Almost everywhere convergence of spectral sums for self-adjoint operators. (English) Zbl 1497.42005
Summary: Let \(L\) be a non-negative self-adjoint operator acting on the space \(L^2(X)\), where \(X\) is a positive measure space. Let \(L=\int_0^{\infty} \lambda dE_L(\lambda)\) be the spectral resolution of \(L\) and \(S_R(L)f=\int_0^R dE_L(\lambda) f\) denote the spherical partial sums in terms of the resolution of \(L\). In this article we give a sufficient condition on \(L\) such that
\[
\lim_{R\rightarrow\infty}S_R(L)f(x)=f(x), \text{ a.e.}
\]
for any \(f\) such that \(\log(2+L)f\in L^2(X)\). These results are applicable to large classes of operators including Dirichlet operators on smooth bounded domains, the Hermite operator and Schrödinger operators with inverse square potentials.
MSC:
| 42A20 | Convergence and absolute convergence of Fourier and trigonometric series |
| 42B15 | Multipliers for harmonic analysis in several variables |
| 42A38 | Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type |
| 42B08 | Summability in several variables |
| 47F05 | General theory of partial differential operators |
| 35J10 | Schrödinger operator, Schrödinger equation |
Keywords:
almost everywhere convergence; spherical partial sums; non-negative self-adjoint operators; Rademacher-Menshov theorem; Plancherel-type estimateReferences:
| [1] | Alexits, G., Convergence problems of orthogonal series, International Series of Monographs in Pure and Applied Mathematics, Vol. 20, ix+350 pp. (1961), Pergamon Press, New York-Oxford-Paris · Zbl 0098.27403 |
| [2] | Benedek, A., The space \(L^p\), with mixed norm, Duke Math. J., 301-324 (1961) · Zbl 0107.08902 |
| [3] | Carbery, Anthony, Almost-everywhere convergence of Fourier integrals for functions in Sobolev spaces, and an \(L^2\)-localisation principle, Rev. Mat. Iberoamericana, 319-337 (1988) · Zbl 0692.42001 · doi:10.4171/RMI/76 |
| [4] | Carleson, Lennart, On convergence and growth of partial sums of Fourier series, Acta Math., 135-157 (1966) · Zbl 0144.06402 · doi:10.1007/BF02392815 |
| [5] | Chen, Peng, Restriction estimates, sharp spectral multipliers and endpoint estimates for Bochner-Riesz means, J. Anal. Math., 219-283 (2016) · Zbl 1448.47033 · doi:10.1007/s11854-016-0021-0 |
| [6] | Christ, Michael, \(L^p\) bounds for spectral multipliers on nilpotent groups, Trans. Amer. Math. Soc., 73-81 (1991) · Zbl 0739.42010 · doi:10.2307/2001877 |
| [7] | Coifman, Ronald R., Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc., 569-645 (1977) · Zbl 0358.30023 · doi:10.1090/S0002-9904-1977-14325-5 |
| [8] | Colzani, Leonardo, Almost everywhere convergence of inverse Fourier transforms, Proc. Amer. Math. Soc., 1651-1660 (2006) · Zbl 1082.42006 · doi:10.1090/S0002-9939-05-08329-2 |
| [9] | Thinh Duong, Xuan, Plancherel-type estimates and sharp spectral multipliers, J. Funct. Anal., 443-485 (2002) · Zbl 1029.43006 · doi:10.1016/S0022-1236(02)00009-5 |
| [10] | H\"{o}rmander, Lars, The analysis of linear partial differential operators. III, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], viii+525 pp. (1985), Springer-Verlag, Berlin · Zbl 1115.35005 |
| [11] | Hulanicki, Andrzej, Almost everywhere summability on nilmanifolds, Trans. Amer. Math. Soc., 703-715 (1983) · Zbl 0516.43010 · doi:10.2307/1999179 |
| [12] | Kunstmann, Peer Christian, Spectral multiplier theorems of H\"{o}rmander type on Hardy and Lebesgue spaces, J. Operator Theory, 27-69 (2015) · Zbl 1358.42008 · doi:10.7900/jot.2013aug29.2038 |
| [13] | Liskevich, Vitali, On the \(L_p\)-theory of \(C_0\)-semigroups associated with second-order elliptic operators. II, J. Funct. Anal., 55-76 (2002) · Zbl 1020.47029 · doi:10.1006/jfan.2001.3909 |
| [14] | Meaney, Christopher, A.E. convergence of spectral sums on Lie groups, Ann. Inst. Fourier (Grenoble), 1509-1520 (2007) · Zbl 1131.22007 |
| [15] | Zworski, Maciej, Semiclassical analysis, Graduate Studies in Mathematics, xii+431 pp. (2012), American Mathematical Society, Providence, RI · Zbl 1252.58001 · doi:10.1090/gsm/138 |
| [16] | Zygmund, A., Trigonometric series: Vols. I, II, Vol. I. xiv+383 pp.; Vol. II: vii+364 pp. (two volumes bound as one) pp. (1968), Cambridge University Press, London-New York |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
