Bulletin T.CXXXIII de l’Académie serbe des sciences et des arts − 2006 Classe des Sciences mathématiques et naturelles Sciences mathématiques, N\underline{o} 31

POSITIVITY IN TWISTED CONVOLUTION ALGEBRA AND FOURIER MODULATION SPACES^{1}

J. TOFT

(Presented at the 1st Meeting, held on February 24, 2006)

A b s t r a c t. Let W^{p,q} be the Fourier modulation space F M^{p,q} and let ∗_{σ} be the twisted convolution. If a ∈ D′ such that (a ∗_{σ} φ, φ) ≥ 0 for every φ ∈ C_{0}^{∞}, and χ ∈ S such that χ(0) ≠ 0, then we prove that χa ∈ W^{p,∞} iff a ∈ W^{p,∞}. We also present some extensions to the case when weighted Fourier modulation spaces are used.

AMS Mathematics Subject Classification (2000): 47B65, 35A21, 35S05 Key Words: Twisted convolution, Fourier modulation, positivity, conti-

nuity

0. Introduction

The aim of the paper is to discuss positivity in the twisted convolution algebra (the ∗_{σ}-algebra) in background of Fourier modulation spaces. At the same time we give a new proof of Bochner-Schwartz theorem in the case of twisted convolutions. (See also Proposition 2.8 in [To3].)

1This paper was presented at the Conference GENERALIZED FUNCTIONS 2004, Topics in PDE, Harmonic Analysis and Mathematical Physics, Novi Sad, September 22– 28, 2004

A motivation for studying positivity and algebraic properties in the ∗_{σ}- algebra is the close relation for such properties between the ∗_{σ}-algebra, operator theory and pseudo-differential calculus. Such questions were briefly investigated in [To1], [To2] and [To3]. In the present paper we continue the analysis in [To3] in terms of Fourier modulation spaces. The modulation spaces were introduced in time-frequency analysis by Feichtinger in [Fe]. Later on they have been used in certain problems in pseudo-differential calculus as well. (See [G], [L], [Ta], [Te] and [To5] and the references therein.) A reason for this is that informations concerning regularity as well as growing and decay properties can be easily obtained when using such spaces.

In order to be more specific, we give some necessary definitions. Let W be a symplectic vector space of dimension 2n < ∞ with the symplectic form σ. (The reader who is not familar with symplectic vector spaces may consider W as R^{2n} and σ(X, Y) = ⟨y, ξ⟩ − ⟨x, η⟩ when X = (x, ξ) ∈ R^{2n} and Y = (y, η) ∈ R^{2n}.)

Then the twisted convolution ∗_{σ} is defined by the formula

(a ∗_{σ} b)(X) ≡ (2/π)^{n/2} ∫ a(X − Y)b(Y)e^{2iσ(X,Y)} dY ,  (0.1)

when a, b ∈ L^{1}(W). Here and in what follows we use the standard notation for the usual functions and distribution spaces, see e. g., [H]. The definition of ∗_{σ} extends in different ways. It extends for example to a continuous and bilinear mapping from D′(W) × C_{0}^{∞}(W) to D′(W).

We are concerned with the set S′_{+}(W) of positive elements in the ∗_{σ}- algebra, i. e., the set of all a ∈ D′(W) such that (a ∗_{σ} φ, φ) ≥ 0 for every φ ∈ C_{0}^{∞}(W). Here (a, φ) ≡ ⟨a, \overline{φ}⟩, where ⟨ · , · ⟩ denotes the duality between elements in appropriate function and distribution spaces, and their duals. (A motivation for using S′_{+} instead of D′_{+} is given by Corollary 2.4 below.)

It might seem hard to find common structures for positivity results in the ∗_{σ}-algebra, but there are indeed such ones. In fact, in many situations the following principle holds. (See [To3].)

Assume that a ∈ S′_{+}. If a satisfies a certain regularity or boundedness property at the origin, then a and its Fourier transform â satisfy the same regularity or boundedness property everywhere.

In the present paper we prove that this principle holds when certain types of Fourier modulation spaces are involved.

1. Preliminaries

In this section we make some necessary preparations. First we discuss some algebraic and continuous properties for the twisted convolution. From the definitions it follows that any L^{p}-estimate which are valid when usual convolutions are involved, also holds if these convolutions are replaced by twisted convolutions. For example, Young’s inequality holds for twisted convolution. Moreover, many algebraic properties which are true for the usual convolution also hold for the twisted convolution. For example, if a, b, c ∈ S(W), then

a ∗_{σ} (b ∗_{σ} c) = (a ∗_{σ} b) ∗_{σ} c, (a ∗_{σ} b, c) = (a, c ∗_{σ} b̃),

where b̃(X) = \overline{b(−X)}.

In contrast to the usual convolution there is a canonical one to one correspondence between the twisted convolution algebra and operator algebras on L^{2}(V), leading to that further properties are available for the twisted convolution algebra. For example, if p ∈ [1, 2], then any L^{p}(W) is an algebra under twisted convolution. (See Corollary 1.4.3 in [To1] or Proposition 1.4 in [To2].)

Proposition 1.1 below is a restatement of certain results in [To2] and [To3]. (See also [To1].) The proof is therefore omitted. Here F_{σ}a denotes the symplectic Fourier transform which is defined by the formula

(F_{σ}a)(X) = π^{−n} ∫ a(Y)e^{2iσ(X,Y)} dY, a ∈ L^{1}(W).

Then F_{σ} is a homeomorphism on S(W) which in a usual way extends to a homeomorphism on S′(W), and to a unitary operator on L^{2}(W).

Proposition 1.1. The following are true:

1. the map (φ, a, ψ) ↦ φ∗_{σ} a∗_{σ} ψ is sequentially continuous from S(W)× S′(W) × S(W) to S(W), and from C_{0}^{∞}(W) × D′(W) × C_{0}^{∞}(W) to C^{∞}(W);

2. if a ∈ S′(W) and b ∈ S(W), then F_{σ}(a ∗_{σ} b) = (F_{σ}a) ∗_{σ} b;

3. if p ∈ [1, 2], then ∗_{σ} on S(W) extends uniquely to a continuous multiplication on L^{p}(W).

Remark 1.2. As remarked in the introduction, there are close relations between positivity in pseudo-differential calculus and positivity in the twisted convolution algebra. In fact, let a ∈ S(R^{2n}). Then the Weyl operator for a is defined by the formula

a^{w}(x, D)f(x) = (2π)^{−n} ∫∫ a((x + y)/2, ξ)f(y)e^{i⟨x−y,ξ⟩} dydξ,

where f ∈ S(R^{n}). Then a^{w}(x, D) is continuous on S, and the definition of a^{w}(x, D) extends to any a ∈ S′(R^{2n}) at which a^{w}(x, D) is continuous from S to S′.

If a ∈ S′(R^{2n}), then a^{w}(x, D) is positive semi-definite as an operator on S, if and only if F_{σ}a ∈ S′_{+}. (See [To1]–[To3].)

Next we discuss modulation spaces and start to consider appropriate conditions for the involved weight functions. Let ω and v be positive and measureable functions on the vector space V of finite dimension. Then ω is called v-moderate if there is a constant C > 0 such that ω(x + y) ≤ Cω(x)v(y) for every x, y ∈ V. We say that ω is a moderate function if ω(x + y) ≤ Cω(x)ω(y). The set of all positive functions ω on V such that ω is v-moderate for some polynomial v on V is denoted by P (V). Also let P_{0}(V) be the set of all ω_{0} ∈ P (V) such that for every multi-index α, there is a constant C_{α} such that |∂^{α}ω_{0}| ≤ C_{α}ω_{0}. If ω ∈ P (V) and φ ∈ C_{0}^{∞} \0 is nonnegative, then it follows that ω_{0} = ω ∗φ ∈ P_{0}(V) and that C^{−1}ω ≤ ω_{0} ≤ Cω for some constant C > 0. (See [To5] or [To6].)

Next we give examples on weight functions which are of particular interest. For any s, t ∈ R and x ∈ R^{n}, let τ_{s}(x) = ⟨x⟩^{s}, where ⟨x⟩ = (1+|x|^{2})^{1/2}. Also let τ_{s,t} = τ_{t} ⊗ τ_{s}. Then

τ_{s}(x, y) = (1 + |x|^{2} + |y|^{2})^{s/2} and τ_{s,t}(x, y) = (1 + |x|^{2})^{t/2}(1 + |y|^{2})^{s/2}.

It follows that τ_{s} ∈ P_{0}, and that τ_{s} is τ_{|s|}-moderate.

Assume that χ ∈ S(W)\0, p, q ∈ [1, ∞], ω ∈ P (W ×W) and a ∈ S′(W), and let (τ_{X}χ)(Y) = χ(Y − X). Then we set

\begin{aligned}∥a∥_{M_{(ω)}^{p,q}} & ≡ ( ∫ ( ∫ |F_{σ}(a τ_{X}χ)(Y)ω(X, Y)|^{p} dX)^{q/p} dY )^{1/q}, \\ ∥a∥_{W_{(ω)}^{p,q}}| & ≡ ( ∫ ( ∫ |F_{σ}(a τ_{X}χ)(Y)ω(X, Y)|^{p} dY )^{q/p} dX)^{1/q} \end{aligned}  (1.1)

(with obvious interpretation when p = ∞ and/or q = ∞). Note here that ∥a∥_{W_{(ω)}^{p,q}} and ∥a∥_{M_{(ω)}^{p,q}} may attain the value +∞.

Definition 1.3. Assume that χ ∈ S(W) \ 0, p, q ∈ [1, ∞], and ω ∈ P (W × W). Then the modulation space M_{(ω)}^{p,q}(W) consists of all a ∈ S′(W) such that ∥a∥_{M_{(ω)}^{p,q}} < ∞. In the same way, W_{(ω)}^{p,q}(W) consists of all a ∈ S′(W) such that ∥a∥_{W_{(ω)}^{√,⨿}} < ∞.

If ω = 1, then the notation W^{p,q} is used instead of W_{(ω)}^{p,q}. Moreover, if p = q, then the notations W_{(ω)}^{p} and W^{p} are used instead of W_{(ω)}^{p,p} and W^{p,p}, respectively. If in addition W = R^{2n}, then we set W_{s}^{p,q} = W_{(ω)}^{p,q} when ω(X, Y) = τ_{s}(X, Y), and W_{s,t}^{p,q} = W_{(ω)}^{p,q} when ω(X, Y) = τ_{s,t}(X, Y). Such spaces are common in many situations. (See [G] and the references therein.)

If ω(X, Y) is a weight function on W ×W which is constant with respect to X or Y, then we set ω(X, Y) = ω(Y) and ω(X, Y) = ω(X), respectively. In this situation, ω is sometimes considered as a function on W instead of a function on W × W.

The convention using parenthesis, when weight functions are involved in the definition of function spaces, are also used in other situations. For example, if ω ∈ P (W), then L_{(ω)}^{p}(W) consists of all measurable functions a such that a ω ∈ L^{p}(W).

The space W_{(ω)}^{p,q} is a Fourier modulation space. In fact, by Parseval’s formula it follows that

F_{σ}(â τ_{Y} (F_{σ}χ̌))(X) = e^{2iσ(X,Y)}F_{σ}(a τ_{X}χ)(Y),  (1.2)

where χ̌(X) = χ(X). This in turn implies that

W_{(ω)}^{p,q} = F_{σ}M_{(ω_{0})}^{p,q} when ω_{0}(Y, X) = ω(X, Y).  (1.3)

Here it is essential that the definitions of M_{(ω)}^{p,q} and W_{(ω)}^{p,q} are independent of the choice of χ ∈ S \ 0. This is a consequence of the following proposition, where some important properties for modulation spaces are recalled. Here and in what follows, p′ denotes the conjugate exponent for p, i. e., 1/p + 1/p′ = 1.

Proposition 1.4. Assume that ω, ω_{1}, ω_{2} ∈ P (W ×W) and p, q ∈ [1, ∞]. Then the following are true:

1. M_{(ω)}^{p,q}(W) and W_{(ω)}^{p,q}(W) are Banach spaces which are independent of the choices of χ ∈ S(W) \ 0 in (1.1). Moreover, different choices of χ give rise to equivalent norms; 2. if p_{1}, p_{2}, q_{1}, q_{2} ∈ [1, ∞] such that p_{1} ≤ p_{2}, q_{1} ≤ q_{2} and ω_{2} ≤ Cω_{1} for

some constant C, then

S(W) ↪ W_{(ω_{1})}^{p_{1},q_{1}}(W) ↪ W_{(ω_{2})}^{p_{2},q_{2}}(W) ↪ S′(W) .

Moreover, W^{1,p} ⊆ C(W) ∩ L^{∞};

3. assume that 1 ≤ q_{1} ≤ min(p, p′) and max(p, p′) ≤ q_{2} ≤ ∞, and that in addition ω(X, Y) = ω(Y). Then

W_{(ω)}^{p,q_{1}} ↪ F_{σ}(L_{(ω)}^{p}) ↪ W_{(ω)}^{p,q_{2}} and W_{(ω)}^{p,q} ∩ E′ = F_{σ}(L_{(ω)}^{p}) ∩ E′ ;

4. let ω_{0}(X, Y) = ω(Y, X), and assume in addition that q ≤ p. Then F_{σ}W_{(ω)}^{p,q} ↪ W_{(ω_{0})}^{q,p}. In particular, F_{σ} is a homeomorphism on W^{p}.

P r o o f. The assertions (1), (2) and (4) follow from [G] and (1.3) (see also [Fe], or [To5] and the references therein). The assertion (3) is an immediate consequence of Proposition 4.3 and Theorem 5.5 in [To6], and (1.3). □

Remark 1.5. Assume that χ ∈ C_{0}^{∞} and that ω ∈ P (W × W), and let W̃_{(ω)}^{p,q} be the set of all a ∈ D′(W) such that ∥a∥_{W_{(ω)}^{p,q}} < ∞. Then it follows

that W̃_{(ω)}^{p,q} ⊆ S′. Consequently, W̃_{(ω)}^{p,q} = W_{(ω)}^{p,q} in view of Proposition 1.4.

We finish the section with the following proposition which concerns multiplications and differentiations of elements in Fourier modulation spaces. We refer to [To6] for the proof. Here and in what follows, if ω ∈ P_{0}(W), then ω(D) is the linear and continuous operator on S′(W) which means a multiplication by ω on the symplectic Fourier transform side.

Proposition 1.6. Assume that p, q ∈ [1, ∞], ω ∈ P (W × W), ω_{0} ∈ P_{0}(W), and set

ω_{1}(X, Y) = ω(X, Y)ω_{0}(X) and ω_{2}(X, Y) = ω(X, Y)ω_{0}(Y).

Then a ↦ a ω_{0} is a homeomorphism from W_{(ω_{1})}^{p,q} to W_{(ω)}^{p,q}, and a ↦ ω_{0}(D)a is a homemorphism from W_{(ω_{2})}^{p,q} to W_{(ω)}^{p,q}.

2. Positive elements in the ∗_{σ}-algebra

In this section we prove that if ω ∈ P is appropriate, a ∈ S′_{+} and χ a ∈ W_{(ω)}^{p,∞} for some χ ∈ C_{0}^{∞} such that χ(0) ≠ 0, then a ∈ W_{(ω)}^{p,∞}. We use these results to prove that S′_{+} ⊆ S′.

In the following proposition we recall some facts for elements in C_{+}(W) ≡ S′_{+}(W)∩C(W), the set of positive elements in ∗_{σ}-algebra which at the same time are continuous functions. The proof is omitted since the result is a consequence of Proposition 1.10, Theorem 3.3 and Corollary 3.7 in [To3].

Proposition 2.1. Assume that a ∈ S′_{+}(W). Then the following are true:

1. if φ ∈ C_{0}^{∞}(W), then φ̃ ∗_{σ} a ∗_{σ} φ ∈ C_{+} ∩ C^{∞}. Moreover, if in addition ∫ φ dX = (π/2)^{n/2} and φ_{ε} = ε^{−2n}φ( · /ε), then φ̃_{ε} ∗_{σ} a ∗_{σ} φ_{ε} → a in D′ as ε → 0;

2. if a is a continuous function near the origin, then a ∈ C_{+};

3. if a ∈ C_{+}, then a ∈ L^{2} vanishes at infinity, and |a(X)| ≤ a(0);

4. a ∈ C_{+} if and only if a = ψ̃ ∗_{σ} ψ for some ψ ∈ L^{2};

5. if b ∈ C_{0}^{∞} ∩ C_{+}, then (a, b) ≥ 0.

As a consequence of Proposition 1.1 (4) and Proposition 2.1, it follows that if a ∈ C_{+}, then (φ, ψ)_{a} ≡ (a ∗_{σ} φ, ψ) is a semi-scalar product on L^{2}. Hence Cauchy-Schwartz inequality holds, i. e.,

|(φ, ψ)_{a}|^{2} ≤ (φ, φ)_{a}(ψ, ψ)_{a}.

The investigations in Section 4 in [To3] as well as in the present section depend on the following result. (Cf. Proposition 4.6 in [To3].) Here a slightly different proof is given.

Proposition 2.2. Assume that a ∈ S′_{+}(W) and that χ ∈ C_{+}(W) ∩ C_{0}^{∞}(W). Then F_{σ}(a χ) is a non-negative function. If u = (F_{σ}(a χ))^{1/2} and X, Y ∈ W, then

|F_{σ}(a τ_{Y} χ)(X)| ≤ u(X + Y)u(X − Y)  (2.1)

P r o o f. By Proposition 2.1 and a simple argument of approximation, it suffices to prove the result in the case a ∈ C_{+} ∩ C^{∞}.

By straight-forward computations it follows that \overline{χ} ∈ C_{+}, and that S′_{+} is invariant under multiplication by exponentials. Hence Proposition 2.1 (5) gives that F_{σ}(a χ)(X) = (e^{2iσ(X, ·)}a, \overline{χ}) is non-negative for every X ∈ W. This proves the first part of the proposition.

Next we prove (2.1). By Proposition 2.1, there is a function ψ ∈ L^{2} such that \overline{χ} = ψ̃ ∗_{σ} ψ. Then \overline{χ}_{Y} = ψ̃_{Y} ∗_{σ} ϕ_{Y}, where ψ_{Y} = ψ( · + Y) and ϕ_{Y} = e^{−2iσ(Y, ·)}ψ. From the fact that e^{2iσ(X, ·)}a ∈ S′_{+} for every X ∈ W, an application of Cauchy-Schwartz inequality gives

\begin{aligned}|F_{σ}(a τ_{Y} χ)(X)|^{2} & = |(e^{2iσ(X, ·)}a, ψ̃_{Y} ∗_{σ} ϕ_{Y})|^{2} \\ & ≤ (e^{2iσ(X, ·)}a, ψ̃_{Y} ∗_{σ} ψ_{Y})(e^{2iσ(X, ·)}a, ϕ̃_{Y} ∗_{σ} ϕ_{Y}). \end{aligned} (2.2)

By simple calculations it follows that

ψ̃_{Y} ∗_{σ} ψ_{Y} = e^{2iσ(Y, ·)}\overline{χ}, and ϕ̃_{Y} ∗_{σ} ϕ_{Y} = e^{−2iσ(Y, ·)}\overline{χ}.

This implies that

(e^{2iσ(X, ·)}a, ψ̃_{Y} ∗_{σ} ψ_{Y}) = (e^{2iσ(X−Y, ·)}a, \overline{χ}) = F_{σ}(a χ)(X − Y),

and similarly (e^{2iσ(X, ·)}a, ϕ̃_{Y} ∗_{σ} ϕ_{Y}) = F_{σ}(a χ)(X + Y). This proves (2.2), and the result follows. The proof is complete. □

The following result is now an immediate consequence of Proposition 1.4 (3), Proposition 2.2 and Hölder’s inequality.

Theorem 2.3. Assume that ω_{0} ∈ P (W × W) such that ω_{0}(X, Y) = ω_{0}(Y), and set

ω_{1}(X, Y) = (ω_{0}(X − Y)ω_{0}(X + Y))^{1/2}.

Also assume that a, χ ∈ S′_{+}(W) such that χ ∈ C_{0}^{∞} \ 0 and a χ ∈ W_{(ω_{0})}^{p,∞} for some p ∈ [1, ∞]. Then a ∈ W_{(ω_{1})}^{p,∞}.

As a consequence of Theorem 2.3 we obtain Bochner-Schwartz theorem in the case when the usual convolutions are replaced by twisted convolutions.

Corollary 2.4. The set S′_{+}(W) is contained in S′(W).

P r o o f. Assume that a, χ ∈ S′_{+} such that χ ∈ C_{0}^{∞} \ 0. Then a χ ∈ E′. Hence F_{σ}(a χ) ∈ L_{(ω_{0})}^{p} for some ω_{0} ∈ P which in turn implies that a ∈ W_{(ω)}^{p,∞} for some ω ∈ P by Proposition 1.4 (3) and Theorem 2.3. Since W_{(ω)}^{p,∞} ⊂ S′, the result follows. □

By Corollary 2.4, it follows that Proposition 2.2 also holds under the weaker assumption for χ that χ ∈ S′_{+} ∩ S. Moreover, by (1.2) it follows that (2.1) is equivalent to

|F_{σ}(â τ_{Y} (F_{σ}χ̌))(X)| ≤ u(X + Y)u(Y − X).  (2.1)′

Theorem 2.3 may now be improved in the following way.

Theorem 2.3′ Assume that ω_{0} and ω_{1} are the same as in Theorem 2.3, and let ω_{2}(X, Y) = ω_{1}(Y, X). Also assume that a, χ ∈ S′_{+}(W) such that χ ∈ S and χ(0) ≠ 0 and a χ ∈ W_{(ω_{0})}^{p,∞} for some p ∈ [1, ∞]. Then a ∈ W_{(ω_{1})}^{p,∞} and F_{σ}a ∈ W_{(ω_{2})}^{p,∞}.

P r o o f. Let Ω be an open neighbourhood of the origin such that χ(X) ≠ 0 as X ∈ Ω, and let χ_{0} = ψ̃ ∗_{σ} ψ, where ψ ∈ C_{0}^{∞}(W) \ 0. Then 0 ≠ χ_{0} ∈ C_{+} ∩ C_{0}^{∞}, and the support of χ_{0} is contained in Ω, provided ψ is chosen with sufficiently small support. This gives

a χ_{0} = (χ_{0}/χ)(a χ) ∈ S · W_{(ω_{0})}^{p,∞} ⊆ W_{(ω_{0})}^{p,∞}.  (2.3)

Here the last step is a consequence of Theorem 5.5 in [To6], and the fact that multiplications are replaced by convolutions on the Fourier transform. Hence Theorem 2.3 and (2.1)′ show that a ∈ W_{(ω_{1})}^{p,∞} and F_{σ}a ∈ W_{(ω_{2})}^{p,∞}. The proof is complete. □

Theorem 2.5. Assume that v_{0} ∈ P (W × W) such that v_{0}(X, Y) = v_{0}(Y) is an even and moderate function, v(X, Y) = v_{0}(X)v_{0}(Y), p ∈ [1, ∞], a ∈ S′_{+}(W), and χ ∈ S(W) such that χ(0) ≠ 0. Then the following conditions are equivalent:

1. F_{σ}(a χ) ∈ L_{(1/v_{0})}^{p}(W);

2. a χ ∈ W_{(1/v_{0})}^{p,∞}(W);

3. a χ ∈ W_{(1/v)}^{p,∞}(W);

4. a ∈ W_{(1/v)}^{p,∞}(W); 5. a, F_{σ}a ∈ W_{(1/v)}^{p,∞}(W).

Before the proof we observe that if v_{0} is even and moderate, then

(v_{0}(X − Y)v_{0}(X + Y))^{1/2} ≤ Cv_{0}(X)v_{0}(Y),  (2.4)

for some constant C.

P r o o f. Let χ_{0} be the same as in the proof of Theorem 2.3′. Then F_{σ}(aχ_{0}) ∈ L_{(1/v_{0})}^{p} (cf. (2.3)). This in turn implies that aχ_{0} ∈ W_{(1/v_{0})}^{p,∞} by Proposition 1.4 (3). Hence Theorem 2.3′ and (2.4) show that (5) holds. By similar arguments it follows that (2) implies (5).

The implication (5) ⇒ (4) is obvious, and the implications (4) ⇒ (2) and (4) ⇒ (1) follow immediately from Theorem 5.5 in [To6] and the inequality

∥F_{σ}(aχ)∥_{L_{(1/v_{0})}^{p}} ≤ C∥a∥_{W_{(1/v)}^{p,∞}},

respectively.

When proving the equivalence (2) ⇔ (3) we may assume that v_{0} ∈ P_{0}. Then the assertion follows immediately from Proposition 1.6 and the fact that the map φ ↦ v_{0} φ is a homeomorphism on S. The proof is complete. □

Corollary 2.6 Assume that s ∈ R, p ∈ [1, ∞], a ∈ S′_{+}(R^{2n}), and χ ∈ S(R^{2n}) such that χ(0) ≠ 0. Then the following are true:

1. if s ≤ 0, then a χ ∈ W_{s,0}^{p,∞} if and only if a ∈ W_{s,s}^{p,∞};

2. if s ≥ 0 and a χ ∈ W_{s,0}^{p,∞}, then a ∈ W_{s^{1/4},s^{1/4}}^{p,∞}.

P r o o f. The assertion (1) follows by letting v_{0} = τ_{s} in Theorem 2.5. The assertion (2) follows from Theorem 2.3′ and the fact that for some constant C we have ⟨X⟩⟨Y ⟩ ≤ C⟨X − Y ⟩^{2}⟨X + Y ⟩^{2}. The proof is complete. □

Remark 2.7 Assume that a ∈ S′_{+}, χ ∈ S such that χ(0) ≠ 0, and that F_{σ}(a χ) ∈ L^{1}. Then Theorem 2.5 shows that a ∈ W^{1,∞}. This is also a consequence of Theorem 1.5 in [To4].

In fact, from the assumptions it follows that a χ is a continuous function near the origin. Hence Proposition 2.1 shows that a belongs to the set of all symbols such that their corresponding Weyl operators are trace-class operators on L^{2}. The result now follows since the latter set is contained in W^{1,∞} in view of Theorem 1.5 in [To4].

Finally we have the following result, parallel to Theorem 2.3′.

Proposition 2.8 Assume that p, q ∈ [1, ∞] and p_{0} ∈ [2^{−1}, ∞] satisfy p ≤ q and 1/p + 1/q = 1/p_{0}. Assume also that v_{0}, χ and a are the same as in Theorem 2.5, and set v(X, Y) = (v_{0}(X − Y)v_{0}(X + Y))^{1/2}. If F_{σ}(a χ) ∈ L_{(1/v_{0})}^{p_{0}}(W), then a, F_{σ}a ∈ W_{(1/v)}^{p,q}(W).

P r o o f. It is no restriction to assume that χ ∈ C_{+} ∩ C_{0}^{∞}. We use the same notations as in Proposition 2.2. Let h = (u/v_{0}^{1/2})^{p} and r = q/p ≥ 1. Then for some constant C, Young’s inequality gives

\begin{matrix}∥a∥_{W_{(1/v)}^{p,q}} = ( ∫ ( ∫ |F_{σ}(a τ_{X}χ)(Y)/v(X, Y)|^{p} dY )^{q/p} dX)^{1/q} \\ ≤ C(∥h ∗ h∥_{L^{r}})^{1/p} ≤ C′(∥h∥_{L^{2p_{0}/p}})^{2/p} = ∥F_{σ}(a χ)∥_{L_{(1/v_{0})}^{p_{0}}}. \end{matrix}

The result is now a consequence of Proposition 1.4. □

Corollary 2.9 Let v_{0}, v, a and χ be the same as in Proposition 2.8, and assume that F_{σ}(a χ) ∈ L_{(1/v_{0})}^{1/2} Then a, F_{σ}a ∈ W_{(1/v)}^{1}.

Acknowledgements. I am very grateful to the professors Stevan Pilipovic, Bogoljub Stankovic and Nenad Teofanov, and the other organizers of GF2004, for a nice meeting, fruitful discussions and nice hospitality during the conference. I also thank Roger Petersson for careful reading of the paper, and the referee for valuable comments.

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Department of Mathematics and Systems Engineering Växjö University 351 95 Växjö Sweden

e-mail: joachim.toft@vxu.se