×

zbMATH — the first resource for mathematics

\(L^2\) 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
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Beals, R., On the boundedness of pseudodifferential operators, Comm. partial differential equations, 2, 1063-1070, (1977) · Zbl 0397.35072
[2] 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
[3] Boulkhemair, A., On canonical transformations of paradifferential operators, Comm. partial differential equations, 18, 917-964, (1993) · Zbl 0786.35157
[4] Boulkhemair, A., Estimations L2 précisées pour LES opérateurs pseudodifférentiels, C. R. acad. sci. Paris Sér. I, 318, 445-448, (1994) · Zbl 0802.47052
[5] Boulkhemair, A., L2 estimates for pseudodifferential operators, Ann. scuola norm. sup. Pisa cl. sci. (4), 22, 155-183, (1995) · Zbl 0844.35145
[6] 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
[7] Bourdaud, G., Sur LES opérateurs pseudodifférentiels à coefficients peu réguliers, (1983), Univ. Paris-Sud Orsay
[8] Bourdaud, G., Une algèbre maximale d’op’erateurs pseudodifférentiels, Comm. partial differential equations, 13, 1059-1083, (1988) · Zbl 0659.35115
[9] 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
[10] Childs, A.G., On the L2 boundedness of pseudodifferential operators, Proc. amer. math. soc., 61, 252-254, (1976) · Zbl 0345.47043
[11] Ching, C.H., Pseudodifferential operators with non regular symbols, J. differential equations, 11, 436-447, (1972)
[12] Coifman, R.; Meyer, Y., Au delà des opérateurs pseudodifférentiels, Astérisque, 57, (1978) · Zbl 0483.35082
[13] 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
[14] Folland, G.B., Harmonic analysis in phase space, (1989), Princeton Univ. Press Princeton · Zbl 0671.58036
[15] Hörmander, L., The Weyl calculus of pseudodifferential operators, Comm. pure appl. math., 32, 355-443, (1979) · Zbl 0388.47032
[16] Hörmander, L., Pseudodifferential operators of type (1, 1), Comm. partial differential equations, 13, 1085-1111, (1988) · Zbl 0667.35078
[17] Hörmander, L., Continuity of pseudodifferential operators of type (1, 1), Comm. partial differential equations, 14, 231-243, (1989) · Zbl 0688.35107
[18] Howe, R., Quantum mechanics and partial differential equations, J. funct. anal., 38, 188-254, (1980) · Zbl 0449.35002
[19] Hwang, I.L., On the L2 boundedness of pseudodifferential operators, Trans. amer. math. soc., 302, 55-76, (1987) · Zbl 0651.35089
[20] Kato, T., The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. rational mech. anal., 58, 181-205, (1975) · Zbl 0343.35056
[21] Kato, T., Boundedness of some pseudodifferential operators, Osaka J. math., 13, 1-9, (1976) · Zbl 0342.47029
[22] Katznelson, Y., An introduction to harmonic analysis, (1976), Dover New York · Zbl 0169.17902
[23] 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
[24] 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.
[25] 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
[26] Sjöstrand, J., An algebra of pseudodifferential operators, Math. res. lett., 1, 185-192, (1994) · Zbl 0840.35130
[27] J. Sjöstrand, Wiener type algebras of pseudodifferential operators, in, Séminaire EDP de l’Ecole Polytechnique, 1994/1995, Exposé No, IV.
[28] Taylor, M.E., Pseudodifferential operators and nonlinear partial differential equations, (1991), Birkhäuser Boston
[29] Unterberger, A., Oscillateur harmonique et opérateurs pseudodifférentiels, Ann. inst. Fourier, 29, 201-221, (1979) · Zbl 0396.47027
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.