On the L-boundedness of Pseudo-differential Operators and Their Commutators with Symbols in Α-modulation Spaces

Since the theory of pseudo-differential operators was established in 1970’s, the L-boundedness of them with symbols in the Hörmander class S ρ,δ has been well investigated by many authors. Among them, Calderón-Vaillancourt [5] first treated the boundedness for the class S 0,0, which means that the boundedness of all the derivatives of symbols assures the L-boundedness of the corresponding operators. It should be mentioned that the boundedness of all the derivatives of symbols is not necessary in their proof. Being motivated by this argument, many authors as Coifman-Meyer [6], Cordes [8], Kato [17], Miyachi [19], Muramatu [20], Nagase [21] contributed to know the minimal assumption on the regularity of symbols for the corresponding operators to be L-bounded. They said that the boundedness of the derivatives of symbols up to a certain order, which exceeds n/2, assures the L(R)-boundedness. Especially, Sugimoto [24] showed that symbols in the Besov space B (∞,∞),(1,1) (n/2,n/2) implies the L -boundedness. In the last decade, new developments in this problem have appeared. Sjöstrand [22] introduced a wider class than S 0,0 which assures the L -boundedness and is now recognized as a special case of modulation spaces introduced by Feichtinger [9, 10, 11]. These spaces are based on the idea of quantum mechanics or timefrequency analysis. Sjöstrand class can be written asM if we follow the notation of modulation spaces. Gröchenig-Heil [16] and Toft [26] gave some related results to Sjöstrand’s one by developing the theory of modulation spaces. Boulkhemir [3] treated the same discussion for Fourier integral operators. We remark that the relation between Besov and modulation spaces is well studied by the works of Gröbner [15], Toft [26] and Sugimoto-Tomita [25], and we know that the spaces B (∞,∞),(1,1) (n/2,n/2) and M ∞,1 have no inclusion relation with each others (see Appendix) although the class S 0,0 is properly included in both spaces. In this sense, the results of Sugimoto [24] and Sjöstrand [22] are independent extension of Calderon-Vaillancourt’s result. The objective of this paper is to show that these two results, which appeared to be independent ones, can be proved based on the same principle. Especially we give another proof to Sjöstrand’s result following the same argument used to prove Sugimoto’s result. For the purpose, we use the notation of α-modulation spaces M s,α (0 ≤ α ≤ 1), a parameterized family of function spaces, which includes Besov spaces B s and modulation spaces M p,q as special cases corresponding to