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.
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