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\(L^2\) estimates for Weyl quantization. (English) Zbl 0934.35217
Sharp \(L^{2}\)-estimates for Weyl quantized pseudodifferential operators are established in the framework of L. Hörmander classes of symbols \(S_{\rho ,\delta }^{m}\), where \[ S_{\rho ,\delta }^{m}=\{ a(x,\xi)\in C^{\infty }(\mathbb{R}^{2n}:(1+|\xi |^{2})^{-(m-\rho |\alpha |+\delta |\beta |)}|\partial_{x}^{\beta }\partial_{\xi }^{\alpha }a(x,\xi)|\leq C_{\alpha \beta } \;\text{for all }\alpha ,\beta \in \mathbb{N}^{n}\}. \] Let \(Op^{w}(a)\) be the Weyl quantization of \(a\) \[ Op^{w}(a)u(x)=(2\pi)^{-n}\int_{\mathbb{R}^{2n}}e^{i(x-y)\xi }a\Biggl( \frac{x+y}{2} ,\xi \Biggr)u(y) dy d\xi ,\quad u\in S(\mathbb{R}^{n}). \] The main results of the paper is given in the following theorems.
Theorem 1. \(Op^{w}(a)\) defines a bounded operator in \(L^{2}(\mathbb{ R}^{n})\) whenever \(\partial_{x}^{\beta }\partial_{\xi }^{\alpha }a(x,\xi)\in L^{\infty }(\mathbb{R}^{2n})\) for all multi-indices \(\alpha ,\beta \) such that \(|\alpha |,|\beta |\leq [ n/2] +1\) (resp. \(\alpha ,\beta \in \{ 0,1\}^{n}).\) Moreover, there exist symbols \(a(x,\xi)\) for which \(Op^{w}(a)\) is not bounded in \(L^{2}( \mathbb{R}^{n})\) and such that \(\partial_{x}^{\beta }\partial_{\xi }^{\alpha }a\) are bounded functions for \(|\beta |\leq n/2\) and arbitrary \(\alpha \) (or symmetrically, for \(|\alpha |\leq n/2\) and arbitrary \(\beta).\)
Theorem 2. If \(0<\delta <1,\) \(Op^{w}(a)\) defines a bounded operator in \(L^{2}(\mathbb{R}^{n})\) whenever \((1+|\xi |^{2})^{-(m-\rho |\alpha |+\delta |\beta |)}\partial_{x}^{\beta }\partial_{\xi }^{\alpha }a(x,\xi)\in L^{\infty }(\mathbb{R}^{2n})\) for all multi-indices \(\alpha ,\beta \) such that \(|\alpha |,|\beta |\leq [ n/2] +1\) (resp. \(\alpha ,\beta \in \{ 0,1\}^{n}).\) Moreover, this result is sharp with respect to the used number of derivatives of the symbol.
Also the author obtains some results on boundedness of \(Op^{w}(a),\) where \(a\in S_{1,1}^{0}\) in \(H^{s}\), \(s>0.\) The obtained results are an extension of well-known results of H. O. Cordes, R. Coifman and Y. Meyer and other authors on sharp estimates in \(L^{2}(\mathbb{R}^{n})\) of the usual quantization \(Op(a)\).

35S05 Pseudodifferential operators as generalizations of partial differential operators
47G30 Pseudodifferential operators
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