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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 \(T : L^2(\mathbb R^n)\times L^2(\mathbb R^n) \to L^1(\mathbb R^n)\) of the form \[ T(f,g)(x) = \int_{\mathbb R^n}\int_{\mathbb R^n}\sigma(x,\xi,\eta)\widehat{f}(\xi) \widehat{g}(\eta) e^{ix(\xi+\eta)}\,d\xi\, d\eta \] under suitable growth conditions on the symbol \(\sigma(x,\xi,\eta)\) 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:

47G30 Pseudodifferential operators
35S05 Pseudodifferential operators as generalizations of partial differential operators
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
42B15 Multipliers for harmonic analysis in several variables
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