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Continuity and Schatten–Von Neumann Properties for Pseudo–Differential Operators and Toeplitz Operators on Modulation Spaces

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Continuity and Schatten–Von Neumann Properties for Pseudo–Differential Operators and Toeplitz Operators on Modulation Spaces The Erwin Schr¨odinger International Boltzmanngasse 9 ESI Institute for Mathematical Physics A-1090 Wien, Austria Continuity and Schatten–von Neumann Properties for Pseudo–Differential Operators and Toeplitz operators on Modulation Spaces Joachim Toft Vienna, Preprint ESI 1732 (2005) November 2, 2005 Supported by the Austrian Federal Ministry of Education, Science and Culture Available via http://www.esi.ac.at CONTINUITY AND SCHATTEN-VON NEUMANN PROPERTIES FOR PSEUDO-DIFFERENTIAL OPERATORS AND TOEPLITZ OPERATORS ON MODULATION SPACES JOACHIM TOFT p,q Abstract. Let M(ω) be the modulation space with parameters p,q and weight function ω. We prove that if p1 = p2, q1 = q2, α ∞ ω1 = ω0ω and ω2 = ω0, and ∂ a/ω0 ∈ L for all α, then the Ψdo p1,q1 p2,q2 p,q at(x, D) : M(ω1) → M(ω2 ) is continuous. If instead a ∈ M(ω) for appropriate p, q and ω, then we prove that the map here above is continuous, and if in addition pj = qj = 2, then we prove that at(x, D) is a Schatten-von Neumann operator of order p. We use these results to discuss continuity for Toeplitz operators. Mathematics Subject Classifications (2000): Primary: 47B10, 35S05, 47B35, 47B37; Secondary: 42B35, 46E35. Key words: Schatten-von Neumann, pseudo-differential operators, Toeplitz oper- ators, modulation spaces, embeddings. 0. Introduction In [G2] and [GH1], Gr¨ochenig and Heil present an alternative method, based on time-frequency analysis when investigating pseudo-differential operators with non-smooth symbols belonging to non-weighted modu- lation spaces. Here they make suitable Gabor expansions of the sym- bols, which in certain extent essentially reduce the problems in such way that the symbols are translations and modulations of a fix and well-known function. In that end, they are able to make a somewhat detailed study of compactness, and prove embedding properties be- tween Schatten-von Neumann classes in pseudo-differential operators acting on L2, and modulation spaces. Furthermore, they also prove that any pseudo-differential operator ∞,1 with symbol in the modulation space M (denoted by Sw in [Sj2] by Sj¨ostrand) is continuous on any non-weighted modulation space M p,q. Since L2 = M 2,2, it follows that in particular that such operators are continuous on L2, a property which was proved by Sj¨ostrand in [Sj1], where modulation spaces were used as symbol classes for the first time. 0 Furthermore, since S0 , the set of functions which are bounded together 1 with all their derivatives, is contained in M ∞,1, it follows from these 0 investigations that any pseudo-differential operator with symbol in S0 is continuous on M p,q. The latter result was remarked in the L2-case in [Sj1], and for general p and q, the result is a special case of Theorem 2.1 in [Ta] by Tachizawa. Some further generalizations can also be found in the independent papers [To5] and [GH2], where pseudo-differential operators with sym- bol class M p,q are considered. In [To6] these results were further ex- tended for Weyl operators where the symbols belong to weighted mod- ulation spaces. Important parts of the investigations here above concern modula- tion spaces, introduced by Feichtinger in Fe2 and [Fe4] during the period 1980–1983 as an appropriate family of function and distribu- tion spaces to have in background when discussing certain problems within time-frequency analysis. The basic theory of such spaces were thereafter established by H. Feichtinger and K. Gr¨ochenig in the pa- pers [Fe3], [FG1]–[FG2] and [G1]. Roughly speaking, for an appropriate p,q weight function ω, the modulation space M(ω) is obtained by imposing p,q a mixed L(ω)-norm on the short-time Fourier transform of a tempered distribution. The non-weighted modulation space M p,q is then obtained by choosing ω = 1. In terms of modulation spaces it is sometimes easy to obtain information concerning growth and decay properties, as well as certain localization and regularity properties for distributions. In this paper we continue the discussions in [To5] and [To6] concern- ing continuity for pseudo-differential operators in background of mod- ulaiton space theory. More precisely, we consider pseudo-differential operators with smooth symbols or with symbols belonging appropriate modulation spaces, and discuss continuity for such operators when act- ing on modulation spaces. Especially we are concerned with a somewhat detailed study of continuity and compactness for pseudo-differential op- erators acting between modulation spaces of Hilbert type in terms of Schatten-von Neumann classes. In particular we investigate trace-class and Hilbert-Schmidt properties. Except for the Hilbert-Schmidt case it is in general a hard task to find complete characterizations of Schatten-von Neumann classes. One is therefore forced to find embeddings between such classes and other spaces which are more convenient. In Section 5 we discuss embeddings between such classes and modulation spaces, and generalize certain results in [GH1], [To2] and [To5]. The situation in Section 5 is more complicated depending on the fact that we consider operators acting on modulation spaces which involve weight functions of general types, instead of operators acting on L2 which is the case in the latter papers. In particular, by choosing the involved weight functions in appropriate 2 ways, we may use our results to discuss Schatten-von Neumann proper- ties for pseudo-differential operators acting between weighted Lebesgue spaces and/or Sobolev spaces of Hilbert type. The general types of modulation spaces which are involved in the continuity investigations cause new problems comparing to [GH1] and [To5]–[To6]. These problems are overcome by using a related Gabor technique as in [GH1], leading to a convenient way to expand the sym- bols, and discretization of certain parts of the problems. The requested results are thereafter obtained by using techniques in modulation space theory, which are well-known within time-frequency analysis, in combi- nation with certain results in harmonic analysis and pseudodifferential calculus. In the last part we apply these results to Toeplitz operators (in the litterature, the terms localization operators and anti-Wick operators also occur), which are convenient to study in background of pseudo- differential calculus. In particular we extend certain continuity and Schatten-von Neumann results in [Bg], [CG] and [To5]. Some attention is also paid to discuss pseudo-differential operators with smooth symbols acting on modulation spaces. These investigations are presented in Section 3, where we extend some results by Tachizawa in [Ta] and in [To6]–[To7]. In order to describe our results in more detail we recall the definition of modulation spaces. Assume that χ ∈ S (Rm) \ 0, p, q ∈ [1, ∞] and 2m that ω is an appropriate function on R , and let τxχ(y) = χ(y − x) when x,y ∈ Rm. (We use the same notation for the usual functions and distribution spaces as in e. g. [H¨o].) Then the modulation space p,q m S ′ m M(ω)(R ) consists of all f ∈ (R ) such that kfk p,q = kfk p,q,χ M(ω) M(ω) (0.1) q/p 1/q p ≡ |F (f τxχ)(ξ)ω(x, ξ)| dx dξ < ∞, Z Z with the obvious modifications when p = ∞ and/or q = ∞. Here F denotes the Fourier transform on S ′(Rm), which takes the form F f(ξ)= f(ξ)=(2π)−m/2 f(x)e−ihx,ξi dx Z b when f ∈ S (Rm). Next assume that t ∈ R is fixed and that a ∈ S (R2m). Then the pseudo-differential operator at(x, D) is the continuous operator on S (Rm) which is defined by the formula (at(x, D)f)(x) = (Opt(a)f)(x) (0.2) = (2π)−m a((1 − t)x + ty,ξ)f(y)eihx−y,ξi dydξ. ZZ 3 ′ 2m The definition of at(x, D) extends to any a ∈ S (R ), and then m ′ m at(x, D) is continuous from S (R ) to S (R ). (See e.g. [H¨o].) If w t = 1/2, then at(x, D) is equal to the Weyl operator a (x, D) for a. If instead t = 0, then the standard (Kohn-Nirenberg) representation a(x, D) is obtained. In Section 5 we discuss continuity for pseudo-differential operators acting on modulation spaces when the operator symbols belong to mod- ulation spaces. Some of these questions were discussed for Weyl oper- ators (i. e. when t = 1/2 in (0.2)) already in Section 5 in [To6]. In the first part of Section 5 we extend the results in [To6] to arbitrary t ∈ R. In particular we find appropriate conditions on ω, ω1, ω2 and p, q, pj ,qj for j = 1, 2, in order for a (x, D) : M p1,q1 → M p2,q2 t (ω1) (ω2) p,q to be continuous when a ∈ M(ω). The second part of Section 5 is devoted to the case when the M pj,qj (ωj) above are Hilbert spaces, i.e. pj = qj = 2 for j = 1, 2. In this case a more detailed continuity study in terms of Schatten-von Neumann properties is performed. These considerations are dependent on some preparations which are made in Section 2 and in Section 4. Recall that an operator T from M 2,2 to M 2,2 belongs to I , the set of Schatten- (ω1) (ω2) p von Neumann operators of order p ∈ [1, ∞], if and only if 1/p p sup |(T fj, gj )M 2 | < ∞, X (ω2) where the supremum is taken over all orthonormal sequences (fj) in M 2,2 and (g ) in M 2,2 . In particular, this implies that I is the (ω1) j (ω2) ∞ set of linear and continuous operators, and that T is compact when T ∈ Ip and p < ∞.
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