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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 $T : L^2(\Bbb R^n)\times L^2(\Bbb R^n) \to L^1(\Bbb R^n)$ of the form $$ T(f,g)(x) = \int_{\Bbb R^n}\int_{\Bbb R^n}\sigma(x,\xi,\eta)\widehat{f}(\xi) \widehat{g}(\eta) e^{ix(\xi+\eta)}\,d\xi\, d\eta $$ 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 {\it A. P. Calderón} and {\it R. Vaillancourt} [J. Math. Soc. Japan 23, 374--378 (1971; Zbl 0203.45903)] and {\it I. L. Hwang} [Trans. Am. Math. Soc. 302, 55--76 (1987; Zbl 0651.35089)] for linear pseudodifferential operators.

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