×

zbMATH — the first resource for mathematics

The uncertainty principle. (English) Zbl 0526.35080

MSC:
35S05 Pseudodifferential operators as generalizations of partial differential operators
65H10 Numerical computation of solutions to systems of equations
35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
35P15 Estimates of eigenvalues in context of PDEs
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Richard Beals and Charles Fefferman, On local solvability of linear partial differential equations, Ann. of Math. (2) 97 (1973), 482 – 498. · Zbl 0256.35002 · doi:10.2307/1970832 · doi.org
[2] Richard Beals and Charles Fefferman, Spatially inhomogeneous pseudodifferential operators. I, Comm. Pure Appl. Math. 27 (1974), 1 – 24. · Zbl 0279.35071 · doi:10.1002/cpa.3160270102 · doi.org
[3] Michael Beals, Charles Fefferman, and Robert Grossman, Strictly pseudoconvex domains in \?\(^{n}\), Bull. Amer. Math. Soc. (N.S.) 8 (1983), no. 2, 125 – 322. · Zbl 0546.32008
[4] Alberto-P. Calderón and Rémi Vaillancourt, A class of bounded pseudo-differential operators, Proc. Nat. Acad. Sci. U.S.A. 69 (1972), 1185 – 1187. · Zbl 0244.35074
[5] T. Carleman, Complete works , Malmo Litös, 1960.
[6] D. Catlin, Necessary conditions for subelliptiticy and hypoellipticity for the \({\o}verline\partial\)-Neumann problem on pseudoconvex domains, Recent Developments in Several Complex Variables, Ann. of Math. Stud., no. 100, 1981.
[7] Shiu Yuen Cheng and Shing Tung Yau, On the existence of a complete Kähler metric on noncompact complex manifolds and the regularity of Fefferman’s equation, Comm. Pure Appl. Math. 33 (1980), no. 4, 507 – 544. · Zbl 0506.53031 · doi:10.1002/cpa.3160330404 · doi.org
[8] Antonio Córdoba and Charles Fefferman, Wave packets and Fourier integral operators, Comm. Partial Differential Equations 3 (1978), no. 11, 979 – 1005. · Zbl 0389.35046 · doi:10.1080/03605307808820083 · doi.org
[9] J. D’Angelo, Real hypersurfaces with degenerate Levi form, Thesis, Princeton Univ., Princeton, N.J., 1976.9a. F. Dyson and A. Lenard, Stability of matter. I, J. Math. Phys. 8 (1967), 423-434.
[10] Ju. V. Egorov, Subelliptic operators, Uspehi Mat. Nauk 30 (1975), no. 3(183), 57 – 104 (Russian). · Zbl 0331.35054
[11] Charles Fefferman, Recent progress in classical Fourier analysis, Proceedings of the International Congress of Mathematicians (Vancouver, B.C., 1974) Canad. Math. Congress, Montreal, Que., 1975, pp. 95 – 118. · Zbl 0332.42021
[12] Charles Fefferman, The Bergman kernel and biholomorphic mappings of pseudoconvex domains, Invent. Math. 26 (1974), 1 – 65. · Zbl 0289.32012 · doi:10.1007/BF01406845 · doi.org
[13] C. Fefferman and D. H. Phong, On positivity of pseudo-differential operators, Proc. Nat. Acad. Sci. U.S.A. 75 (1978), no. 10, 4673 – 4674. · Zbl 0391.35062
[14] C. Fefferman and D. H. Phong, The uncertainty principle and sharp Gȧrding inequalities, Comm. Pure Appl. Math. 34 (1981), no. 3, 285 – 331. · Zbl 0458.35099 · doi:10.1002/cpa.3160340302 · doi.org
[15] C. Fefferman and D. H. Phong, On the asymptotic eigenvalue distribution of a pseudodifferential operator, Proc. Nat. Acad. Sci. U.S.A. 77 (1980), no. 10, 5622 – 5625. · Zbl 0443.35082
[16] C. Fefferman and D. H. Phong, Subelliptic eigenvalue problems, Conference on harmonic analysis in honor of Antoni Zygmund, Vol. I, II (Chicago, Ill., 1981) Wadsworth Math. Ser., Wadsworth, Belmont, CA, 1983, pp. 590 – 606. · Zbl 0503.35071
[17] C. Fefferman and D. H. Phong, Symplectic geometry and positivity of pseudodifferential operators, Proc. Nat. Acad. Sci. U.S.A. 79 (1982), no. 2, 710 – 713. · Zbl 0553.35089
[18] R. Fefferman, personal communication.
[19] G. B. Folland and E. M. Stein, Estimates for the \partial _\? complex and analysis on the Heisenberg group, Comm. Pure Appl. Math. 27 (1974), 429 – 522. · Zbl 0293.35012 · doi:10.1002/cpa.3160270403 · doi.org
[20] R. Greiner and E. M. Stein, Estimates for the \({\o}verline\partial\)-Neumann problem, Math. Notes 19 (1977).
[21] Richard A. Hunt, An extension of the Marcinkiewicz interpolation theorem to Lorentz spaces, Bull. Amer. Math. Soc. 70 (1964), 803 – 807. · Zbl 0145.38203
[22] Lars Hörmander, Hypoelliptic second order differential equations, Acta Math. 119 (1967), 147 – 171. · Zbl 0156.10701 · doi:10.1007/BF02392081 · doi.org
[23] Lars Hörmander, The spectral function of an elliptic operator, Acta Math. 121 (1968), 193 – 218. · Zbl 0164.13201 · doi:10.1007/BF02391913 · doi.org
[24] Lars Hörmander, Subelliptic operators, Seminar on Singularities of Solutions of Linear Partial Differential Equations (Inst. Adv. Study, Princeton, N.J., 1977/78) Ann. of Math. Stud., vol. 91, Princeton Univ. Press, Princeton, N.J., 1979, pp. 127 – 208.
[25] D. Iagolnitzer, Appendix: Microlocal essential support of a distribution and decomposition theorems -an introduction, Lecture Notes in Math., Vol. 449, Springer-Verlag, 1975, pp. 121-132.
[26] J. J. Kohn, Harmonic integrals on strongly pseudo-convex manifolds. II, Ann. of Math. (2) 79 (1964), 450 – 472. · Zbl 0178.11305 · doi:10.2307/1970404 · doi.org
[27] J. J. Kohn, Subellipticity of the \partial -Neumann problem on pseudo-convex domains: sufficient conditions, Acta Math. 142 (1979), no. 1-2, 79 – 122. · Zbl 0395.35069 · doi:10.1007/BF02395058 · doi.org
[28] A. Kolmogoroff, Zufällige Bewegungen (zur Theorie der Brownschen Bewegung), Ann. of Math. (2) 35 (1934), no. 1, 116 – 117 (German). · Zbl 0008.39906 · doi:10.2307/1968123 · doi.org
[29] R. Lee and R. Melrose, to appear.
[30] Elliott H. Lieb, The stability of matter, Rev. Modern Phys. 48 (1976), no. 4, 553 – 569. · doi:10.1103/RevModPhys.48.553 · doi.org
[31] R. D. Moyer, On the Nirenberg-Treves condition for local solvability, J. Differential Equations 26 (1977), no. 2, 223 – 239. · Zbl 0344.35002 · doi:10.1016/0022-0396(77)90192-9 · doi.org
[32] A. Nagel, E. M. Stein and S. Wainger, to appear.
[33] L. Nirenberg and F. Treves, Solvability of a first order linear partial differential equation, Comm. Pure Appl. Math. 16 (1963), 331 – 351. · Zbl 0117.06104 · doi:10.1002/cpa.3160160308 · doi.org
[34] Louis Nirenberg and François Trèves, On local solvability of linear partial differential equations. I. Necessary conditions, Comm. Pure Appl. Math. 23 (1970), 1 – 38. , https://doi.org/10.1002/cpa.3160230102 Louis Nirenberg and François Trèves, On local solvability of linear partial differential equations. II. Sufficient conditions, Comm. Pure Appl. Math. 23 (1970), 459 – 509. · Zbl 0208.35902 · doi:10.1002/cpa.3160230314 · doi.org
[35] O. Oleinik and E. Radkevitch, Second-order equations with non-negative characteristic form.
[36] D. H. Phong, On integral representations for the Neumann operator, Proc. Nat. Acad. Sci. U.S.A. 76 (1979), no. 4, 1554 – 1558. · Zbl 0402.35074
[37] Linda Preiss Rothschild and E. M. Stein, Hypoelliptic differential operators and nilpotent groups, Acta Math. 137 (1976), no. 3-4, 247 – 320. · Zbl 0346.35030 · doi:10.1007/BF02392419 · doi.org
[38] A. Sanchez, Estimates for kernels associated to some subelliptic operators, Thesis, Princeton Univ., 1983.
[39] B. Simon, to appear.
[40] Barry Simon, Functional integration and quantum physics, Pure and Applied Mathematics, vol. 86, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1979. · Zbl 0434.28013
[41] Hermann Weyl, Ramifications, old and new, of the eigenvalue problem, Bull. Amer. Math. Soc. 56 (1950), 115 – 139. · Zbl 0041.21003
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.