Bilinear pseudo-differential operators with symbols in \(BS^m_{1,1}\) on Triebel-Lizorkin spaces
The authors consider symbols in the (bilinear) Hörmander symbol class \(BS_{1,1} ^m\), that is the class of \(\sigma(\cdot,\cdot,\cdot) \in C^\infty(\mathbb R^n\times \mathbb R^n\times \mathbb R^n)\) such that \[ |\partial_x ^\alpha \partial_\xi ^\beta \partial_\eta ^\gamma \sigma(x,\xi,\eta)|\lesssim_{\alpha,\beta,\gamma} (1+|\xi|+|\eta|)^{m+|\alpha|-|\beta|-|\gamma|}. \] The corresponding pseudo-differential operator is the bilinear operator \(T_\sigma\) defined as \[ T_\sigma (f,g):=\frac{1}{(2\pi)^n}\int_{\mathbb R^{2n}} e^{ix\cdot(\xi+\eta)}\sigma(x,\xi,\eta) \hat f (\xi)\hat g(\eta) d\xi d\eta, \] initially defined for nice functions \(f,g\). The purpose of the paper under review is to study the boundedness property \[ \|T_\sigma(f,g)\|_{F_s ^{p,q}} \lesssim \|f\|_{L^{p_1,q} _{m+s}} \|g\|_{h^{p_2}}+\|f\|_{h^{\tilde {p_1}}} \|g\|_{F_{m+s} ^{\tilde{p_2},q}}, \] for appropriate range of indices. Here \(h^p\) is a local Hardy space defined by means of a suitable maximal operator in the range \(0< p\leq \infty\). The inhomogeneous Triebel-Lizorkin space \(F_s ^{p,q}\), with \(0<p<\infty,\, 0<q\leq \infty\), and \(s>0\), is defined via a smooth Littlewood-Paley decomposition, as usual, with \(s\) playing the roles of smoothness, and \((p,q)\) amounts to an \(L^p(\ell^q)\) norm of the weighted Littlewood-Paley pieces; it is known that \(F_0 ^{p,2}\equiv h^p\). The main theorem of the paper proves the bilinear norm-inequality in the display above, for \(\sigma\in BS_{1,1} ^m\), \(0 < p_1, p_2 , \tilde{p}_1, \tilde{p}_2 <\infty\), \(0 < p < \infty\), \(1/p_1+1/p_2=1/{\tilde{p}_1}+1/\tilde p_2=1/p\), \(0 < q \leq \infty\), and \(s>\tau_{p,q}\); the theorem is also true if either \(p_2\) or \(\tilde p_1\) is equal to \(\infty\). As a consequence the authors obtain an improvement of the Kato-Ponce inequality \[ \| (I-\Delta)^\frac s2(fg)\|_{h^p}\lesssim\|(I-\Delta)^\frac s2f \|_{h^{p_1}}\|g\|_{h^{p_2}}+\|f\|_{h^{\tilde {p}_1}}\|(I-\Delta)^\frac s2 g \|_{h^{\tilde p_1}} \] for appropriate range of parameters. This improves (for \(p\leq 1\)) the classical version of the Kato-Ponce inequality (with Lebesgue spaces \(L^{p_j}\) instead of Hardy spaces). Bilinear estimates for \(T_\sigma\) were previous established by e.g. [\textit{Á. Bényi} and \textit{R. H. Torres}, Commun. Partial Differ. Equations 28, No. 5--6, 1161--1181 (2003; Zbl 1103.35370)], for Sobolev spaces, [\textit{V. Naibo}, J. Fourier Anal. Appl. 21, No. 5, 1077--1104 (2015; Zbl 1334.47048)], with Triebel-Lizorkin spaces of the type \(F_s ^{p_1,q}\), \(F_0 ^{p_2,1}\), and symmetric, on the right hand side.
scientific article; zbMATH DE number 6849512
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