𝐿^{𝑝} bounds for spectral multipliers on nilpotent groups

Christ, Michael

doi:10.1090/S0002-9947-1991-1104196-7

$L^ p$ bounds for spectral multipliers on nilpotent groups
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by Michael Christ PDF

Trans. Amer. Math. Soc. 328 (1991), 73-81 Request permission

Abstract:

A criterion is given for the ${L^p}$ boundedness of a class of spectral multiplier operators associated to left-invariant, homogeneous subelliptic second-order differential operators on nilpotent Lie groups, generalizing a theorem of Hörmander for radial Fourier multipliers on Euclidean space. The order of differentiability required is half the homogeneous dimension of the group, improving previous results in the same direction.

References

F. M. Christ and C. D. Sogge, The weak type $L^1$ convergence of eigenfunction expansions for pseudodifferential operators, Invent. Math. 94 (1988), no. 2, 421–453. MR 958838, DOI 10.1007/BF01394331
E. B. Davies, Explicit constants for Gaussian upper bounds on heat kernels, Amer. J. Math. 109 (1987), no. 2, 319–333. MR 882426, DOI 10.2307/2374577
G. B. Folland and Elias M. Stein, Hardy spaces on homogeneous groups, Mathematical Notes, vol. 28, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1982. MR 657581
Lars Hörmander, Estimates for translation invariant operators in $L^{p}$ spaces, Acta Math. 104 (1960), 93–140. MR 121655, DOI 10.1007/BF02547187
Andrzej Hulanicki and Joe W. Jenkins, Almost everywhere summability on nilmanifolds, Trans. Amer. Math. Soc. 278 (1983), no. 2, 703–715. MR 701519, DOI 10.1090/S0002-9947-1983-0701519-0
David S. Jerison and Antonio Sánchez-Calle, Estimates for the heat kernel for a sum of squares of vector fields, Indiana Univ. Math. J. 35 (1986), no. 4, 835–854. MR 865430, DOI 10.1512/iumj.1986.35.35043
S. Kusuoka and D. Stroock, Applications of the Malliavin calculus. III, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 34 (1987), no. 2, 391–442. MR 914028
S. Kusuoka and D. Stroock, Long time estimates for the heat kernel associated with a uniformly subelliptic symmetric second order operator, Ann. of Math. (2) 127 (1988), no. 1, 165–189. MR 924675, DOI 10.2307/1971418
Giancarlo Mauceri and Stefano Meda, Vector-valued multipliers on stratified groups, Rev. Mat. Iberoamericana 6 (1990), no. 3-4, 141–154. MR 1125759, DOI 10.4171/RMI/100
Richard Melrose, Propagation for the wave group of a positive subelliptic second-order differential operator, Hyperbolic equations and related topics (Katata/Kyoto, 1984) Academic Press, Boston, MA, 1986, pp. 181–192. MR 925249

On Riesz means of eigenfunction expansions for the Kohn-Laplacian

Michael Reed and Barry Simon, Methods of modern mathematical physics. I, 2nd ed., Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1980. Functional analysis. MR 751959
Elias M. Stein, Topics in harmonic analysis related to the Littlewood-Paley theory. , Annals of Mathematics Studies, No. 63, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1970. MR 0252961
Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
Nicholas Th. Varopoulos, Semi-groupes d’opérateurs sur les espaces $L^p$, C. R. Acad. Sci. Paris Sér. I Math. 301 (1985), no. 19, 865–868 (French, with English summary). MR 823141