Bulletin T.CXXIII de l’Académie Serbe des Sciences et des Arts - 2002 Classe des Sciences mathématiques et naturelles Sciences mathématiques, N\underline{o} 27

ON A SYMBOL CLASS OF ELLIPTIC PSEUDODIFFERENTIAL OPERATORS

S. PILIPOVIĆ, N. TEOFANOV^{∗}

(Presented at the 5th Meeting, held on June 21, 2002)

A b s t r a c t. We consider a class of symbols with prescribed smoothness and growth conditions and give examples of such symbols. The introduced class contains certain polynomial symbols and symbols with more than polynomial growth in phase space. The corresponding pseudodifferential operators defined as the Weyl transforms of the symbols are elliptic. As an application, we give a result on isomorphisms between modulation spaces. In particular, we show that the Bessel potentials establish such isomorphisms.

AMS Mathematics Subject Classification (2000): 47G30

Key Words: pseudodifferential operators, ellipticity, modulation spaces

1. Introduction

Theory of pseudodifferential operators has been established some thirty years ago, with important applications in diverse fields of theoretical and applied mathematics such as partial differential equations and quantum mechanics [17], [15], [18]. In the last decade it has been successfully applied in

∗This research was supported by MNTR of Serbia, project no. 1835

time-frequency analysis and communication theory [5], [7], [6]. In this context new classes of symbols and the corresponding operators are introduced (see [16], [8], [12]). There are different ways to define a pseudodifferential operator by the means of its symbol. In this paper we consider the so called Weyl correspondence (see (1)). As noted in [7], Feichtinger’s modulation spaces introduced in [2] are the most natural framework for time-frequency analysis. Therefore, it is of particular importance to study the action of pseudodifferential operators on modulation spaces. Operators with symbols in modulation spaces are studied in [9], [8] and [12] while in [16] and [1] symbols with at most polynomial growth are considered. However, in quantum field theory it is of interest to study symbols with more than polynomial growth in momentum space in the framework of the corresponding spaces of ultradistributions [11]. A relationship between modulation spaces and ultradistributions is given in [13].

In this paper we define a class of symbols which can grow almost exponentially in phase space. It also contains a large class of polynomials, such as the symbols of the Bessel potentials. In particular, it contains the Schröedinger-type operators with appropriate almost exponentially bounded potentials.

As an application we prove that a class of partial differential operators with constant coefficients and, in particular, the Bessel potentials establish isomorphism between certain modulation spaces.

2. Notation

If x = (x_{1}, …, x_{d}) ∈ R^{d} d ∈ N, then |x| = \sqrt{x_{1}^{2} + … + x_{d}^{2}}, and ⟨x⟩ = (1 + |x|^{2})^{1/2}. For multi-indices α, β ∈ N_{0}^{d}, we have |α| = α_{1} + … + α_{d}, α! = α_{1}! ⋯ α_{d}!, x^{α} = x_{1}^{α_{1}} ⋯ x_{d}^{α_{d}} and, if β ≤ α, i.e., β_{j} ≤ α_{j},

j = 1, 2, …, d, \binom{α}{β} = \binom{α_{1}}{β_{1}} · · · · · \binom{α_{d}}{β_{d}}. We write D^{α} = D_{1}^{α_{1}} ⋯ D_{d}^{α_{d}}

= (\frac{1}{2πi} \frac{∂}{∂x_{1}} )^{α_{1}} ⋯ (\frac{1}{2πi} \frac{∂}{∂x_{d}} )^{α_{d}} . We denote by C a positive constant, not necessarily the same at every occurrence. The symbol γ is reserved for a real number in (0, 1) unless otherwise is indicated. The translation and the modulation of a test function f is given by T_{x}f(·) = f(· − x), x ∈ R^{d}, and M_{ξ}f(·) = e^{2πiξ·}f(·), ξ ∈ R^{d} respectively, and extended to a distribution via duality. Dual pairing is denoted by ⟨·, ·⟩. For functions φ, ψ ∈ S (S is the space of rapidly decreasing functions), ⟨φ, ψ⟩ = ∫ φψdx. The Fourier transform of ψ ∈ L^{2}(R^{d}) is given by Fψ(ξ) = ψ̂(ξ) = ∫_{R^{d}} e^{−2πixξ}ψ(x)dx, and F^{−1}ϕ(x) = ∫_{R^{d}} e^{2πixξ}ϕ(ξ)dξ, is the inverse Fourier transform of ϕ ∈ L^{2}(R^{d}). We denote the norm in L^{2} by ∥ · ∥, and ∥ · ∥_{∞} denotes L^{∞} norm. Recall, Gelfand Shilov type space S^{(γ)} is defined by S^{(γ)} = proj lim_{h→∞} S_{h}^{(γ)}, where

S_{h}^{(γ)}, h ≥ 0, is the space of smooth functions f on R^{d} such that

sup_{α,β∈N_{0}^{d}} \frac{h^{α+β}}{α!^{1/γ}β!^{1/γ}} ∥x^{α}D^{β}f(x)∥_{∞} < ∞.

It is a Banach space and the Fourier transform is an isomorphism of S^{(γ)} into itself. For fixed γ ∈ (0, 1), the space D^{(γ)}(Ω) is defined by D^{(γ)}(Ω) = ind lim_{K⊂⊂Ω} D^{(γ)}(K), where Ω is an open subset in R^{d} and D^{(γ)}(K) is the set

of all complex valued infinitely differentiable functions φ(t) supported by K such that for every h > 0 there exists a positive constant C > 0 such that

sup_{t∈K} |D^{α}φ(t)| ≤ Ch^{α}α!^{1/γ}, α ∈ N_{0}^{d}.

We call D^{′(γ)}(Ω) the Beurling–Gevrey ultradistribution space and S^{′(γ)} the Beurling–Gevrey tempered ultradistribution space.

We observe pseudodifferential operators σ(x, D) as the Weyl transforms of symbols σ(x, ξ), i.e.

σ(x, D)f(x) = ∫_{R^{d}} ∫_{R^{d}} σ (\frac{x + y}{2} , ξ) e^{2πi(x−y)ξ}f(y)dydξ, f ∈ S^{(γ)}(R^{d}). (1)

3. A Class of Symbols

Throughout this section γ ∈ (0, 1) is fixed. We consider a class of symbols σ ∈ C^{∞}(R^{d} × R^{d}) satisfying:

(S1) σ(z) ≥ 1, z = (x, ξ) ∈ R^{d} × R^{d}.

(S2) (∃C > 0) (∃η ≥ 0) such that σ(z + w) ≤ Ce^{η|z|^{γ}}σ(w), z, w ∈ R^{2d}.

(S3) (∀h ≥ 0) (∃C > 0) (∃s ≥ 0) such that

sup_{α∈N_{0}^{2d},|α|≥1} |\frac{h^{|α|}}{α!^{1/γ}} D^{α}σ(z)| ≤ C \frac{σ(z)}{(1 + |z|)^{s}} , z ∈ R^{2d}.

60 S. Pilipović, N. Teofanov

(S4) σ(x, ξ) ≤ σ(x, ξ′) for all ξ, ξ′ ∈ R^{d} such that |ξ| ≤ |ξ′|.

Note that instead (S1) we could consider the condition σ(z) ≥ C, z ∈ R^{2d}, for some C > 0. We put C = 1 for the sake of simplicity. Also, (S2)

implies σ(z) ≥ \frac{σ(0)}{C}e^{−η|z|^{γ}}, z ∈ R^{2d}.

If σ_{1}, σ_{2} satisfy (S1)-(S4), then it is clear that σ_{1} · σ_{2} satisfies (S1)-(S4) as well.

Theorem 1. The following functions satisfy conditions (S1)-(S4).

a) σ(z) = ∑_{k=0}^{n} a_{k}⟨z⟩^{2k}, z = (x, ξ) ∈ R^{d} × R^{d}, where a_{0} ≥ 1, a_{k} > 0;

b) σ(x, ξ) = (1 + |x|^{2} + |ξ|^{2})^{s/2}, s ≥ 0, x, ξ ∈ R^{d}. In particular, σ(ξ) = (1 + |ξ|^{2})^{s/2}, s ≥ 0, ξ ∈ R^{d};

c) σ(x, ξ) = |ξ|^{2} + V (x), x, ξ ∈ R^{d}, where (V1) V ∈ C^{∞}(R^{d}), V ≥ 1, V (x) → ∞ when |x| → ∞. (V2) (∃C > 0) (∃η > 0) such that V (x + y) ≤ Ce^{η|x|^{γ}}V (y), x, y ∈ R^{d}. (V3) (∀h ≥ 0) (∃C > 0) such that

sup_{α∈N_{0}^{d},|α|≥1} |\frac{h^{|α|}}{α!^{1/γ}} D^{α}V (x)| ≤ CV (x), x ∈ R^{d};

d) σ(x, ξ) = e^{(1+|x|^{2}+|ξ|^{2})^{γ/2}}, x, ξ ∈ R^{d};

e) σ(x, ξ) = (f ∗ ϕ)(x, ξ), x, ξ ∈ R^{d}, where

f(x, ξ) = \{ \begin{matrix}e^{μ(|x|^{γ}+|ξ|^{γ})} & |ξ| ≥ |ξ_{0}| > 1 \\ e^{μ|x|^{γ}+r|ξ|} & |ξ| < |ξ_{0}|, \end{matrix}

where μ > 0, r = μ|ξ_{0}|^{γ−1}, and ϕ ∈ D^{(γ)}(Ω), ϕ ≥ 0, ∫_{R^{2d}} ϕ(x, ξ)dxdξ = 1;

f) σ(x, ξ) = C −φ(x, ξ), x, ξ ∈ R^{d}, where φ ∈ S^{(γ)}(Ω) is an even function such that φ(x, ξ) ≤ φ(x, ξ′) if |ξ| ≤ |ξ′|, and C > sup_{x,ξ} φ(x, ξ) + 1.

Remark: The function σ(z) = ∑_{k=0}^{n} a_{k}z^{k}, z = (x, ξ) ∈ R^{d} × R^{d}, a_{0} ≥ 1,

a_{k} > 0 satisfies the conditions (S1), (S2), (S3).

P r o o f. It is not difficult to prove a), b) and f), so we skip these parts of the proof.

Proof of c) easily follows from the assumptions (V1), (V2) and (V3). Note that |ξ|^{2} + V (x) is the symbol of the Schrödinger operator −△ + V, with the increasing potential V (see also [16] in this context).

d) We show that σ satisfies (S3). To that end we use the Cauchy integral formula [10, Chapter 2] for polydisc K ⊂ C^{n} given by

K = ∏_{j=1}^{n} K_{j} = {z = (z_{1}, …, z_{n}) | z_{j} ∈ K_{j}, j = 1, …, n},

where K_{j} are discs in C with the boundaries ∂K_{j}, j = 1, …, n. We have

∂^{α}σ(z) = \frac{α!}{(2πi)^{2n}}  (2)

× ∫_{∂K_{1}} ⋯ ∫_{∂K_{2n}} \frac{σ(y, η)dydη}{∏_{j=1}^{n}(y_{j} − x_{j})^{α_{j}+1} · ∏_{j=1}^{n}(η_{j} − ξ_{j})^{α_{n+j}+1}} ,

where z = (x, y) ∈ C^{2n} and ∂K_{j} is the circle with center z_{j} and radius r < 1/\sqrt{2n}. Let r = 1/(2n). From (2) it follows

\frac{|∂^{α}σ(z)|}{α!} ≤ \frac{1}{(2πi)^{2n}(1/(2n))^{|α|+1}}

×(2πi)^{2n} max_{θ_{j}∈[0,2π],j=1,...,2n} e^{(1+|x_{1}+ \frac{1}{2n} e^{θ_{1}i}|^{2}+···+|ξ_{n}+ \frac{1}{2n} e^{θ_{2n}i}|^{2})^{γ/2}}

≤ \frac{1}{(1/(2n))^{|α|+1}} e^{(1+|x|^{2}+|ξ|^{2}+ \frac{1}{n})^{γ/2}} ≤ C \frac{1}{(1/(2n))^{|α|+1}} e^{(1+|x|^{2}+|ξ|^{2})^{γ/2}}.

Since (α!)^{ε} > a^{|α|} for all a, ε > 0 and |α| large enough, there exists C > 0 such that, for every h ≥ 0,

\frac{h^{|α|}}{α!^{1/γ}} ≤ C \frac{1}{α!}, for γ ∈ (0, 1).

This implies (S3). The function σ obviously satisfies (S1), (S2) and (S4).

e) Conditions (S1) and (S2) are obviously satisfied. We show (S3) for |ξ| ≥ |ξ_{0}|. Assume, for simplicity, that d = 1, and Ω = [−1, 1]×[−1, 1]. Then we have

|\frac{h^{|α|+|β|}}{α!^{1/γ}β!^{1/γ}} D_{x}^{α}D_{ξ}^{β}σ(x, ξ)|

= |\frac{h^{|α|+|β|}}{α!^{1/γ}β!^{1/γ}} ∫_{R} ∫_{R} (e^{μ(|y|^{γ}+|η|^{γ})}D_{x}^{α}D_{ξ}^{β}ϕ(x − y, ξ − η)) dydη|

≤ C ∫_{|x−y|≤1} ∫_{|ξ−η|≤1} e^{μ(|y|^{γ}+|η|^{γ})}dydη ≤ C ∫_{|y|≤1} ∫_{|η|≤1} e^{μ(|x−y|^{γ}+|ξ−η|^{γ})}dydη

≤ Ce^{μ((|x|+1)^{γ}+(|ξ|+1)^{γ})} ≤ C ∫_{|y|≤1} ∫_{|η|≤1} e^{μ((|x|−y+2)^{γ}+(|ξ|−η+2)^{γ})}ϕ(y, η)dydη

≤ C ∫_{R} ∫_{R} e^{μ(|x−y|^{γ}+|ξ−η|^{γ})}ϕ(y, η)dydη ≤ Cσ(x, ξ),

that is, σ satisfies (S3). We now show (S4) for |ξ_{0}| ≥ |ξ|. If |ξ_{0}| ≥ |ξ| ≥ 1, after an easy computation, we obtain

σ(x, ξ) = \frac{2}{r}e^{r|ξ|} ∫_{R} e^{μ(|x−y|)^{γ}} (∫_{0}^{1} sinh(rη)ϕ(x − y, η)dη) dy.

This implies (S4). Let now |ξ| ≤ |ξ′| ≤ 1. If, for example, −1 ≤ ξ′ ≤ −ξ ≤ 0, then

σ(x, ξ) = ∫_{R} e^{μ|x−y|^{γ}} (e^{rξ} ∫_{−ξ}^{1} f(x, η)dη + e^{−rξ} ∫_{ξ}^{1} f(x, η)dη) dy,

where f(x, η) = e^{r|η|}ϕ(x, η). Hence, for any fixed x ∈ R, we have

σ(x, ξ′) − σ(x, ξ)

≥ C (e^{rξ′} ∫_{−ξ′}^{1} f(x, η)dη + e^{−rξ′} (∫_{ξ′}^{−ξ}f(x, η)dη + ∫_{−ξ}^{−ξ′}f(x, η)dη + ∫_{−ξ′}^{1} f(x, η)dη)

− e^{rξ} ( ∫_{−ξ}^{−ξ′}f(x, η)dη + ∫_{−ξ′}^{1} f(x, η)dη) − e^{−rξ} ( ∫_{ξ}^{−ξ′}f(x, η)dη + ∫_{−ξ′}^{1} f(x, η)dη))

= 2(cosh rξ′ − cosh rξ) ∫_{−ξ′}^{1} f(x, η)dη + (e^{−rξ′} − e^{rξ}) ∫_{−ξ}^{−ξ′}f(x, η)dη+

+e^{−rξ′} ∫_{ξ′}^{−ξ}f(x, η)dη − e^{−rξ} ∫_{ξ}^{−ξ′}f(x, η)dη = I_{1} + I_{2} + I_{3}.

I_{1} and I_{2} are obviously nonnegative. After a change of variables we obtain

I_{3} = e^{−rξ′} ∫_{ξ′}^{−ξ}f(x, η)dη − e^{−rξ} ∫_{ξ′}^{−ξ}f(x, η)dη ≥ 0.

The other cases can be treated in a similar way.

4. An Application

In this section we observe the polynomial symbol P (ξ) = ∑_{|α|≤s} a_{α}ξ^{α}, a_{α} ∈ R, assuming that it satisfies (S1) and (S4). Note that the symbol P (ξ) obviously satisfies (S2) and (S3). In particular, we consider the symbol σ(ξ) = (1 + |ξ|^{2})^{s/2} = ⟨ξ⟩^{s}, s ≥ 0, ξ ∈ R^{d} (see Theorem 1 b)). Then the corresponding pseudodifferential operator given by (1) is called the Bessel potential of order s. It is known that the Bessel potentials define isomorphisms between Sobolev spaces [18]. However, the question which operators establish isomorphisms between modulation spaces is still open and important in time-frequency analysis. In this section we prove that the Weyl transforms of P (ξ), and ⟨ξ⟩^{s}, s ≥ 0, establish isomorphism between certain modulation spaces.

We recall the notion of moderate weight function. A locally integrable function v is called submultiplicative weight if v(z_{1} + z_{2}) ≤ v(z_{1})v(z_{2}), z_{1}, z_{2} ∈ R^{2d}, and a locally integrable function m is moderate weight with respect to a submultiplicative weight v if

m(x + y, ξ + η) ≤ Cv(x, ξ)m(y, η), x, y, ξ, η ∈ R^{d}.

Weights m_{1} and m_{2} are equivalent if C_{1}m_{1} ≤ m_{2} ≤ C_{2}m_{1} for some positive constants C_{1} and C_{2}. Every submultiplicative weight is equivalent to a continuous weight.

Any function w which satisfies (S1)-(S4) is moderate with respect to e^{η(|x|^{γ}+|ξ|^{γ})}. In particular, σ(ξ) = ⟨ξ⟩^{s}, s ≥ 0, ξ ∈ R^{d}, the symbol of the Bessel potential of order s is a moderate weight.

Definition 1.Let w be a moderate weight, 1 ≤ p, q < ∞ and 0 ≢ g ∈ S. Modulation space M_{p,q}^{w,t} is given by

M_{p,q}^{w,t} = {f ∈ S′ : ∥f∥_{M_{p,q}^{w,t}} < ∞},

where

∥f∥_{M_{p,q}^{w,t}} = [∫_{R^{d}} (∫_{R^{d}} |⟨\overline{T_{x}M_{ξ}g}, f⟩|^{p}w(x, ξ)^{p}(1 + |x| + |ξ|)^{tp}dx)^{q/p} dξ]^{1/q} .

Modulation spaces are Banach spaces [7, Theorem 11.3.5.] independent of the choice of the analyzing function 0 ≢ g ∈ S [2]. It can be shown that M_{2,2}^{1,t} = H_{2}^{t} ∩ L_{2}^{t}, where H_{2}^{t} is Sobolev space and L_{2}^{t} is weighted L^{2} space with the weight (1 + |x|)^{t} [4]. Therefore F(M_{2,2}^{1,t}) = M_{2,2}^{1,t}. Obviously M_{2,2}^{1,t+μ} ⊂ M_{2,2}^{1,t}, for any μ > 0.

Proof of the following theorem can be found in [14], [16].

Theorem 2.Let 1 ≤ p, q < ∞, t ≥ 0 and let σ(x, D) be the Weyl transform of a symbol σ(x, ξ) satisfying (S1)-(S4). For every f ∈ M_{p,q}^{σ,t} there exist positive constants C_{1}, C_{2} and C_{3} such that

C_{1}∥f∥_{M_{p,q}^{σ,t}} ≤ ∥σ(x, D)f∥_{M_{p,q}^{1,t}} + C_{2}∥f∥_{M_{p,q}^{1,0}} ≤ C_{3}∥f∥_{M_{p,q}^{σ,t}}.  (3)

If, additionally, σ(z) ≥ C⟨z⟩^{μ} for |z| ≥ K, for some positive constants C, μ and K, and if σ(x, D)f ∈ M_{2,2}^{1,s}, then f belongs to M_{2,2}^{1,s+μ}.

Immediate consequence of Theorem 2 is the continuity of the mapping σ : M_{p,q}^{σ,t} ↦ M_{p,q}^{1,t}, and σ(M_{p,q}^{σ,t}), the image of M_{p,q}^{σ,t} under σ, is a Banach subspace of M_{p,q}^{1,t}.

It is an open question to find the conditions under which the operator σ(x, D) isomorphically maps M_{p,q}^{σ,t} onto M_{p,q}^{1,t}. Here we give only a partial answer, namely we observe operators of the form ∑_{|α|≤s} a_{α}D^{α}, a_{α} ∈ R, whose symbols satisfy (S1)-(S4). More general cases, including the so called ultra-modulation spaces [13] will be considered in a separate paper.

If σ = σ(ξ), ξ ∈ R^{d} (σ = σ(x), x ∈ R^{d}, respectively), we denote the corresponding modulation space by M_{p,q}^{σ(ξ),t} (M_{p,q}^{σ(x),t} respectively). If P (·) =

On a Symbol Class of Elliptic Pseudodifferential Operators 65

∑_{|α|≤s} a_{α}(·)^{α}, a_{α} ∈ R, s ≥ 0, then F (M_{p,q}^{P (x),t}) = M_{p,q}^{P (−ξ),t} [3, page 365.].

Lemma 1.Let there be given P (ξ) = ∑_{|α|≤s} a_{α}ξ^{α}, a_{α} ∈ R, satisfying (S4) and P (ξ) ≥ C(1+|ξ|)^{s}, ξ ∈ R^{d}, for some C > 0 (Take C = 1 as in (S1).) Let f ∈ M_{2,2}^{1,t}. Then there exists a function f_{1} ∈ M_{2,2}^{P (ξ),t} such that P (D)f_{1} = f. In particular, if σ(ξ) = ⟨ξ⟩^{s}, s ≥ 0, and f ∈ M_{2,2}^{1,t}, then f_{1} ∈ M_{2,2}^{σ(ξ),t}.

P r o o f. We formally put P (D)f_{1} = f ∈ M_{2,2}^{1,t}. Since f̂ ∈ M_{2,2}^{1,t} ⊂ L^{2} and \frac{f̂(ξ)}{P (ξ)} ∈ L^{2} it follows f̂_{1} ∈ L^{2}, i.e. f_{1} ∈ L^{2}. We than have

⟨\overline{T_{x}M_{ξ}g}, f⟩ = ⟨\overline{T_{x}M_{ξ}g}, P (D)f_{1}⟩
= e^{2πiξx} ∫_{R^{d}} P (−D_{t}) (e^{−2πiξt}ḡ(t − x)) f_{1}(t)dt
= e^{2πiξx} ∫_{R^{d}} ∑_{|α|≤s} a_{α}(−D_{t})^{α} (e^{−2πiξt}ḡ(t − x)) f_{1}(t)dt
= e^{2πiξx} ∑_{|α|≤s} a_{α} ∫_{R^{d}} ∑_{β≤α} \binom{α}{β}(−D_{t})^{α−β}e^{−2πiξt}(−D_{t})^{β}ḡ(t − x)f_{1}(t)dt
= e^{2πiξx} ∑_{|α|≤s} a_{α}ξ^{α} ∫_{R^{d}} e^{−2πiξt}ḡ(t − x)f_{1}(t)dt
+ e^{2πiξx} ∑_{|α|≤s} a_{α} ∑_{1≤|β|, β≤α} \binom{α}{β} ∫_{R^{d}} ξ^{α−β}e^{−2πiξt}(−D_{t})^{β}ḡ(t − x)f_{1}(t)dt
= P (ξ)⟨\overline{T_{x}M_{ξ}g}, f_{1}⟩ + ∑_{|α|≤s} a_{α} ∑_{1≤|β|, β≤α} \binom{α}{β}ξ^{α−β}⟨\overline{T_{x}M_{ξ}D_{t}^{β}g}, f_{1}⟩
= P (ξ)⟨\overline{T_{x}M_{ξ}g}, f_{1}⟩ + ∑_{1≤|β|≤s} P_{β}(ξ)⟨\overline{T_{x}M_{ξ}D_{t}^{β}g}, f_{1}⟩,

where P_{β}(ξ) = ∑_{|j|≤s−|β|} b_{β,j}ξ^{j}, 1 ≤ |β| ≤ s.

If g ∈ S, then D_{t}^{β}g ∈ S wherefrom ∥⟨\overline{T_{x}M_{ξ}(−D_{t})^{β}g}, f_{1}⟩P_{β}(ξ)(1 + |x|^{2} + |ξ|^{2})^{t/2}∥

≤ C_{β}∥⟨\overline{T_{x}M_{ξ}g}, f_{1}⟩⟨ξ⟩^{s−|β|}(1 + |x|^{2} + |ξ|^{2})^{t/2}∥.

This implies

∞ > ∥⟨\overline{T_{x}M_{ξ}g}, f⟩(1 + |x|^{2} + |ξ|^{2})^{t/2}∥  (4)
≥ ∥⟨\overline{T_{x}M_{ξ}g}, f_{1}⟩P (ξ)(1 + |x|^{2} + |ξ|^{2})^{t/2}∥

−∥ ∑_{1≤|β|≤s} C_{β}⟨\overline{T_{x}M_{ξ}g}, f_{1}⟩⟨ξ⟩^{s−|β|}(1 + |x|^{2} + |ξ|^{2})^{t/2}∥.

In order to prove that f_{1} ∈ M_{2,2}^{P (ξ),t}, we split R_{ξ}^{d} in orthants. For the sake of simplicity we show the two dimensional case and note that the case d > 2 can be treated in a completely analogous way.

Let R_{ξ}^{2} = R_{(ξ_{1}^{+},ξ_{2}^{+})}^{2} ⋃ R_{(ξ_{1}^{−},ξ_{2}^{+})}^{2} ⋃ R_{(ξ_{1}^{−},ξ_{2}^{−})}^{2} ⋃ R_{(ξ_{1}^{+},ξ_{2}^{−})}^{2}, where ξ_{j}^{+} (ξ_{j}^{−}), j = 1, 2, are non-negative (resp. non-positive) real numbers. Consider, for example, R_{(ξ_{1}^{+},ξ_{2}^{−})}^{2}. We take h = (h_{1}, h_{2}) where h_{1} > 0, h_{2} < 0 are chosen such that

\frac{1}{2}P (ξ +h)(1+|x|^{2} +|ξ +h|^{2})^{t/2} ≥ ∑_{1≤|β|≤s} C_{β}⟨ξ +h⟩^{s−|β|}(1+|x|^{2} +|ξ +h|^{2})^{t/2}

holds for all ξ ∈ R_{(ξ_{1}^{+},ξ_{2}^{−})}^{2}, x ∈ R^{2}. By (4) we have

∞ > ∥⟨\overline{T_{x}M_{ξ+h}g}, f⟩(1 + |x|^{2} + |ξ + h|^{2})^{t/2}∥_{L^{2}(R^{2}×R_{(ξ{+ \atop 1},ξ{− \atop 2})}^{2})}
≥ ∥⟨\overline{T_{x}M_{ξ+h}g}, f_{1}⟩P (ξ + h)(1 + |x|^{2} + |ξ + h|^{2})^{t/2}∥_{L^{2}(R^{2}×R_{(ξ{+ \atop 1},ξ{− \atop 2})}^{2})}
− ∥ ∑_{1≤|β|≤s} C_{β}⟨\overline{T_{x}M_{ξ+h}g}, f_{1}⟩⟨ξ + h⟩^{s−|β|}(1 + |x|^{2} + |ξ + h|^{2})^{t/2}∥_{L^{2}(R^{2}×R_{(ξ{+ \atop 1},ξ{− \atop 2})}^{2})}

≥ \frac{1}{2}∥⟨\overline{T_{x}M_{ξ+h}g}, f_{1}⟩P (ξ + h)(1 + |x|^{2} + |ξ + h|^{2})^{t/2}∥_{L^{2}(R^{2}×R_{(ξ{+ \atop 1},ξ{− \atop 2})}^{2})}.

After a change of variables we obtain f_{1} ∈ M_{2,2}^{P (ξ),t}(R^{2} ×R_{(ξ_{1}^{+},ξ_{2}^{−})}^{2}). The same procedure could be done for all of the other orthants, so we conclude that f_{1} ∈ M_{2,2}^{P (ξ),t} if f ∈ M_{2,2}^{1,t}.

Theorem 3.Let there be given P (ξ) = ∑_{|α|≤s} a_{α}ξ^{α}, a_{α} ∈ R, satisfying

(S4) and P (ξ) ≥ C(1 + |ξ|)^{s}, ξ ∈ R^{d}, for some C > 0. The corresponding pseudodifferenital operator defined by (1) establishes an isomorphism between M_{2,2}^{P (ξ),t} and M_{2,2}^{1,t}, t ≥ 0. In particular, the Bessel potential of order s, s ≥ 0, establishes an isomorphism between M_{2,2}^{σ(ξ),t} and M_{2,2}^{1,t}, t ≥ 0, where σ(ξ) = ⟨ξ⟩^{s}.

P r o o f. The proof that the mapping is injective is based on the properties of the Fourier transform. Let P (D) be the Weyl transform of the symbol P (ξ), f ∈ M_{2,2}^{P (ξ),t}, and let P (D)f = 0. Then we have

P (D)f(x) = ∫_{R^{d}} ∫_{R^{d}} e^{2πi(x−y)ξ}P (ξ)f(y)dydξ = F^{−1} (P (ξ)f̂(ξ)) (x) = 0

which implies P (ξ)f̂(ξ) = 0. Since P (ξ) ≥ 1 we have f̂(ξ) = 0, wherefrom f = 0.

Let f belong to M_{2,2}^{1,t}. We define f_{1} by f̂_{1} := \frac{f̂}{P}. It remains to show that

f_{1} ∈ M_{2,2}^{P (ξ),t} and P (D)f_{1} = f. As already mentioned in the proof of Lemma 1, f_{1} ∈ L^{2}. Since

F(P (D)f_{1})(ξ) = P (ξ)f̂_{1}(ξ) = f̂(ξ),

and the Fourier transforms in an isomorphism on M_{2,2}^{1,t} we obtain P (D)f_{1} =

f. Finally, f_{1} ∈ M_{2,2}^{P (ξ),t} by Lemma 1, which completes the proof.

REFERENCES

[1] W. C z a j a, Z. R z e s z o t n i k, Pseudodifferential Operators and Gabor frames: Spectral Asymptotics, Math. Nach., 233-234 (2002), 77–88.

[2] H. G. F e i c h t i n g e r, Modulation Spaces on Locally Compact Abelian Groups, Technical Report, Vienna (1983).

[3] H. G. F e i c h t i n g e r, K. G r ö c h e n i g, Gabor Wavelets and the Heinseberg Group: Gabor Expansion and Short Time Fourier Transform from the Group Theoretical Point of View, in Wavelets – A Tutorial in Theory and Applications, C.H. Chiu (ed.), Academic Press, New York (1992), 359–397.

[4] H. G. F e i c h t i n g e r, K. G r ö c h e n i g, D. W a l n u t, Wilson Bases and Modulation Spaces, Math. Nachr., 155 (1992) 7–17.

[5] H. G. F e i c h t i n g e r, T. S t r o h m e r, editors, Gabor analysis and algorithms: theory and applications, Birkhäuser Boston Inc., Boston, MA (1998).

[6] H. G. F e i c h t i n g e r, T. S t r o h m e r, editors, Advances in Gabor analysis, Birkhäuser Boston Inc., Boston, MA (2002).

[7] K. G r ö c h e n i g, Foundations of Time-Frequency Analysis, Birkhäuser, Boston, Basel, Berlin (2000).

[8] K. G r ö c h e n i g, C. H e i l, Modulation Spaces and Pseudodifferential Operators, Integral Equations Operator Theory, 34 (1999), 439–457.

[9] C. E. H e i l, J. R a m a n a t h a n, P. T o p i w a l a, Singular Values of Compact Pseudodifferential Operators, J. Funct. Anal., 150 (1997), 426–452.

[10] L. H ö r m a n d e r, An Introduction to Complex Analysis in Several Variables, Pricneton, New Jersey (1966).

[11] A. J a f f e, High Energy Behaviour of Strictly Localizable Fields, Phys. Rev., 158 (1967), 1454-1461.

[12] D. L a b a t e, Time–Frequency Analysis of Pseudodifferential Operators, Monats. f. Math., 133 (2001), 143–156.

[13] S. P i l i p o v i ć, N. T e o f a n o v, Wilson Bases and Ultramodulation Spaces, Math. Nach., 242 (2002), 179-196.

[14] S. P i l i p o v i ć, N. T e o f a n o v, Pseudodifferential Operators and Ultramodulation Spaces, preprint (2002).

[15] M. A. S h u b i n, Pseudodifferential Operators and Spectral Theory, Springer-Verlag, Heidelberg (2001).

[16] K. T a c h i z a w a, The Pseudodifferential Operators and Wilson Bases, J. Math. Pures Appl., 75 (1996), 509–529.

[17] A. U n t e r b e r g e r, Pseudo-differential operators and applications: an introduction, Lecture Notes Series, 46 (1976), Matematisk Institut, Aarhus Universitet.

[18] M. W. W o n g, An Introduction To Pseudodifferential Operators, World Scientific, Singapore, (1991).

Institute of Mathematics University of Novi Sad Trg Dositeja Obradovića 4 21000 Novi Sad Yugoslavia E-mail address: pilipovic@im.ns.ac.yu, tnenad@im.ns.ac.yu