Miyachi, Akihiko; Tomita, Naohito Bilinear pseudo-differential operators with exotic symbols. (Opérateurs pseudo-différentiels bilinéaires avec symboles exotiques.) (English. French summary) Zbl 1470.35456 Ann. Inst. Fourier 70, No. 6, 2737-2769 (2020). Let \(m\in\mathbb R, 0\leq\delta\leq\rho\leq 1\). A function \(\sigma(x,\xi,\eta)\in C^{\infty}(\mathbb R^n\times\mathbb R^n\times\mathbb R^n)\) belongs to the bilinear Hörmander symbol class \(BS^m_{\rho,\delta}(\mathbb R^n)\equiv BS^m_{\rho,\delta}\) if for every triple of multi-indices \(\alpha,\beta,\gamma\in\mathbb N^n_0=\{0,1,2,\dots\}^n\) there exists a constant \(C_{\alpha,\beta,\gamma}>0\) such that \[ |\partial^{\alpha}_x\partial^{\beta}_{\xi}\partial^{\gamma}_{\eta}\sigma(x,\xi,\eta)|\leq C_{\alpha,\beta,\gamma}(1+|\xi|+|\eta|)^{m+\delta|\alpha|-\rho(|\beta|+|\gamma|)}. \] Such a symbol \(\sigma\in BS^m_{\rho,\delta}\) generates the bilinear pseudo-differential operator \(T_{\sigma}\) in the following way \[ T_{\sigma}(f,g)(x)=\frac{1}{(2\pi)^{2n}}\int\limits_{\mathbb R^n\times\mathbb R^n}e^{ix\cdot(\xi+\eta)}\sigma(x,\xi,\eta)\hat f(\xi)\hat g(\eta)d\xi d\eta,~~~f,g\in S(\mathbb R^n). \] Main results of the paper are the following.Theorem 1.2. Let \(0\leq\rho<1, m=-(1-\rho)n/2\). The the operator \(T_{\sigma}, \sigma\in BS^m_{\rho,\rho}\) is a bounded operator \(L^2(\mathbb R^n)\times L^{\infty}(\mathbb R^n)\rightarrow L^2(\mathbb R^n)\).Theorem 1.3. Let \(0\leq\rho<1, m=-(1-\rho)n\). The the operator \(T_{\sigma}, \sigma\in BS^m_{\rho,\rho}\) is a bounded operator \(L^{\infty}(\mathbb R^n)\times L^{\infty}(\mathbb R^n)\rightarrow\mathrm{BMO}(\mathbb R^n)\).There are also very important Corollary 1.4 regarding to boundedness \(T_{\sigma}: L^p(\mathbb R^n)\times L^q(\mathbb R^n)\rightarrow L^r(\mathbb R^n)\) for \(1\leq p,q,r\leq\infty\) and good introduction with historical remarks and explanations in the paper. Reviewer: Vladimir Vasilyev (Belgorod) Cited in 5 Documents MSC: 35S05 Pseudodifferential operators as generalizations of partial differential operators 42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.) 47G30 Pseudodifferential operators Keywords:bilinear pseudo-differential operators; bilinear Hörmander symbol classes; exotic symbols PDFBibTeX XMLCite \textit{A. Miyachi} and \textit{N. Tomita}, Ann. Inst. Fourier 70, No. 6, 2737--2769 (2020; Zbl 1470.35456) Full Text: DOI arXiv References: [1] Bényi, Árpád; Bernicot, Frédéric; Maldonado, Diego; Naibo, Virginia; Torres, Rodolfo H., On the Hörmander classes of bilinear pseudodifferential operators II, Indiana Univ. Math. J., 62, 6, 1733-1764 (2013) · Zbl 1297.35300 · doi:10.1512/iumj.2013.62.5168 [2] Bényi, Árpád; Maldonado, Diego; Naibo, Virginia; Torres, Rodolfo H., On the Hörmander classes of bilinear pseudodifferential operators, Integral Equations Oper. 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