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Bilinear pseudo-differential operators with exotic symbols. (Opérateurs pseudo-différentiels bilinéaires avec symboles exotiques.) (English. French summary) Zbl 1470.35456

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.

MSC:

35S05 Pseudodifferential operators as generalizations of partial differential operators
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
47G30 Pseudodifferential operators
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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. Theory, 67, 3, 341-364 (2010) · Zbl 1213.47053 · doi:10.1007/s00020-010-1782-y
[3] Bényi, Árpád; Torres, Rodolfo H., Symbolic calculus and the transposes of bilinear pseudodifferential operators, Commun. Partial Differ. Equations, 28, 5-6, 1161-1181 (2003) · Zbl 1103.35370 · doi:10.1081/pde-120021190
[4] Bényi, Árpád; Torres, Rodolfo H., Almost orthogonality and a class of bounded bilinear pseudodifferential operators, Math. Res. Lett., 11, 1, 1-11 (2004) · Zbl 1067.47062 · doi:10.4310/mrl.2004.v11.n1.a1
[5] Calderón, Alberto-P.; Vaillancourt, Rémi, A class of bounded pseudo-differential operators, Proc. Natl. Acad. Sci. USA, 69, 1185-1187 (1972) · Zbl 0244.35074
[6] Coifman, Ronald R.; Meyer, Yves, Au delà des opérateurs pseudo-différentiels, 57, i+185 p. pp. (1978), Société Mathématique de France · Zbl 0483.35082
[7] Grafakos, Loukas, Classical Fourier analysis, 249, xvi+489 p. pp. (2008), Springer · Zbl 1220.42001
[8] Grafakos, Loukas; Torres, Rodolfo H., Multilinear Calderón-Zygmund theory, Adv. Math., 165, 1, 124-164 (2002) · Zbl 1032.42020 · doi:10.1006/aima.2001.2028
[9] Michalowski, Nicholas; Rule, David; Staubach, Wolfgang, Multilinear pseudodifferential operators beyond Calderón-Zygmund theory, J. Math. Anal. Appl., 414, 1, 149-165 (2014) · Zbl 1348.47038 · doi:10.1016/j.jmaa.2013.12.062
[10] Miyachi, Akihiko; Tomita, Naohito, Calderón-Vaillancourt-type theorem for bilinear operators, Indiana Univ. Math. J., 62, 4, 1165-1201 (2013) · Zbl 1429.42018 · doi:10.1512/iumj.2013.62.5059
[11] Naibo, Virginia, On the \({L}^\infty \times{L}^\infty \rightarrow{BMO}\) mapping property for certain bilinear pseudodifferential operators, Proc. Am. Math. Soc., 143, 12, 5323-5336 (2015) · Zbl 1336.47053 · doi:10.1090/proc12775
[12] Stein, Elias M., Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, 43, xiv+695 p. pp. (1993), Princeton University Press · Zbl 0821.42001
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