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Almost orthogonality and a class of bounded bilinear pseudodifferential operators. (English) Zbl 1067.47062

In this work, the authors prove two theorems on the boundedness of bilinear pseudodifferential operators T:L 2 ( n )×L 2 ( n )L 1 ( n ) of the form

T(f,g)(x)= n n σ(x,ξ,η)f ^(ξ)g ^(η)e ix(ξ+η) dξdη

under suitable growth conditions on the symbol σ(x,ξ,η) and its derivatives. Moreover, they explain how their methods are linked to the boundedness theorems of A. P. Calderón and R. Vaillancourt [J. Math. Soc. Japan 23, 374–378 (1971; Zbl 0203.45903)] and I. L. Hwang [Trans. Am. Math. Soc. 302, 55–76 (1987; Zbl 0651.35089)] for linear pseudodifferential operators.

MSC:
47G30Pseudodifferential operators
35S05General theory of pseudodifferential operators
42B20Singular and oscillatory integrals, several variables
42B15Multipliers, several variables