CUBO A Mathematical Journal
Vol.11, No¯ 04, (87–107). September 2009

Pseudo-differential operators with smooth symbols

on modulation spaces

Joachim Toft

Department of Mathematics and Systems Engineering,
Växjö University, Sweden

email: joachim.toft@vxu.se

ABSTRACT

Let M
p,q
(ω0)

be the modulation space with parameters p,q and weight function ω0. If

∂
α
a/ω ∈ L∞ for all α, then we prove that the pseudo-differential operator at(x,D)

is continuous from M
p,q
(ω0ω)

to M
p,q
(ω0)

. More generally, if B is a translation invariant

BF-space, then we prove that at(x,D) is continuous from M(ω0ω)(B) to M(ω0)(B). We

use these results to establish identifications between such spaces with different weights.

RESUMEN

Sea M
p,q
(ω0)

el espacio de modulación con parámetros p,q y función de peso ω0. Si

∂
α
a/ω ∈ L∞ para todo α, entonces probamos que el operador pseudo-diferencial

at(x,D) es continuo de M
p,q
(ω0ω)

a M
p,q
(ω0)

. En general, si B es una translación inva-

riante en el espacio-BF, entonces probamos que at(x,D) es continuo de M(ω0ω)(B) en

M(ω0)(B). Usamos estos resultados para establecer las identificaciones entre dichos

espacios con diferentes pesos.

Key words and phrases: Pseudo-differential operators, Modulation spaces, Coorbit spaces, BF-
spaces, Sobolev spaces, Besov spaces.

Math. Subj. Class.: 35S05, 47B37, 47G30, 42B35.



88 Joachim Toft CUBO
11, 4 (2009)

1 Introduction

In this paper we establish continuity properties for certain pseudo-differential operators with

smooth symbols when acting on general class of modulation spaces. These modulation spaces

involve the usual modulation spaces, as well as certain type of weighted spaces related to Wiener

amalgam spaces. Furthermore, we establish bijectivity properties for multiplication operators and

Fourier multipliers, and use these properties to establish identification properties between modu-

lation spaces with different weights.

In particular we cover Theorem 2.1 in [30], where Tachizawa considers pseudo-differential

operators with symbols in S(ω)(R
2d

), the set of all smooth functions a on R2d such that (∂αa)/ω ∈
L
∞

(R2d). Here ω is an appropriate weight function on R2d, which takes the form of

ω(x,ξ) = 〈x〉t〈ξ〉s (1)

in [30], where s,t ∈ R and 〈x〉 = (1 + |x|2)1/2. (We use the usual notation for function and
distribution spaces, see e. g. [22].) In this context, Tachizawa extends Calderon-Vaillancourt’s

theorem, and proves that if ω0 is appropriate, and p,q ∈ [1,∞], then the corresponding pseudo-
differential operators are continuous from the modulation space M

p,q
(ω0ω)

to M
p,q
(ω0)

. (Cf. Section 2

for the definition of modulation spaces and pseudo-differential operators.) Tachizawa’s result were

thereafter generalized in Theorem 3.2 in the report [38], where the conditions on the weight ω are

relaxed in the sense that it is only assumed that ω should be v-moderate for some polynomial v.

A similar and interesting result comparing to [30, 38], concerns [32, Theorem 5.3], where

Teofanov discuss similar properties in context of ultra-modulation spaces. In this approach, the

condition on v here above is relaxed in the sense that v is permitted to grow subexponentially,

instead of polynomially. This in turn implies that symbols to the pseudo-differential operators

might grow subexponentially. However, the classes of pseudo-differential operators in [32] do not

contain those in [30] or [38], because the symbols in [32] have to fulfill certain conditions of Gelfand-

Shilov type, which is not the case in [30, 38].

Other related results are Theorem 3 in [25], and Theorem 3, Corollary 2 and Remark 3 in [26],

where Pilipović and Teofanov consider mapping properties for pseudo-differential operators with

symbols in ultra-modulation space and which fulfill certain ellipticity conditions.

In Section 3 we generalize [38, Theorem 3.2], and prove continuity for such pseudo-differential

operators on a broad class of modulation spaces, which contains the modulation spaces in [38],

and their Fourier transforms. These modulation spaces are in turn special cases of so called

coorbit spaces (see [11, 12] for the definition of coorbit spaces, and [9] for an updated definition of

modulation spaces). (See Theorem 3.2 and Theorem 3.2′.) Furthermore we establish bijectivity

properties for pseudo-differential operators, if they, in addition, are appropriate multiplication

operators or Fourier multipliers. (See Corollary 3.6.) Thereafter we give links on how these

results can be used to establish identification properties between modulation spaces with different

weights. (See Remark 3.7 and Theorems 3.9.) Here we also present some immediate consequences



CUBO
11, 4 (2009)

Pseudo-differential operators with smooth symbols ... 89

in modulation space theory and for spaces related to Wiener amalgam spaces. (See Corollary 3.6′,

Theorem 3.9′ and Theorem 3.9′′.)

The (classical) modulation spaces Mp,q, p,q ∈ [1,∞], as introduced by Feichtinger in [6],
consist of all tempered distributions whose short-time Fourier transforms (STFT) have finite mixed

L
p,q norm. It follows that the parameters p and q to some extent quantify the degrees of asymptotic

decay and singularity of the distributions in Mp,q. The theory of modulation spaces was developed

further and generalized in [11–13, 16], where Feichtinger and Gröchenig established the theory of

coorbit spaces. In particular, the modulation space M
p,q
(ω)

, where ω denotes a weight function on

phase (or time-frequency shift) space, appears as the set of tempered (ultra-) distributions whose

STFT belong to the weighted and mixed Lebesgue space L
p,q
(ω)

.

A major idea behind the design of these spaces was to find useful Banach spaces, which are

defined in a way similar to Besov spaces, in the sense of replacing the dyadic decomposition on the

Fourier transform side, characteristic to Besov spaces, with a uniform decomposition. From the
construction of these spaces, it turns out that modulation spaces and Besov spaces in some sense

are rather similar, and sharp embeddings between these spaces can be found in [1, 29, 35, 37]. (See

also [15, 23] for other embeddings.)

During the last 15 years many results have been proved which confirm the usefulness of the

modulation spaces in time-frequency analysis, where they occur naturally. For example, in [13, 17,

21], it is shown that all modulation spaces admit reconstructible sequence space representations

using Gabor frames.

Parallel to this development, modulation spaces have been incorporated into the calculus

of pseudo-differential operators, in the sense of (i) the study of continuity of pseudo-differential

operators with smooth symbols acting on modulation spaces, and (ii) the use of modulation spaces

as symbol classes. Tachizawa pioneered topic (i) in [30]. (See at the above.)

In [28], Sjöstrand founded topic (ii) and introduced the modulation space M∞,1, which con-

tains non-smooth functions, as a symbol class. He proved that the symbol class M∞,1 corresponds

to an algebra of operators which are bounded on L2.

Gröchenig and Heil thereafter proved in [17, 18] that each operator with symbol in M∞,1 is

continuous on all modulation spaces Mp,q, p,q ∈ [1,∞]. This extends Sjöstrand’s result since
M

2,2
= L

2. Some generalizations to operators with symbols in general unweighted modulation

spaces were obtained in [19, 35], and in [36, 38, 39] some further extensions involving weighted

modulation spaces are presented. Modulation spaces in pseudodifferential calculus is currently an

active field of research (see e. g. [18–20, 25, 31, 32]).



90 Joachim Toft CUBO
11, 4 (2009)

2 Preliminaries

In this section we discuss basic properties for modulation spaces and other related spaces. The

proofs are in many cases omitted since they can be found in [4–6, 11–13, 17, 33–36].

We start by recalling some properties of the involved weight functions. The positive function

ω ∈ L∞loc(Rd) is called v-moderate for some appropriate function v ∈ L∞loc(Rd), if there is a constant
C > 0 such that

ω(x1 + x2) ≤ Cω(x1)v(x2), x1,x2 ∈ Rd. (2)
If v can be chosen as polynomial, then ω is called polynomially moderate. The function v is called

submultiplicative, if (2) holds for ω = v.

As in [36] we let P(Rd) be the set of all polynomially moderate functions on Rd. We also let

P0(R
d
) be the set of all smooth ω ∈ P(Rd) such that (∂αω)/ω is bounded for every α.

Note that if ω ∈ P(Rd), then ω(x) + ω(x)−1 ≤ P(x), for some polynomial P on Rd.
In most of the applications, it is no restriction to assume that the weight functions belong to

P0, which is a consequence of the following lemma. (See also [36].)

Lemma 2.1. Assume that ω ∈ P(Rd). Then there is a function ω0 ∈ P(Rd) and a constant
C > 0 such that C−1ω ≤ ω0 ≤ Cω.

Proof. The assertion follows by letting ω0 = ω ∗ ϕ for some 0 ≤ ϕ ∈ S (Rd) \ 0.

The duality between a topological vector space and its dual is denoted by 〈 · , · 〉. For admissible
a and b in S ′(Rd), we set (a,b) = 〈a,b〉, and it is obvious that ( · , · ) on L2 is the usual scalar
product.

Next let V1 and V2 be vector spaces such that V1 ⊕ V2 = Rd and V2 = V ⊥1 , and assume that
v0 ∈ S ′(V1) and v ∈ S ′(Rd) are such that v(x1,x2) = (v0 ⊗ 1)(x1,x2), when xj ∈ Vj for j = 1, 2.
Then v(x1,x2) is identified with v0(x1), and we set v(x1,x2) = v(x1).

In order to discuss modulation spaces, we recall the definition of short-time Fourier transform.

Assume that χ ∈ S ′(Rd) \ 0 and let τxχ(y) = χ(y − x) when x,y ∈ Rd. Then the short-time
Fourier transform Vχf of f ∈ S ′(Rd) with respect to the window function χ is the distribution
in S ′(R2d), defined by the formula

(Vχf)(x,ξ) = F (f · τxχ)(ξ).

Here F denotes the Fourier transform on S ′(Rd), which takes the form

Ff(ξ) = f̂(ξ) = (2π)
−d/2

∫
f(y)e

−i〈y,ξ〉
dy

when f ∈ S (Rd). We note that Vχf is well-defined (as an element in S ′), since it is the partial
Fourier transform of the tempered distribution (x,y) 7→ f(y)χ(y − x) with respect to the y-variable.



CUBO
11, 4 (2009)

Pseudo-differential operators with smooth symbols ... 91

(Cf. [14].) If f,χ ∈ S (Rd), then Vχf is given by the formula

(Vχf)(x,ξ) = (2π)
−d/2

∫
f(y)χ(y − x)e−i〈y,ξ〉 dy.

Assume that χ ∈ S (Rd) \ 0, p,q ∈ [1,∞] and ω ∈ P(R2d) are fixed. Then the modulation
space Mp,q

(ω)
(Rd) consists of all f ∈ S ′(Rd) such that

‖f‖M p,q
(ω)

≡
( ∫ ( ∫

|Vχf(x,ξ)ω(x,ξ)|p dx
)q/p

dξ

)1/q
< ∞ (3)

(with the obvious modifications when p = ∞ and/or q = ∞). Furthermore, the space W p,q
(ω)

(Rd)

consists of all f ∈ S ′(Rd) such that

‖f‖W p,q
(ω)

≡
( ∫ ( ∫

|Vχf(x,ξ)ω(x,ξ)|q dξ
)p/q

dx

)1/p
< ∞. (4)

Note that the latter space is related to certain types of Wiener amalgam spaces. (Cf. Definition 4

in [13].)

We recall that W
p,q
(ω)

= FM
q,p
(ω0)

when ω0(x,ξ) = ω(−ξ,x) ∈ P(R2d). In fact, let χ̌(x) = χ(−x)
as usual. Then Parseval’s formula and a change of the order of integration shows that

|F −1(f̂ τξχ̂)(x)| = |F (f τxχ̌)(ξ)|. (5)

Hence, by applying the L
q,p
(ω)

norm, the assertion follows.

The convention of indicating weight functions with parenthesis is used also in other situations.

For example, if ω ∈ P(Rd), then Lp
(ω)

(Rd) is the set of all measurable functions f on Rd such

that fω ∈ Lp(Rd), i. e. such that ‖f‖Lp
(ω)

≡ ‖fω‖Lp is finite.
The following proposition is a consequence of well-known facts in [6, 17]. Here and in what

follows, we let p′ denotes the conjugate exponent of p, i. e. 1/p + 1/p′ = 1.

Proposition 2.2. Assume that p,q,pj,qj ∈ [1,∞] for j = 1, 2, ω,ω1,ω2,v ∈ P(R2d) are such
that ω is v-moderate, χ ∈ M1

(v)
(Rd) \ 0, and let f ∈ S ′(Rd). Then the following is true:

1. f ∈ Mp,q
(ω)

(Rd) if and only if (3) holds, i. e. Mp,q
(ω)

(Rd) is independent of the choice of χ.
Moreover, Mp,q

(ω)
is a Banach space under the norm in (3), and different choices of χ give rise

to equivalent norms;

2. f ∈ W p,q
(ω)

(Rd) if and only if (4) holds, i. e. W p,q
(ω)

(Rd) is independent of the choice of χ.
Moreover, W p,q

(ω)
is a Banach space under the norm in (4), and different choices of χ give rise

to equivalent norms;

3. if p1 ≤ p2, q1 ≤ q2 and ω2 ≤ Cω1 for some constant C, then

S (R
d
) ⊆Mp1,q1

(ω1)
(R

d
) ⊆Mp2,q2

(ω2)
(R

d
) ⊆ S ′(Rd),

S (R
d
) ⊆W p1,q1

(ω1)
(R

d
) ⊆W p2,q2

(ω2)
(R

d
) ⊆ S ′(Rd).



92 Joachim Toft CUBO
11, 4 (2009)

Proposition 2.2 permits us to be rather vague about to the choice of χ ∈ M1
(v)

\ 0 in (3) and
(4). For example, if C > 0 is a constant and Ω is a subset of S ′, then ‖a‖M p,q

(ω)
≤ C for every a ∈ Ω,

means that the inequality holds for some choice of χ ∈ M1
(v)

\ 0 and every a ∈ Ω. Evidently, for
any other choice of χ ∈ M1

(v)
\ 0, a similar inequality is true although C may have to be replaced

by a larger constant, if necessary.

Next we discuss weight functions which are common in the applications. For any s,t ∈ R, set

σt(x) = 〈x〉t, σs,t(x,ξ)〈ξ〉s〈x〉t, (6)

when x,ξ ∈ Rd. Then it follows that σt ∈ P0(Rd) and σs,t ∈ P0(R2d) for every s,t ∈ R, and
σt is σ|t|-moderate and σs,t is σ|s|,|t|-moderate. Obviously, σs(x,ξ) = (1 + |x|2 + |ξ|2)s/2, and
σs,t = σt ⊗ σs. Moreover, if ω ∈ P(Rd), then ω is σt-moderate provided t is chosen large enough.

For conveniency we use the notations Lps , M
p,q
s and M

p,q
s,t instead of L

p
(σs)

, M
p,q
(σs)

and M
p,q
(σs,t)

respectively.

Remark 2.3. Assume that p,q,q1,q2 ∈ [1,∞] and ω ∈ P(R2d). Then the following properties
for modulation spaces hold:

1. if q1 ≤ min(p,p′) and q2 ≥ max(p,p′) and ω(x,ξ) = ω(x), then

M
p,q1
(ω)

⊆ Lp
(ω)

⊆ Mp,q2
(ω)

, W
p,q1
(ω)

⊆ Lp
(ω)

⊆ W p,q2
(ω)

;

2. S0
0

=
⋂

s∈R M
∞,1
s,0 ;

3. if p ≥ q, then Mp,q
(ω)

⊆ W p,q
(ω)

. If instead q ≥ p, then W p,q
(ω)

⊆ Mp,q
(ω)

;

4. M1,∞(Rd) and W 1,∞(Rd) are convolution algebras such that if M(Rd) is the set of all

measures on Rd with bounded mass, then M ⊆ W 1,∞ ⊆ M1,∞;

5. if Ω is a subset of P(R2d) such that for any polynomial P on R2d, there is an element ω ∈ Ω
such that P/ω is bounded, then

S (R
d
) =

⋂

ω∈Ω

M
p,q
(ω)

(R
d
), S

′
(R

d
) =

⋃

ω∈Ω

M
p,q
(1/ω)

(R
d
);

6. if s,t ∈ R are such that t ≥ 0, then

M
2

s,0 = H
2

s , M
2

0,s = L
2

s, and M
2

t = L
2

t ∩ H2t .

(See e. g. [4–6, 10–13, 17, 35, 36].)



CUBO
11, 4 (2009)

Pseudo-differential operators with smooth symbols ... 93

We refer to [6, 11–13, 17, 27] for more facts about modulation spaces and W
p,q
(ω)

-spaces.

As anounced in the introduction we consider in Section 3 mapping properties for pseudo-

differential operators when acting on certain types of coorbit spaces, which are defined by imposing

certain types of translation invariant solid BF-space norms on the short-time Fourier transforms.

(Cf. [6, 8, 11, 12].) This familly of coorbit spaces contains the modulation and Wiener amalgam

spaces. In the following we recall the definition of these spaces.

Definition 2.4. Assume that B is a Banach space of complex-valued measurable functions on Rd

and v ∈ P(Rd). Then B is called a translation invariant BF-space on Rd (with respect to v), if
there is a constant C such that the following conditions are fulfilled:

1. S (Rd) ⊆ B ⊆ S ′(Rd) (continuous embeddings);

2. if x ∈ Rd and f ∈ B, then τxf ∈ B, and

‖τxf‖B ≤ Cv(x)‖f‖B; (7)

3. if f,g ∈ L1loc(Rd) satisfy g ∈ B and |f| ≤ |g|, then f ∈ B and

‖f‖B ≤ C‖g‖B.

Here the condition (3) in Definition 2.4 means that a translation invariant BF-space is a solid

BF-space in the sense of (A.3) in [8]. It follows from this condition that if f ∈ B and h ∈ L∞,
then f · h ∈ B, and

‖f · h‖B ≤ C‖f‖B‖h‖L∞. (8)
Example 2.5. Assume that p,q ∈ [1,∞], and let Lp,q

1
(R2d) be the set of all f ∈ L1loc(R2d) such

that

‖f‖Lp,q1 ≡
( ∫ ( ∫

|f(x,ξ)|p dx
)q/p

dξ

)1/q

if finite. Also let L
p,q
2

(R2d) be the set of all f ∈ L1loc(R2d) such that

‖f‖Lp,q2 ≡
( ∫ ( ∫

|f(x,ξ)|q dξ
)p/q

dx

)1/p

is finite. Then it follows that L
p,q
1

and L
p,q
2

are translation invariant BF-spaces with respect to

v = 1.

More generally, assume that ω,v ∈ P(R2d) are such that ω is v-moderate, and let Lp,q
j,(ω)

(R2d),

for j = 1, 2, be the set of all f ∈ L1loc(R2d) such that ‖f‖Lp,qj,(ω) ≡ ‖f ω‖Lp,qj is finite. Then L
p,q
j,(ω)

is

a translation invariant BF-space with respect to v.

Remark 2.6. The conclusion in the latter part of Example 2.5 is also a consequence of the first

part in that example and the following observation. Assume that ω0,v,v0 ∈ P(Rd) are such that
ω is v-moderate, and assume that B is a translation invariant BF-space on Rd with respect to v0.

Also let B0 be the Banach space which consists of all f ∈ L1loc(Rd) such that ‖f‖B0 ≡ ‖f ω‖B is
finite. Then B0 is a translation invariant BF-space with respect to v0v.



94 Joachim Toft CUBO
11, 4 (2009)

For translation invariant BF-spaces we make the following observation.

Proposition 2.7. Assume that v ∈ P(Rd), and that B is a translation invariant BF-space with
respect to v. Then the convolution mapping (ϕ,f) 7→ ϕ ∗ f from C∞

0
(Rd) × B to S ′ extends

uniquely to a continuous mapping from L1
(v)

(Rd) × B to B, and

‖ϕ ∗ f‖B ≤ C‖ϕ‖L1
(v)

‖f‖B,

for some constant C which is independent of ϕ ∈ L1
(v)

and f ∈ B.

Proposition 2.9 is a consequence of the results in [6, 8]. In order to be more self-contained we

give here a short motivation.

Proof. First assume that ϕ ∈ C∞
0

and that f ∈ B. Then Minkowski’s inequality and (8) give

‖ϕ ∗ f‖B =
∥∥∥

∫
f( · − y)ϕ(y) dy

∥∥∥
B

≤
∫

‖f( · − y)ϕ(y)‖B dy =
∫

‖f( · − y)‖B|ϕ(y)|dy

≤ C
∫

‖f‖B v(y)|ϕ(y)|dy = C‖f‖B‖ϕ‖L1
(v)
,

which proves the result in this case. For general ϕ ∈ L1
(v)

, the result follows from the fact that C0
is dense in L1

(v)
.

Next we consider the general type of modulation spaces which we are especially interested in.

Definition 2.8. Assume that B is a translation invariant BF-space on R2d, ω ∈ P(R2d), and
that χ ∈ S (Rd) \ 0. Then the modulation space M(ω) = M(ω)(B) consists of all f ∈ S ′(Rd) such
that

‖f‖M(ω) = ‖f‖M(ω)(B) ≡ ‖Vχf ω‖B
is finite.

Assume that ω ∈ P(R2d) is fix, and consider the familly of distribution spaces which consists
of all spaces of the form M(ω)(B) such that B is a translation invariant BF-space on R

2d. Then

it follows by Remark 2.6 that this familly is invariant under ω. Consequently we do not increase

the set of possible spaces in Definition 2.8 by permitting ω that are not identically 1.

From this observation it seems to be superfluous to include the weight ω in Definition 2.8. How-

ever, it will be convenient for us to permit such ω dependency when investigating mapping prop-

erties for pseudo-differential operators in Section 3, when acting on spaces of the form M(ω)(B).

Obviously, we have

M
p,q
(ω)

(R
d
) = M(ω)(B1) and W

p,q
(ω)

(R
d
) = M(ω)(B2)



CUBO
11, 4 (2009)

Pseudo-differential operators with smooth symbols ... 95

when B1 = L
p,q
1

(R2d) and B2 = L
p,q
2

(R2d) (cf. Example 2.5). It follows that many properties

which are valid for the modulation spaces also hold for the spaces of the form M(ω)(B). For example

we have the following proposition, which shows that the definition of M(ω)(B) is independent of

the choice of χ. We omit the proof since it can be found in e. g. [8,11,12]. It also follows by similar

arguments as in the proof of Proposition 11.3.2 in [17].

Proposition 2.9. Assume that B is a translation invariant BF-space with respect to v0 ∈ P(R2d)
for j = 1, 2. Also assume that ω,v ∈ P(R2d) are such that ω is v-moderate, M(ω)(B) is the same
as in Definition 2.8, and let χ ∈ M1

(v0v)
(Rd) \ 0 and f ∈ S ′(Rd). Then f ∈ M(ω)(B) if and only

if Vχf ω ∈ B, and different choices of χ gives rise to equivalent norms in M(ω)(B).

Next we recall some facts in Chapter XVIII in [22] concerning pseudo-differential operators.

Assume that t ∈ R is fixed and that a ∈ S (R2d). Then the pseudo-differential operator at(x,D)
is the continuous operator on S (Rd), defined by the formula

(at(x,D)f)(x) = (Opt(a)f)(x)

= (2π)
−d

∫ ∫
a((1 − t)x + ty,ξ)f(y)ei〈x−y,ξ〉 dydξ.

(9)

The definition of at(x,D) extends to any a ∈ S ′(R2d), and then at(x,D) is continuous from S (Rd)
to S ′(Rd). Moreover, for every fixed t ∈ R, it follows that there is a one to one correspondance
between such operators, and pseudo-differential operators of the form at(x,D). (See e. g. [22].) If

t = 1/2, then at(x,D) is equal to the Weyl operator a
w
(x,D) for a. If instead t = 0, then the

standard (Kohn-Nirenberg) representation a(x,D) is obtained.

Consequently, for every a ∈ S ′(R2d) and s,t ∈ R, there is a unique b ∈ S ′(R2d) such that
as(x,D) = bt(x,D). By straight-forward applications of Fourier’s inversion formula, it follows that

as(x,D) = bt(x,D) ⇐⇒ b(x,ξ) = ei(t−s)〈Dx,Dξ〉a(x,ξ). (10)

(Cf. [22].)

In the next section we discuss continuity for pseudo-differential operators with symbols in

S(ω)(R
2d

), the set of all smooth functions a on R2d such that ∂αa/ω ∈ L∞(R2d). Here ω ∈
P(R2d). If ω = 1, then we use the notation S0

0
(R2d) instead of S(ω)(R

2d
).

3 Continuity for pseudo-differential operators with symbols

in S(ω)

In this section we discuss continuity for operators in Op(S(ω0)) when acting on modulation spaces.

In the first part we prove in Theorem 3.2 below that if ω,ω0 ∈ P, t ∈ R and a ∈ S(ω), then
at(x,D) is continuous from M(ω0ω)(B) to M(ω0)(B). In particular, Theorem 2.1 in [30] as well as

Theorem 2.2 in [36] are covered.



96 Joachim Toft CUBO
11, 4 (2009)

In the second part we present some applications and prove that certain properties which are

valid for Sobolev spaces carry over to modulation spaces.

We start by giving some remarks on S(ω)(R
2d

) when ω ∈ P(R2d). By straight-forward
computations it follows that S(ω)(R

2d
) agrees with S(ω,g) when g(x,ξ)(y,η) = |y|2 + |η|2 is the

standard euclidean metric on R2d. (See Section 18.4–18.6 in [22].) Since the metric g is constant it

follows that it is trivially slowly varying and σ-temperate, where σ denotes the standard symplectic

form on R2d. Moreover, from the fact that ω is σt-moderate when t is large enough, it follows

by straight-forward computations that ω is σ,g-temperate. The following lemma is therefore a

consequence of Theorem 18.5.10 in [22].

Lemma 3.1. Assume that ω ∈ P(R2d), s,t ∈ R, and that a,b ∈ S ′(R2d) are such that as(x,D) =
bt(x,D). Then

a ∈ S(ω)(R2d) ⇐⇒ b ∈ S(ω)(R2d).

We have now the following result.

Theorem 3.2. Assume that t ∈ R, ω,ω0 ∈ P(R2d), a ∈ S(ω)(R2d), t ∈ R, and that B is a
translation invariant BF-space on R2d. Then at(x,D) is continuous from M(ω0ω)(B) to M(ω0)(B).

We need some preparations for the proof, and start by recalling Minkowski’s inequality in a

somewhat general form. Assume that dµ is a positive measure, and that f ∈ L1(dµ; B) for some
Banach space B. Then Minkowski’s inequality asserts that

∥∥∥
∫
f(x) dµ(x)

∥∥∥
B

≤
∫

‖f(x)‖B dµ(x).

We also need some lemmas.

Lemma 3.3. Assume that ω ∈ P(R2d), a ∈ S(ω)(R2d), f ∈ S (Rd), χ ∈ S (Rd), χ2 = σsχ and
0 ≤ s ∈ R. If

Φ(x,ξ,z,ζ) =
a(x + z,ξ + ζ)

ω(x,ξ)〈z〉s〈ζ〉s (11)

and

H(x,ξ,y) =

∫ ∫
Φ(x,ξ,z,ζ)χ2(z)〈ζ〉sei〈y−x−z,ζ〉 dzdζ,

then
Vχ(a(·,D)f)(x,ξ) = (2π)−d(f,ei〈 · ,ξ〉H(x,ξ, · ))ω(x,ξ). (12)

Proof. For simplicity we assume that a is real-valued. By straight-forward computations we get

Vχ(a(·,D)f)(x,ξ) = (a(·,D)f,τxχei〈·,ξ〉)

= (f,a(·,D)∗(τxχei〈·,ξ〉))

= (2π)
−d

(f,e
i〈 · ,ξ〉

H̃(x,ξ, · ))ω(x,ξ),

(13)



CUBO
11, 4 (2009)

Pseudo-differential operators with smooth symbols ... 97

where

H̃(x,ξ,y) = (2π)
d
e
−i〈y,ξ〉

(a(·,D)∗(τxχei〈·,ξ〉))(y)/ω(x,ξ)

=

∫ ∫
a(z,ζ)

ω(x,ξ)
χ(z − x)ei〈y−z,ζ−ξ〉 dzdζ

=

∫ ∫
Φ(x,ξ,z − x,ζ − ξ)χ2(z − x)〈ζ − ξ〉sei〈y−z,ζ−ξ〉 dzdζ.

If z − x and ζ − ξ are taken as new variables of integrations, it follows that the right-hand side is
equal to H(x,y,ξ). This proves the assertion.

Lemma 3.4. Let s ≥ 0 be an even integer, Φ and H be the same as in Lemma 3.3, and set

Φβ (x,ξ,z,ζ) = ∂
β
z Φ(x,ξ,z,ζ), χ2,γ = ∂

γ
χ2. (14)

Also let Ψβ (x,ξ,y, · ) be the inverse partial Fourier transform of Φβ (x,ξ,y,η) with respect to the
η variable, and let

Hβ,γ (x,ξ,y) =

∫
Ψβ (x,ξ,y − z − x,z)χ2,γ (y − z − x) dz. (15)

Then there are constants Cβ,γ which depend on β, γ, s and d only such that

H(x,ξ,y) =

∑

|β+γ|≤s

Cβ,γHβ,γ (x,ξ,y).

Proof. By integrating by parts we get

H(x,ξ,y) =

∫ ∫
Φ(x,ξ,z,ζ)χ2(z)〈ζ〉s/2ei〈y−x−z,ζ〉 dzdζ

=

∫ ∫
Φ(x,ξ,z,ζ)χ2(z)(1 − ∆z)s/2(ei〈y−x−z,ζ〉) dzdζ

=

∑

|β+γ|≤N

Cβ,γH̃β,γ (x,ξ,y),

where

H̃β,γ (x,ξ,y) = (2π)
−d/2

∫ ∫
Φβ (x,ξ,z,ζ)χ2,γ (z)e

i〈y−x−z,ζ〉
dzdζ.

If we take y−x−z and ζ as new variables of integrations, and perform the integration with respect
to the ζ variable, it follows that H̃β,γ = Hβ,γ , which gives the result.

For the next lemma we observe that if f ∈ S ′(Rd) is fixed, then there are positive constants
s0, N and C0 such that

|Vχ0f(x,ξ)| ≤ C0〈x,ξ〉N, where χ0 = σ−s and s ≥ s0. (16)



98 Joachim Toft CUBO
11, 4 (2009)

Lemma 3.5. Assume that ω ∈ P(R2d), a ∈ S(ω)(R2d), ϕ ∈ S (Rd), and f ∈ S ′(Rd). If s ≥ 0
is large enough and χ0 = σ−s, then there is a constant C such that

|Vϕ(at( · ,D)f)(x,ξ)| ≤ C(F(x, · ) ∗ χ0)(ξ), (17)

where
F(x,ξ) = |Vχ0f(x,ξ)ω(x,ξ)|. (18)

Proof. It is no restriction to assume that a is real-valued, and by Lemmas 2.1 and 3.1 it follows
that we may assume that t = 0 and that ω ∈ P0. Furthermore, by Lemma 3.3, Lemma 3.4 and
(13), the result follows if we prove the following:

1. the right-hand side of (12) is well-defined for the fixed f ∈ S ′(Rd) when s is chosen large
enough, and that (12) holds also in this case;

2. for each multi-indices β and γ, there is a constant C such that

Iβ,γ (x,ξ) ≡ |(f,ei〈 · ,ξ〉Hβ,γ (x,ξ, · ))ω(x,ξ)| ≤ C(F(x, · ) ∗ σ−s)(ξ). (9)

Let C0, s0 and N be chosen such that (16) is fulfilled, let N1 be an even and large integer,

and let Φβ be as in (14). The assertion (1) follows if we prove that for each multi-indices α and β,

there is a constant Cα,β = CN1,α,β such that

|(∂αΦβ )(x,ξ,z,ζ)| ≤ Cα,β〈z〉−N1〈ζ〉−N1. (19)

In order to prove (19) we choose M ≥ 0 and s ≥ M + N1 such that ω ∈ P0 is σM -moderate,
and assume first that α = β = 0. Then (11) and the facts that a ∈ S(ω) give

|Φ(x,ξ,z,ζ)| = |a(x + z,ξ + ζ)|
ω(x,ξ)〈z〉s〈ζ〉s ≤ C1

|a(x + z,ξ + ζ)|〈z,ζ〉M
ω(x + z,ξ + ζ)〈z〉s〈ζ〉s ≤ C2〈z〉

−N1〈ζ〉−N1.

For general α and β, (19) follows from these computations in combination with Leibnitz rule,

using the facts that (∂γa)/ω ∈ L∞ and (∂γω)/ω ∈ L∞ for each multi-index γ. This gives (1).
Assume that N2 ≥ 0 is arbitrary. Then it follows by choosing N1 in (19) large enough, that

for some constant C it holds

|∂αΨβ (x,ξ,y − z,z)| ≤ C〈y〉−N2〈z〉−N2 (20)

for every multi-index α such that |α| ≤ N2.
If N3 is a fixed integer, then it follows from (15) and (20) that

Hβ,γ (x,ξ,y) = σ−N3 (y − x)ϕβ,γ (x,ξ,y − x), (21)

where ϕβ,γ satisfies

|∂αϕβ,γ (x,ξ,y)| ≤ C〈y〉−N3, |α| ≤ N3,



CUBO
11, 4 (2009)

Pseudo-differential operators with smooth symbols ... 99

for some constant C, provided N2 was chosen large enough. Hence, for any fixed s ≥ 0, it follows
by choosing N3 large enough that

|F (ϕβ,γ (x,ξ, · ))(η)| ≤ C〈η〉−s, (22)

for some constant C.

By choosing s > d, it follows from (21), (22) and straight-forward computations that

Iβ,γ (x,ξ) = |(f,ei〈·,ξ〉χ0(· − x)ϕβ,γ (x,ξ, · − x))|

= |F ((f τxχ0)ϕβ,γ (x,ξ, · − x))(ξ)|

≤ (2π)−d/2
∫

|F (f τxχ0)(ξ − η)||F (ϕβ,γ (x,ξ, · − x))(η)|dη,

≤ C
∫

|Vχ0f(x,ξ − η)|χ0(η) dη,

(23)

where

C = (2π)
−d/2

∫
sup
x,ξ

|
(
F (ϕβ,γ (x,ξ, · − x))(η)|

)
dη

= (2π)
−d/2

∫
sup
x,ξ

|
(
F (ϕβ,γ (x,ξ, · ))(η)|

)
dη

≤ C1
∫
〈η〉−s dη < ∞.

This gives (17), and the proof is complete.

Proof of Theorem 3.2. We use the same notations as in Lemma 3.5, and set

G = |Vχ(at( · ,D)f)|.

Since ω0 ∈ P, it follows that ω0(x,ξ) ≤ Cω0(x,ξ − η)〈η〉s0 , for some constants C and s0. By
Lemma 3.5 we get

G(x,ξ)ω0(x,ξ) ≤ C1
∫
F(x,ξ − η)〈η〉−sω0(x,ξ) dη

≤ C2
∫
F(x,ξ − η)ω0(x,ξ − η)〈η〉s0−s dη,

= C2

∫
Fη,ω0 (x,ξ)〈η〉s0−s dη,

for some constants C1 and C2, where

Fη,ω0 (x,ξ) = F(x,ξ − η)ω0(x,ξ − η).



100 Joachim Toft CUBO
11, 4 (2009)

Now choose s1,s2 ∈ R in such way that s1 = s−s0 and B is a translation invariant BF-space
with respect to σs2 , and let ω1 = ω0ω. Then it follows for some constant C and Minkowki’s

inequality that

‖at(x,D)f‖M(ω0)(B) = ‖Gω0‖B ≤ C1
∫

‖Fη,ω0‖B〈η〉−s1 dη

≤ C2
∫

‖F ω0‖B〈η〉s2−s1 dη = C3‖f‖M(ω0ω)(B),

where

C3 = C2‖σs2−s1‖L1.

Since s can be chosen arbitrary large, it follows that s1 can be chosen larger than s2 + d, which

implies that C3 < ∞. This gives the result.

Next we show that [36, Theorem 2.2] is essentially a consequence of Theorem 3.2.

Corollary 3.6. Assume that t ∈ R, ω ∈ P0(R2d), ω0 ∈ P(R2d) are such that ω(x,ξ) = ω(x)
or ω(x,ξ) = ω(ξ), and that B is a translation invariant BF-space on R2d. Then ωt(x,D) is a
homeomorphism from M(ω0ω)(B) to M(ω0)(B).

Proof. Since it follows from the assumptions that ω ∈ S(ω), Theorem 3.2 shows that ωt(x,D) is
continuous from M(ω0ω)(B) to M

p,q
(ω)

(B). On the other hand, since ω(x,ξ) = ω(x) or ω(x,ξ) = ω(ξ),

it follows that the inverse of ωt(x,D) on S
′
(Rd) is equal to (1/ω)t(x,D). Hence Theorem 3.2

together with the obvious fact that 1/ω ∈ P0 give

‖f‖M(ω0ω)(B) = ‖(1/ω)t(x,D)(ωt(x,D)f)‖M(ω0ω)(B)

≤ C‖ωt(x,D)f‖M(ω0)(B)

for some constant C. This proves that ωt(x,D) is a bijective map from M(ω0ω)(B) to M(ω0)(B),

and the result follows.

Remark 3.7. We remark that an immediate consequence of Corollary 3.6 is that if B is a trans-

lation invariant BF-space on R2d, ω(x,ξ) = ω1(x)ω2(ξ) where ωj ∈ P0(Rd) for j = 1, 2, and
ω0 ∈ P(R2d), then

M(ω0ω)(B) = {f ∈ S ′(Rd) ; ω1(x)ω2(D)f ∈ M(ω0)(B) }

= {f ∈ S ′(Rd) ; ω2(D)(ω1f) ∈ M(ω0)(B) }.

In particular, if s,t ∈ R, B = Lp,q
1

or B = L
p,q
2

, and ω(x,ξ) = σs,t(x,ξ) = 〈x〉t〈ξ〉s, then

M
p,q
(σs,tω0)

(R
d
) = {f ∈ S ′(Rd) ; 〈x〉t〈D〉sf ∈ Mp,q

(ω0)
(R

d
) }

= {f ∈ S ′(Rd) ; 〈D〉s(〈 · 〉tf) ∈ Mp,q
(ω0)

(R
d
) }



CUBO
11, 4 (2009)

Pseudo-differential operators with smooth symbols ... 101

and

W
p,q
(σs,tω0)

(R
d
) = {f ∈ S ′(Rd) ; 〈x〉t〈D〉sf ∈ W p,q

(ω0)
(R

d
) }

= {f ∈ S ′(Rd) ; 〈D〉s(〈 · 〉tf) ∈ W p,q
(ω0)

(R
d
) }.

Remark 3.8. For certain ω it is possible to use Remark 2.12 in [36] to prove that the continuity

assertions in Theorem 3.2 also holds when the symbols for the pseudo-differential operators belong

to M
∞,1
(ω)

(R2d).

Note that σs,t(x,D) here above, appears frequently in harmonic analysis and in the pseudo-

differential calculus. For example, if p ∈ [1,∞], then recall that f ∈ S ′(Rd) belongs to the Sobolev
space Hps (R

d
) if and only if ‖f‖Hps ≡ ‖σs(D)f‖Lp is finite. It is well-known that if s = N is a

positive integer and 1 < p < ∞, then Hps agrees with

{f ∈ Lp ; ∂αf ∈ Lp when |α| ≤ N }.

(See [2].)

In the following theorems we prove that similar properties in a somewhat extended form also

hold for general spaces of the form M(ω)(B).

Theorem 3.9. Assume that N1,N2 ≥ 0 are integers, ω ∈ P(R2d), B is a translation invariant
BF-spaces on R2d, and assume that f ∈ S ′(Rd). Then the following conditions are equivalent:

1. f ∈ M(σN1,N2 ω)(B);

2. xβ∂αf ∈ M(ω)(B) for all multi-indices α and β such that |α| ≤ N1 and |β| ≤ N2;

3. ∂α(xβf) ∈ M(ω)(B) for all multi-indices α and β such that |α| ≤ N1 and |β| ≤ N2;

4. f,xN2j f, ∂
N1
k f, x

N2
j ∂

N1
k f ∈ M(ω)(B) for all 1 ≤ j,k ≤ d;

5. f,xN2j f, ∂
N1
k f, ∂

N1
k (x

N2
j f) ∈ M(ω)(B) for all 1 ≤ j,k ≤ d.

Proof. We only prove the equivalences

(1) ⇐⇒ (2) ⇐⇒ (4).

The equivalences

(1) ⇐⇒ (3) ⇐⇒ (5)

follow by similar arguments and are left for the reader.



102 Joachim Toft CUBO
11, 4 (2009)

Let M0 be the set of all f ∈ M(ω)(B) such that xβ∂αf ∈ M(ω)(B) when |α| ≤ N1 and
|β| ≤ N2, and let M̃0 be the set of all f ∈ M(ω)(B) such that

x
N2
j f,∂

N1
k f,x

N2
j ∂

N1
k f ∈ M(ω)(B)

for j,k = 1, . . . ,d. We shall prove that M0 = M̃0 = M(σN1,N2 ω)(B). Obviously, M0 ⊆ M̃0. Since
the symbol ξα of the operator Dα belongs to S(σN1,N2 ) when |α| ≤ N, it follows from Theorem
3.2 that the embedding M(σN1,N2 ω)(B) ⊆ M0 holds. The result therefore follows if we prove that
M̃0 ⊆ M(σN1,N2 ω)(B).

In order to prove this, assume first that N1 = N, N2 = 0, f ∈ M̃0, and choose open sets

Ω0 = {ξ ∈ Rd ; |ξ| < 2 }, and Ωj = {ξ ∈ Rd ; 1 < |ξ| < d|ξj| }.

Then
⋃d

j=0 Ωj = R
d, and there are non-negative functions ϕ0, . . . ,ϕd in S

0

0
such that supp ϕj ⊆ Ωj

and
∑d

j=0 ϕj = 1. In particular, f =
∑d

j=0 fj when fj = ϕj (D)f. The result follows if we prove

that fj ∈ M(σN,0ω)(B) for every j.
Now set ψ0(ξ) = σN (ξ)ϕ0(ξ) and ψj (ξ) = ξ

−N
j σN (ξ)ϕj (ξ) when j = 1, . . . ,d. Then ψj ∈ S00

for every j. Hence Theorem 3.2 gives

‖fj‖M(σN,0ω)(B) ≤ C1‖σN (D)fj‖M(ω)(B)

= C1‖ψj (D)∂Nj f‖M(ω)(B) ≤ C2‖∂
N
j f‖M(ω)(B) < ∞

and

‖f0‖M(σN,0ω)(B) ≤ C1‖σN (D)f0‖M(ω)(B)

= C1‖ψ0(D)f‖M(ω)(B) ≤ C2‖f‖M(ω)(B) < ∞

for some constants C1 and C2. This proves that

‖f‖M(σN,0ω)(B) ≤ C
(
‖f‖M(ω)(B) +

d∑

j=1

‖∂Nj f‖M(ω)(B)
)
, (24)

and the result follows in this case.

If we instead split up f into
∑
ϕjf, then similar arguments show that

‖f‖M(σ0,N ω)(B) ≤ C
(
‖f‖M(ω)(B) +

d∑

k=1

‖xNk f‖M(ω)(B)
)
, (25)

and the result follows in the case N1 = 0 and N2 = N from this estimate.

The general case now follows if we combine (24) with (25). The proof is complete.



CUBO
11, 4 (2009)

Pseudo-differential operators with smooth symbols ... 103

We finish the section by stating the previous results in the special cases of modulation spaces

and corresponding Wiener amalgam related spaces. In fact, by letting B = L
p,q
1

or B = L
p,q
2

, the

following results are immediate consequences of the previous ones.

Theorem 3.2′. Assume that ω,ω0 ∈ P(R2d), a ∈ S(ω)(R2d), t ∈ R, and that p,q ∈ [1,∞]. Then
at(x,D) is continuous from M

p,q
(ω0ω)

(Rd) to Mp,q
(ω0)

(Rd), and from W p,q
(ω0ω)

(Rd) to W p,q
(ω0)

(Rd).

We note that if t = 0 and ω0 = σs1,s2 where s1,s2 ∈ R, then Theorem 3.2′ agrees with
Theorem 1.1 in [30].

Corollary 3.6′. Assume that ω ∈ P0(R2d), ω0 ∈ P(R2d) are such that ω(x,ξ) = ω(x) or
ω(x,ξ) = ω(ξ), and that p,q ∈ [1,∞]. Then ωt(x,D) is a homeomorphism from Mp,q(ω0ω)(R

d
) to

M
p,q
(ω0)

(Rd), and from W p,q
(ω0ω)

(Rd) to W p,q
(ω0)

(Rd).

Theorem 3.9′. Assume that N1,N2 ≥ 0 are integers, ω ∈ P(R2d), p,q ∈ [1,∞], and that
f ∈ S ′(Rd). Then the following conditions are equivalent:

1. f ∈ Mp,q
(σN1,N2 ω)

(Rd);

2. xβ∂αf ∈ Mp,q
(ω)

(Rd) for all multi-indices α and β such that |α| ≤ N1 and |β| ≤ N2;

3. ∂α(xβf) ∈ Mp,q
(ω)

(Rd) for all multi-indices α and β such that |α| ≤ N1 and |β| ≤ N2;

4. f,xN2j f, ∂
N1
k f, x

N2
j ∂

N1
k f ∈ M

p,q
(ω)

(Rd) for all 1 ≤ j,k ≤ d;

5. f,xN2j f, ∂
N1
k f, ∂

N1
k (x

N2
j f) ∈ M

p,q
(ω)

(Rd) for all 1 ≤ j,k ≤ d.

Theorem 3.9′′. Assume that N1,N2 ≥ 0 are integers, ω ∈ P(R2d), p,q ∈ [1,∞], and that
f ∈ S ′(Rd). Then the following conditions are equivalent:

1. f ∈ W p,q
(σN1,N2 ω)

(Rd);

2. xβ∂αf ∈ W p,q
(ω)

(Rd) for all multi-indices α and β such that |α| ≤ N1 and |β| ≤ N2;

3. ∂α(xβf) ∈ W p,q
(ω)

(Rd) for all multi-indices α and β such that |α| ≤ N1 and |β| ≤ N2;

4. f,xN2j f, ∂
N1
k f, x

N2
j ∂

N1
k f ∈ W

p,q
(ω)

(Rd) for all 1 ≤ j,k ≤ d;

5. f,xN2j f, ∂
N1
k f, ∂

N1
k (x

N2
j f) ∈ W

p,q
(ω)

(Rd) for all 1 ≤ j,k ≤ d.

The following result was presented in [39, Remark 1.3]. Since the facts here do not seems to

be well-known, we give some explicit motivations.



104 Joachim Toft CUBO
11, 4 (2009)

Corollary 3.10. Assume that p,q ∈ [1,∞] and ω ∈ P(R2d) is such that ω(x,ξ) = ω(x). Then
the following is true:

1. Mp,q
(ω)

(Rd) →֒ C(Rd) if and only if q = 1;

2. W p,q
(ω)

(Rd) →֒ C(Rd) if and only if q = 1.

Proof. By Corollary 3.6′ it follows that we may assume that ω = 1. If f ∈ W∞,1, then it follows
that F (fϕ) ∈ L1 for every ϕ ∈ S , which implies that fϕ is a continuous function. Since ϕ ∈ S
is arbitrary chosen, it follows that f is continuous. This gives

M
p,1 ⊆ W p,1 ⊆ W∞,1 ⊆ C, (26)

which proves one part of the assertion.

Next assume that q > 1, and let f be the characteristic function of the cube [0, 1]d. Then

f /∈ C, and it follows by straight-forward computations that f ∈ W 1,q ⊆ M1,q. Since Mp,q and
W

p,q increases with the parameters p and q, it follows that

M
p,q

* C, and W p,q * C, when q > 1. (27)

Hence (26) and (27) give the result.

Remark 3.11. By using techniques of ultra-distributions, Pilipović and Teofanov prove in [24–

26, 31, 32] parallel results comparing to Theorem 3.2′. Here they consider generalized modulation

spaces, where less growth restrictions are assumed on the weight function ω. It is for example not

necessary that ω should be bounded by polynomials.

Received: May 2008. Revised: September 2008.

References

[1] Baoxiang, W. and Chunyan, H., Frequency-uniform decomposition method for the gen-

eralized BO, KdV and NLS equations, J. Differential Equations, 239 (2007), 213–250.

[2] Bergh, J. and Löfström, J., Interpolation Spaces, An Introduction, Springer-Verlag,

Berlin Heidelberg NewYork, 1976.

[3] Feichtinger, H. G., Un espace de Banach de distributions temperees sur les groupes locale-

ment compacts abeliens (French), C. R. Acad. Sci. Paris SŐr. A-B 290 17 (1980), A791–A794.



CUBO
11, 4 (2009)

Pseudo-differential operators with smooth symbols ... 105

[4] Feichtinger, H. G., Banach spaces of distributions of Wiener’s type and interpolation,

in: Ed. P. Butzer, B. Sz. Nagy and E. Görlich (Eds), Proc. Conf. Oberwolfach, Functional

Analysis and Approximation, August 1980, Int. Ser. Num. Math. 69 Birkhäuser Verlag, Basel,

Boston, Stuttgart, 1981, pp. 153–165.

[5] Feichtinger, H. G., Banach convolution algebras of Wiener’s type, in: Proc. Functions,

Series, Operators in Budapest, Colloquia Math. Soc. J. Bolyai, North Holland Publ. Co.,

Amsterdam Oxford NewYork, 1980.

[6] Feichtinger, H. G., Modulation spaces on locally compact abelian groups. Technical report,

University of Vienna, Vienna, 1983; also in: M. Krishna, R. Radha, S. Thangavelu (Eds)

Wavelets and their applications, Allied Publishers Private Limited, NewDehli Mumbai Kolkata

Chennai Hagpur Ahmedabad Bangalore Hyderbad Lucknow, 2003, pp.99–140.

[7] Feichtinger, H. G., Atomic characterizations of modulation spaces through Gabor-type

representations, in: Proc. Conf. on Constructive Function Theory, Rocky Mountain J. Math.

19 (1989), 113–126.

[8] Feichtinger, H. G., Wiener amalgams over Euclidean spaces and some of their applications,

in: Function spaces (Edwardsville, IL, 1990), Lect. Notes in pure and appl. math., 136, Marcel

Dekker, New York, 1992, pp. 123–137.

[9] Feichtinger, H. G., Modulation spaces: Looking back and ahead, Sampl. Theory Signal

Image Process. 5 (2006), 109–140.

[10] Feichtinger, H. G. and Gröbner, P., Banach Spaces of Distributions Defined by Decom-

position Methods, I, Math. Nachr. 123 (1985), 97–120.

[11] Feichtinger, H. G. and Gröchenig, K. H., Banach spaces related to integrable group

representations and their atomic decompositions, I, J. Funct. Anal. 86 (1989), 307–340.

[12] Feichtinger, H. G. and Gröchenig, K. H., Banach spaces related to integrable group

representations and their atomic decompositions, II, Monatsh. Math. 108 (1989), 129–148.

[13] Feichtinger, H. G. and Gröchenig, K. H., Gabor frames and time-frequency analysis of

distributions, J. Functional Anal. (2) 146 (1997), 464–495.

[14] Folland, G. B., Harmonic analysis in phase space, Princeton U. P., Princeton, 1989.

[15] Gröbner, P., Banachräume Glatter Funktionen und Zerlegungsmethoden, Thesis, University
of Vienna, Vienna, 1992.

[16] Gröchenig, K. H., Describing functions: atomic decompositions versus frames, Monatsh.

Math.112 (1991), 1–42.

[17] Gröchenig, K. H., Foundations of Time-Frequency Analysis, Birkhäuser, Boston, 2001.



106 Joachim Toft CUBO
11, 4 (2009)

[18] Gröchenig, K. H. and Heil, C., Modulation spaces and pseudo-differential operators,

Integral Equations Operator Theory (4) 34 (1999), 439–457.

[19] Gröchenig, K. H. and Heil, C., Modulation spaces as symbol classes for pseudodifferential

operators in: M. Krishna, R. Radha, S. Thangavelu (Eds) Wavelets and their applications,

Allied Publishers Private Limited, NewDehli Mumbai Kolkata Chennai Hagpur Ahmedabad

Bangalore Hyderbad Lucknow, 2003, pp. 151–170.

[20] Gröchenig, K. H. and Heil, C., Counterexamples for boundedness of pseudodifferential

operators, Osaka J. Math. 41 (2004), 681–691.

[21] Gröchenig, K. and Leinert, M., Wiener’s lemma for twisted convolution and Gabor

frames, J. Amer. Math. Soc. (1) 17 (2004), 1–18.

[22] Hörmander, L., The Analysis of Linear Partial Differential Operators, vol I, III, Springer-

Verlag, Berlin Heidelberg NewYork Tokyo, 1983, 1985.

[23] Okoudjou, K., Embeddings of some classical Banach spaces into modulation spaces, Proc.

Amer. Math. Soc., 132 (2004), 1639–1647.

[24] Pilipović, S. and Teofanov, N., Wilson Bases and Ultramodulation Spaces, Math. Nachr.

242 (2002), 179–196.

[25] Pilipović, S. and Teofanov, N., On a symbol class of Elliptic Pseudodifferential Operators,

Bull. Acad. Serbe Sci. Arts 27 (2002), 57–68.

[26] Pilipović, S. and Teofanov, N., Pseudodifferential operators on ultra-modulation spaces,

J. Funct. Anal.208 (2004), 194–228.

[27] Ruzhansky, M. Sugimoto, M. Tomita, N. and Toft, J., Changes of variables in mod-

ulation and Wiener amalgam spaces, Preprint, 2008, Available at arXiv:0803.3485v1.

[28] Sjöstrand, J., An algebra of pseudodifferential operators, Math. Res. L. 1 (1994), 185–192.

[29] Sugimoto, M. and Tomita, N., The dilation property of modulation spaces and their

inclusion relation with Besov Spaces, J. Funct. Anal. (1), 248 (2007), 79–106.

[30] Tachizawa, K., The boundedness of pseudo-differential operators on modulation spaces,

Math. Nachr. 168 (1994), 263–277.

[31] Teofanov, N., Ultramodulation spaces and pseudodifferential operators, Endowment An-

drejević, Beograd, 2003.

[32] Teofanov, N., Modulation spaces, Gelfand-Shilov spaces and pseudodifferential operators,

Sampl. Theory Signal Image Process, 5 (2006), 225–242.

[33] Toft, J., Continuity and Positivity Problems in Pseudo-Differential Calculus, Thesis, De-

partment of Mathematics, University of Lund, Lund, 1996.



CUBO
11, 4 (2009)

Pseudo-differential operators with smooth symbols ... 107

[34] Toft, J., Subalgebras to a Wiener type Algebra of Pseudo-Differential operators, Ann. Inst.

Fourier (5) 51 (2001), 1347–1383.

[35] Toft, J., Continuity properties for modulation spaces with applications to pseudo-differential

calculus, I, J. Funct. Anal. (2), 207 (2004), 399–429.

[36] Toft, J., Continuity properties for modulation spaces with applications to pseudo-differential

calculus, II, Ann. Global Anal. Geom., 26 (2004), 73–106.

[37] Toft, J., Convolution and embeddings for weighted modulation spaces in: P. Boggiatto,

R. Ashino, M. W. Wong (eds) Advances in Pseudo-Differential Operators, Operator Theory:

Advances and Applications 155, Birkhäuser Verlag, Basel 2004, pp. 165–186.

[38] Toft, J., Continuity and Schatten-von Neumann Properties for Pseudo-Differential Oper-

ators and Toeplitz operators on Modulation Spaces, The Erwin Schrödinger International

Institute for Mathematical Physics, Preprint ESI 1732 (2005).

[39] Toft, J., Continuity and Schatten properties for pseudo-differential operators on modulation

spaces in: J. Toft, M. W. Wong, H. Zhu (Eds) Modern Trends in Pseudo-Differential Opera-

tors, Operator Theory: Advances and Applications 172, Birkhäuser Verlag, Basel, 2007, pp.

173–206.


	Articulo 7