Riesz means of eigenfunction expansions of elliptic differential operators on compact manifolds
Tóm tắt
Let {λ
2} and {ϕ
λ
} be the eigenvalues and an orthonormal system of eigenvectors of a second order elliptic differential operator Δ on a compact manifoldM of dimensionN. We prove that the Riesz means of order δ, defined by
$$R_\Lambda ^\delta f = \sum\limits_{\lambda< \Lambda } {\left( {1 - \frac{{\lambda ^2 }}{{\Lambda ^2 }}} \right)^\delta \hat f(} \lambda ) \varphi _\lambda $$
, are uniformly bounded from the Hardy spaceH
p
(M) into Weak-L
p
(M), if 0
Tài liệu tham khảo
[Alimov-Il'in-Nikishin]S. A. Alimov, V. A. Il'in, E. M. Nikishin,Convergence problems of multiple Fourier series and spectral decompositions. I, II. Russian Math. Surveys 31 (1976), 29–86; 32 (1977), 115–139.
[Bérard-1]P. Bérard,On the wave equation on a Riemannian manifold without conjugate points. Math. Z. 155 (1977), 249–276.
[Bérard-2]P. Bérard,Riesz means on Riemannian manifolds. A.M.S. Proc. Symp. Pure Math. XXXVI (1980), 1–12.
[Bochner]S. Bochner,Summation of multiple Fourier series by spherical means. Trans. A.M.S. 40 (1936), 175–207.
[Bonami-Clerc]A. Bonami, J. L. Clerc,Sommes de Cesàro et multiplicateurs des developpements en harmoniques spheriques. Trans. A.M.S. 183 (1973), 223–263.
[Carleson-Sjölin]L. Carleson, P. Sjölin,Oscillatory integrals and a multiplier problem for the disc. Studia Math. 44 (1972), 287–299.
[Christ-Sogge]M. Christ, C. Sogge,The weak type L1 convergence of eigenfunction expansions for pseudodifferential operators. Invent. Math. 94 (1988), 421–453.
[Christ-1]M. Christ,Weak type (1,1)bounds for rugh operators. Ann. Math. 128 (1988), 19–42.
[Christ-2]M. Christ,Weak type endpoints bounds for Bochner-Riesz multipliers. Rev. Mat. Ibero-Am. 3 (1987), 25–31.
[Clerc-1]J. L. Clerc,Sommes de Riesz et multiplicateurs sur un groupe de Lie compact. Ann. Inst. Fourier 24 (1974), 149–172.
[Clerc-2]J. L. Clerc,Bochner-Riesz means of Hp functions (0<p<1)on compact Lie groups. In “Noncommutative harmonic analysis and Lie groups”. Lecture Notes in Math. 1243 (1987), 86–107.
[Coifman-Weiss]R. Coifman, G. Weiss,Extensions of Hardy spaces and their use in analysis. Bull. A.M.S. 83 (1977), 569–645.
[Colzani-Taibleson-Weiss]L. Colzani, M. Taibleson, G. Weiss,Maximal estimates for Cesàro and Riesz means on spheres. Indiana J. Math. 33 (1984), 173–189.
[Colzani-Travaglini] L. Colzani, G. Travaglini,Hardy-Lorentz spaces and expansions in eigenfunctions of the Laplace-Beltrami operator on compact manifolds. Quaderno n. 11/1988. Dipartimento di Matematica, Università degli Studi di Milano. To appear in Coll. Math.
[Colzani-1]L. Colzani,Cesàro means of powers series. Boll. U.M.I. (6) 3-A (1984), 147–149.
[Colzani-2]L. Colzani,Taylor coefficients of functions in certain weak Hardy spaces. Boll. U.M.I. (6) 4-A (1985), 57–66.
[Fefferman-1]C. Fefferman,The multiplier problem for the ball. Ann. Math. 94 (1971), 330–336.
[Fefferman-2]C. Fefferman,A note on spherical summation multipliers. Israel J. Math. 15 (1973), 44–52.
[Goldberg]D. Goldberg,A local version of real Hardy spaces. Duke Math. J. 46 (1979), 27–42.
[Hardy-Littlewood]G. H. Hardy, J. E. Littlewood,Theorems concerning Cesàro means of power series. Proc. London Math. Soc. 36 (1934), 516–531.
[Hörmander-1]L. Hörmander,On the Riesz means of spectral functions and eigenfunction expansions for elliptic differential operators. In “Some recent advances in the basic sciences”, Yeshiva University, New York (1966), 155–202.
[Hörmander-2]L. Hörmander,The spectral function of an elliptic operator. Acta Math. 121 (1968), 193–218.
[Hörmander-3]L. Hörmander,The analysis of linear partial differential operators. Vol. I–IV. Springer Verlag, New York, 1983, 1985.
[Kenig-Staton-Tomas]C. Kenig, R. Stanton, P. Tomas,Divergence of eigenfunction expansions. J. Funct. Anal. 46 (1982), 28–44.
[Stein-Taibleson-Weiss]E. Stein, M. Taibleson, G. Weiss,Weak type estimates for maximal operators on certain H p classes. Suppl. Rendiconti Circ. Mat. Palermo 1 (1981), 81–97.
[Stein]E. Stein,Localization and summability of multiple Fourier series. Acta Math. 100 (1958), 93–147.
[Sogge-1]C. Sogge,On the convergence of Riesz means on compact manifolds. Ann. Math. 126 (1987), 439–447.
[Sogge-2]C. Sogge,Concerning the L p norm of spectral clusters for second order elliptic operators on compact manifolds. J. Funct. Anal. 77 (1988), 123–138.
[Taylor]M. Taylor,Pseudodifferential operators. Princeton University Press, Princeton, 1981.