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Sobolev space estimates and symbolic calculus for bilinear pseudodifferential operators. (English) Zbl 1110.47039

The present paper deals with bilinear pseudodifferential operators, defined a priori from \(S (\mathbb R^{n}) \times S (\mathbb R^{n})\) into \(S' (\mathbb R^{n}),\) of the form \[ T_{\sigma} (f,g) (x) = \int_{\mathbb R^{n}} \int_{\mathbb R^{n}} \sigma (x,\xi,\eta) \widehat{f} (\xi) \widehat{g} (\eta) e^{i x(\xi + \eta)} d \xi\, d \eta, \] where their symbols \(\sigma\) satisfy estimates of the form (the class \(B S_{\rho,\delta}^{m})\) \[ | \partial_{x}^{\alpha} \partial_{\xi}^{\beta} \partial_{\eta}^{\gamma} \sigma (x,\xi,\eta) | \leq C_{\alpha \beta \gamma} (1 + | \xi | + | \eta | )^{m + \delta | \alpha | - \rho (| \beta | + | \gamma |)}, \] or (the class \(B S_{\rho,\delta;\theta}^{m}\)) \[ | \partial_{x}^{\alpha} \partial_{\xi}^{\beta} \partial_{\eta}^{\gamma} \sigma (x,\xi,\eta) | \leq C_{\alpha \beta \gamma;\theta} (1 + | \eta - \xi \tan \theta | )^{m + \delta | \alpha | - \rho (| \beta | + | \gamma |)} \] for all \((x,\xi,\eta) \in \mathbb R^{3n}\), all multi-indices \(\alpha,\beta\) and \(\gamma,\) and some constants \(C_{\alpha \beta \gamma}\) or, respectively, \(C_{\alpha \beta \gamma;\theta}\). It is assumed that \(\theta \in (-\frac{\pi}{2},\frac{\pi}{2})\), with the convention that \(\theta = \pi/2\) corresponds to the decay in terms of \(1 + | \xi |\) only. \(S (\mathbb R^{n})\) denotes the Schwartz space of functions and \(S' (\mathbb R^{n})\) is the space of tempered distributions. By \(\widehat{f}\) is denoted the Fourier transform of the function \(f\in S (\mathbb R^{n}).\)
The authors study mainly the boundedness properties of such operators \(T_{\sigma}\). Among many other results, the authors prove that every operator \(T_{\sigma}\) with a symbol in the class \(B S_{1,1}^{m}\), \(m \geq 0,\) has a bounded extension from \(L_{m + s}^{p} \times L_{m + s}^{q}\) into \(L_{s}^{r},\) provided that \(1/p + 1/q = 1/r\), \(1 < p,q,r < \infty,\) and \(s > 0.\) Moreover,
\[ \| T_{\sigma} (f,g) \|_{L_{s}^{r}} \leq C (p,q,r,s,n,m,\sigma) \left(\| f \|_{L_{m + s}^{p}} \| g \|_{L^{q}} + \| f \|_{L^{p}} \| g \|_{L_{m + s}^{q}}\right). \] A symbolic calculus for the transposes of bilinear pseudodifferential operators and for the composition of linear pseudodifferential operators is also presented.

MSC:

47G30 Pseudodifferential operators
42B15 Multipliers for harmonic analysis in several variables
42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
35S99 Pseudodifferential operators and other generalizations of partial differential operators
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