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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 $$Op(a)$$.

##### MSC:
 35S05 Pseudodifferential operators as generalizations of partial differential operators 47G30 Pseudodifferential operators
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##### References:
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