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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:
  Beals, R., On the boundedness of pseudodifferential operators, Comm. partial differential equations, 2, 1063-1070, (1977) · Zbl 0397.35072  Bony, J.-M., Calcul symbolique et propagation des singularités pour LES solutions des EDP non linéaires, Ann. sci. école norm. sup., 14, 209-246, (1981) · Zbl 0495.35024  Boulkhemair, A., On canonical transformations of paradifferential operators, Comm. partial differential equations, 18, 917-964, (1993) · Zbl 0786.35157  Boulkhemair, A., Estimations L2 précisées pour LES opérateurs pseudodifférentiels, C. R. acad. sci. Paris Sér. I, 318, 445-448, (1994) · Zbl 0802.47052  Boulkhemair, A., L2 estimates for pseudodifferential operators, Ann. scuola norm. sup. Pisa cl. sci. (4), 22, 155-183, (1995) · Zbl 0844.35145  Boulkhemair, A., Remarque sur la quantification de Weyl pour la classe de symboles S01, 1, C. R. acad. sci. Paris Sér. I, 321, 1017-1022, (1995) · Zbl 0842.35144  Bourdaud, G., Sur LES opérateurs pseudodifférentiels à coefficients peu réguliers, (1983), Univ. Paris-Sud Orsay  Bourdaud, G., Une algèbre maximale d’op’erateurs pseudodifférentiels, Comm. partial differential equations, 13, 1059-1083, (1988) · Zbl 0659.35115  Calderon, A.P.; Vaillancourt, R., A class of bounded pseudodifferential operators, Proc. nat. acad. sci. U.S.A., 69, 1185-1187, (1972) · Zbl 0244.35074  Childs, A.G., On the L2 boundedness of pseudodifferential operators, Proc. amer. math. soc., 61, 252-254, (1976) · Zbl 0345.47043  Ching, C.H., Pseudodifferential operators with non regular symbols, J. differential equations, 11, 436-447, (1972)  Coifman, R.; Meyer, Y., Au delà des opérateurs pseudodifférentiels, Astérisque, 57, (1978) · Zbl 0483.35082  Cordes, H.O., On compactness of commutators of multiplications and convolutions, and boundedness of pseudodifferential operators, J. funct. anal., 18, 115-131, (1975) · Zbl 0306.47024  Folland, G.B., Harmonic analysis in phase space, (1989), Princeton Univ. Press Princeton · Zbl 0671.58036  Hörmander, L., The Weyl calculus of pseudodifferential operators, Comm. pure appl. math., 32, 355-443, (1979) · Zbl 0388.47032  Hörmander, L., Pseudodifferential operators of type (1, 1), Comm. partial differential equations, 13, 1085-1111, (1988) · Zbl 0667.35078  Hörmander, L., Continuity of pseudodifferential operators of type (1, 1), Comm. partial differential equations, 14, 231-243, (1989) · Zbl 0688.35107  Howe, R., Quantum mechanics and partial differential equations, J. funct. anal., 38, 188-254, (1980) · Zbl 0449.35002  Hwang, I.L., On the L2 boundedness of pseudodifferential operators, Trans. amer. math. soc., 302, 55-76, (1987) · Zbl 0651.35089  Kato, T., The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. rational mech. anal., 58, 181-205, (1975) · Zbl 0343.35056  Kato, T., Boundedness of some pseudodifferential operators, Osaka J. math., 13, 1-9, (1976) · Zbl 0342.47029  Katznelson, Y., An introduction to harmonic analysis, (1976), Dover New York · Zbl 0169.17902  Meyer, Y., Régularité des solutions d’équations aux dérivées partielles non linéaires, Lecture notes in mathematics, (1980), Springer-Verlag New York/Berlin, p. 293-302  Y. Meyer, Remarques sur un thérème de J. M. Bony, in, Supplemento ai rendiconti del circolo matematico di Palermo, atti del Seminario di Analisis Armonica, Pisa, April 8-17 1980, Vol, II, No.  Rouhuai, W.; Chengzhang, L., On the Lp boundedness of several classes of pseudodifferential operators, Chinese ann. math. ser. B, 5, 193-213, (1984) · Zbl 0595.35117  Sjöstrand, J., An algebra of pseudodifferential operators, Math. res. lett., 1, 185-192, (1994) · Zbl 0840.35130  J. Sjöstrand, Wiener type algebras of pseudodifferential operators, in, Séminaire EDP de l’Ecole Polytechnique, 1994/1995, Exposé No, IV.  Taylor, M.E., Pseudodifferential operators and nonlinear partial differential equations, (1991), Birkhäuser Boston  Unterberger, A., Oscillateur harmonique et opérateurs pseudodifférentiels, Ann. inst. Fourier, 29, 201-221, (1979) · Zbl 0396.47027
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