ⊂ ℓ∞ a Banach space satisfying (7.1), (7.4). Let
a ∈ S ′(E).
Definition 7.2 We say that a ∈ S̃(m,B) if the function
Γ ∋ γ 7→
‖χwγ a‖
belongs to B. Here χγ is the partiction of unity (2.2).
Proposition 2.2 extends to
Proposition 7.3 S̃(m,B) is a Banach space with the natural norm. If we replace
Γ, χ, B by Γ̃, χ̃, B̃, having the same properties, and with B̃ ⊂ ℓ∞(Γ̃) equivalent to B,
and if we further replace the L2 norm by the Lp norm for any p ∈ [1,∞], we get the
same space, equipped with an equivalent norm.
Proof It suffices to follow the proof of Proposition 2.2: From the estimate (2.4) we
get for any N ≥ 0,
m(γ̃)
eγ a‖Lp ≤ Cp,N
〈γ̃ − γ〉−n
‖χwγ a‖L2 ,
where we also used that m is an order function. Hence, since B, B̃ are equivalent,
‖χ̃wa · ‖Lp‖ eB ≤ ‖
‖χw· a‖L2‖B.
The reverse estimate is obtained the same way. ✷
As a preparation for the use of Bargmann transforms, we next develop a “con-
tinuous” version of B-spaces; a kind of amalgam spaces in the sense of [22, 13, 15].
Let Γ be a lattice in a d-dimensional real vector space E and let B ⊂ ℓ∞(Γ) satisfy
(7.1), (7.4). Let 0 ≤ χ ∈ C∞0 (E) satisfy
γ∈Γ τγχ > 0.
Definition 7.4 We say that the locally bounded measurable function u : E → C
is of class [B], if there exists v ∈ B such that
|u(x)| ≤
v(γ)τγχ(x). (7.9)
The space of such functions is a Banach space that we shall denote by [B],
equipped with the norm
‖u‖[B] = inf{‖v‖B; (7.9) holds }. (7.10)
This space does not depend on the choice of χ and we may actually characterize it
as the space of all locally bounded measurable functions u on E such that
|u(x)| ≤
w(γ)〈x− γ〉−N , for some w ∈ B, (7.11)
where N > d is any fixed number. Clearly (7.8) implies (7.11). Conversely, if u
satisfies (7.11) and χ is as in Definition 7.4, then
〈x〉−N ≤ C
〈α〉−Nταχ(x),
so if (7.11) holds, we have,
|u(x)| ≤ C
〈α〉−Nχ(x− (γ + α))
(〈·〉−N ∗ w)(β)χ(x− β),
and 〈·〉−N ∗ w ∈ B.
Similarly, the definition does not change if we replace B ⊂ ℓ∞(Γ) by an equivalent
space B̃ ⊂ ℓ∞(Γ̃).
Let m1, m2, m3 be order functions on E1 × E2, E2 × E3, E1 × E3 respectively,
where Ej is a real vectorspace of dimension dj . Let Γj ⊂ Ej be lattices and let
B1 ⊂ ℓ
∞(Γ1 × Γ2), B2 ⊂ ℓ
∞(Γ2 × Γ3), B3 ⊂ ℓ
∞(Γ1 × Γ3)
be Banach spaces satisfying (7.1), (7.4). Introduce the
Assumption 7.5 If kj ∈ mjBj , j = 1, 2, then
k3(α, β) :=
k1(α, γ)k2(γ, β)
converges absolutely for every (α, β) ∈ Γ1 × Γ3. Moreover, k3 ∈ m3B3 and
‖k3/m3‖B3 ≤ C‖k1/m1‖B1‖k2/m2‖B2
where C is independent of k1, k2.
Again, it is an easy exercise to check that the assumption is invariant under
changes of the lattices Γj and the passage to corresponding equivalent B-spaces.
Proposition 7.6 We make the Assumption 7.5, where Bj satisfy (7.1), (7.4). Let
Kj ∈ mj[Bj ] for j = 1, 2 in the sense that Kj/mj ∈ [Bj]. Then the integral
K3(x, y) :=
K1(x, z)K2(z, y)dz, (x, y) ∈ E1 ×E3,
converges absolutely and defines a function K3 ∈ m3[B3]. Moreover,
‖K3/m3‖[B3] ≤ C‖K1/m1‖[B1]‖K2/m2‖[B2],
where C is independent of K1, K2.
Proof Write
|K1(x, z)| ≤
Γ1×Γ2
k1(α, γ)χ
(1)(x− α, z − γ)
|K2(z, y)| ≤
Γ2×Γ3
k2(γ, β)χ
(2)(z − γ, y − β),
with χ(1) ∈ C∞0 (E1 × E2), χ
(2) ∈ C∞0 (E2 × E3) as in Definition 7.4 and with kj ∈
mjBj . Then
|K3(x, y) ≤
|K1(x, z)||K2(z, y)|dz
(α,β)∈Γ1×Γ3
γ,γ′∈Γ2
k1(α, γ)k2(γ
′, β)F (x− α, y − β; γ − γ′),
where
F (x, y; γ − γ′) =
χ(1)(x, z − γ)χ(2)(z − γ′, y)dz
χ(1)(x, z − (γ − γ′))χ(2)(z, y)dz.
We notice that 0 ≤ F (x, y; γ) ∈ C∞0 (E1 × E3) and that F (x, y; γ) 6≡ 0 only for
finitely many γ ∈ Γ. Hence for some R0 > 0,
|K3(x, y)| ≤
|γ|≤R0
(α,β)∈Γ1×Γ3
k1(α, γ
′ + γ)k2(γ
′, β)
F (x− α, y − β; γ).
Since
m1(·, ··)
k1(·, · ·+γ) ∈ B1,
for every fixed γ, and k2/m2 ∈ B2, the assumption 7.5 implies that
k3(α, β; γ) :=
k1(α, γ
′ + γ)k2(γ
′, β) ∈ m3B3,
for every γ ∈ Γ.
The proposition follows. ✷
We next generalize (5.1). Let F = Rd and define T : L2(F ) → HΦ(F
C) as in
(3.8)–(3.11). Let m be an order function on F × F ∗, let Γ ⊂ F × F ∗ be a lattice
and let B ⊂ ℓ∞(Γ) satisfy (7.1), (7.4). Then we get
Proposition 7.7 we have
S̃(m,B) = {u ∈ S ′(F );
(e−ΦTu) ◦ π ◦ κT
∈ [B]}, (7.12)
where π : ΛΦ ∋ (x, ξ) 7→ x ∈ F
C is the natural projection.
Proof This will be a simple extension of the proof of (5.1). As there, we identify
FC with F ×F ∗ by means of π ◦κT and work on the latter space. Assume first that
u ∈ S̃(m,B) and write u =
γ∈Γ ψ
γ u as in Lemma 2.3, so that (‖χ
γ u‖)γ∈Γ ∈
mB. Using (5.2), we see that
|e−Φ/hTu(x)| ≤ CN
‖χwγ u‖〈x− γ〉
and hence e−ΦTu ∈ m[B], i.e. u belongs to the right hand side of (7.12) (with the
identification π ◦ κT ).
Conversely, if e−ΦTu ∈ m[B], then since the effective kernel of χwγ satisfies (5.2),
we see that
|e−ΦTχwγ u(x)| ≤ CN
〈x− γ〉−N〈y − γ〉−N
〈y − α〉−Naαdy,
where (aα) ∈ mB. It follows that
|e−ΦTχwγ u(x)| ≤ C̃N〈x− γ〉
〈γ − α〉−Naα = C̃N〈x− γ〉
−Nbγ ,
where (bγ)γ∈Γ ∈ mB, and hence ‖χ
γ u‖ ≤ ĈNbγ, so u ∈ S̃(m,B). ✷
From this, we deduce as in (5.16) that if a ∈ S ′(E), E = F×F ∗, then a ∈ S̃(m,B)
Keffaw (t−
Jt∗, t+
Jt∗) ∈ m[B], (7.13)
where Keffaw is the effective kernel of a
w in (5.6), (5.7) after identification of Cd = FC
with E via the map π ◦ κT = E → F
C. We recall the identity (5.17) for the
composition of two symbols.
(7.13) can also be written
Keffaw(x, y) ∈ m̃[B̃], where m̃ = m ◦ q, [B̃] = [B] ◦ q, (7.14)
where q is given in (4.15).
The following generalization of Theorem 4.2 now follows from Proposition 7.6.
Theorem 7.8 For j = 1, 2, 3, let mj be an order function E × E
∗, where E =
Rn × (Rn)∗, let Γj ⊂ E × E
∗ be a lattice and let Bj ⊂ ℓ
∞(Γj) satisfy (7.1), (7.4).
Let m̃j = mj ◦ q, Γ̃j = q
−1(Γj), ℓ
∞(Γ̃j) ⊃ B̃j = Bj ◦ q. Assuming (as we may
without loss of generality) that Γ̃j = Γ × Γ where Γ ⊂ E is a lattice, we make the
Assumption 7.5 for m̃jB̃j.
Then if aj ∈ S̃(mj , Bj), j = 1, 2, the composition a3 = a1#a2 is well defined and
belongs to S̃(m3, B3), in the sense that the corresponding composition of effective
kernels in (5.17) is given by an absolutely convergent integral and Keffaw3
∈ m̃3[B̃3].
We next consider the action of pseudodifferential operators on generalized symbol
spaces. Our result will be essentially a special case of the preceding theorem. We
start by “contracting” Assumption 7.5 to the case when E3 = 0.
Let m1, m2, m3 be order functions on E1×E2, E2, E1 respectively. Let Γj ⊂ Ej,
j = 1, 2 be lattices and let
B1 ⊂ ℓ
∞(Γ1 × Γ2), B2 ⊂ ℓ
∞(Γ2), B3 ⊂ ℓ
∞(Γ1)
be Banach spaces satisfying (7.1), (7.4). Assumption 7.5 becomes
Assumption 7.9 If kj ∈ mjBj , j = 1, 2, then
k3(α) =
k1(α, β)k2(β)
converges absolutely for every α ∈ Γ1, and we have k3 ∈ m3B3. Moreover,
‖k3/m3‖B3 ≤ C‖k1/m1‖B1‖k2/m2‖B2
where C is independent of k1, k2.
The corresponding “contraction” of Proposition 7.6 becomes
Proposition 7.10 Let Assumption 7.9 hold, where Bj satisfy (7.1), (7.4). Let
Kj ∈ mj[Bj ] for j = 1, 2. Then the integral
K3(x) :=
K1(x, z)K2(z)dz, x ∈ E1,
converges absolutely and defines a function K3 ∈ m3[B3]. Moreover,
‖K3/m3‖[B3] ≤ C‖K1/m1‖[B1]‖K2/m2‖[B2],
where C is independent of K1, K2.
We get the following result for the action of pseudodifferential operators on
generalized symbol spaces.
Theorem 7.11 Let m2, m3 be order functions on E = R
n× (Rn)∗ and let m1 be an
order function on E×E∗. Let Γ̂ ⊂ E×E∗ be a lattice such that Γ̃ := q−1(Γ̂) = Γ×Γ
where Γ ⊂ E is a lattice. Let B̂1 ⊂ ℓ
∞(Γ̂), B2, B3 ⊂ ℓ
∞(Γ) satisfy (7.1), (7.4). We
make the Assumption 7.9 with Γ1,Γ2 = Γ and with m1, B1 replaced with m̃1 = m1◦q,
B̃1 = B1 ◦ q, where q is given in (4.15).
Then, if a1 ∈ S̃(m1, B1), u ∈ S̃(m2, B2), the distribution v = a
1 (u) is well-
defined in S̃(m3, B3) in the sense that
e−Φ(x)Tv(x) =
Keffaw1
(x, y)e−Φ(y)Tu(y)L(dy),
with Keffaw1
(x, y) as in (5.6), converges absolutely for every x ∈ Cn and
((e−ΦTv) ◦ π ◦ κT ) ∈ [B3],
as in (7.12).
We shall finally generalize Theorem 6.2.
Theorem 7.12 Let p ∈ [1,∞] and let m be an order function on E × E∗ where
E = Rn × (Rn)∗. Let Γ ⊂ E be a lattice and B ⊂ ℓ∞(q(Γ × Γ)) a Banach space
satisfying (7.1), (7.4). Assume that
if (aα,β)α,β∈Γ ∈ (m ◦ q)B ◦ q, then (aα,β) ∈ Cp(ℓ
2(Γ), ℓ2(Γ)) (7.15)
and ‖(aα,β)‖Cp ≤ C‖(aα,β)‖(m◦q)B◦q ,
where q is given in (4.15) and C > 0 is independent of (aα,β). Then there is a (new)
constant C > 0 such that
If a ∈ S̃(m,B), then aw ∈ Cp(L
2, L2) and ‖aw‖Cp ≤ C‖a‖eS(m,B). (7.16)
The proof of Proposition 6.1 shows that the property (7.15) is invariant under
changes (Γ, B) 7→ (Γ̃, B̃) with B̃ ⊂ ℓ∞(q(Γ̃× Γ̃)) equivalent to B.
Proof We follow the proof of Theorem 6.2. Assume that (7.15) holds and let
a ∈ S̃(m,B) be of norm ≤ 1. It suffices to show that Aeff : L
2(Cn) → L2(Cn) is in
Cp with norm ≤ C, where Aeff is given in (6.4) and K
eff there belongs to m◦q[B ◦q],
provided that we identify Cn with E via π ◦ κT .
We see that we still have (6.9) where (6.10) should be replaced by
|∇kx∇
yKα,β(x, y)| ≤ Ck,ℓaα,β, |x− α|, |y − β| ≤ C0, (7.17)
(aα,β)α,β∈Γ ∈ (m ◦ q)B ◦ q, α, β ∈ Γ.
Write Aeff =W
∗AW as in (6.12),
L2(Ωβ) →
L2(Ωβ), A = (Aα,β).
The matrix elements Kα,j;β;k of Aα,β now obey the estimate (cf. (6.13)):
|Kα,j;β,k| ≤ CN〈j〉
−N〈k〉−Naα,β (7.18)
with aα,β as in (7.18). Using (7.15), this leads to (6.14) and from that point on the
proof is identical to that of Theorem 7.12. ✷
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	Introduction
	Symbol spaces
	Effective kernels and L2-boundedness
	Composition
	More direct approach using Bargmann transforms
	Cp classes
	Further generalizations
