Let $\mathcal{L}$ be the Laplace-Beltrami operator. On an $n$-dimensional $ (n\geq 2)$, complete, noncompact Riemannian manifold $\mathbb{M}$, we prove that if $0<\alpha<1, s>\alpha/2$ and $f \in H^s(\mathbb{M})$, then the fractional Schrödinger propagator ${\rm e}^{{\rm i}t|\mathcal{L}|^{\alpha/2}}(f)(x)\rightarrow f(x)$ a.e. as $t\rightarrow0$. In addition, for when $\mathbb{M}$ is a Lie group, the rate of the convergence is also studied. These results are a non-trivial extension of results on Euclidean spaces and compact manifolds.
Yali PAN
. SOME CONVERGENCE PROBLEMS REGARDING THE FRACTIONAL SCHRÖDINGER PROPAGATOR ON NONCOMPACT MANIFOLDS*[J]. Acta mathematica scientia, Series B, 2023
, 43(5)
: 2309
-2319
.
DOI: 10.1007/s10473-023-0522-8
[1] Bisshop R L, Crittenden R J.Geometry of Manifolds. New York: Academic Press, 1964
[2] Bourgain J. A note on the Schrödinger maximal function. J Anal Math, 2016, 130: 393-396
[3] Cao Z, Fan D, Wang M. The rate of convergence on Schrödinger operator. Illinois Journal of Mathematics, 2018, 62: 365-380
[4] Carleson L.Some analytic problems related to statistical mechanics//Benedetto J. Euclidean Harmonic Analysis. Berlin, Heidelberg: Springer, 1980: 5-45
[5] Chen J, Fan D, Zhao F. On the rate of almost everywhere convergence of combinations and multivariate averages. Potential Anal, 2019, 51(3): 397-423
[6] Chen J, Fan D, Zhao F. Approximation of the fractional Schrödinger propagator on compact manifolds. Acta Math Sinica, 2021, 37(10): 1485-1496
[7] Coifman R, Weiss G. Extensions of Hardy spaces and their use analysis. Bull Amer math Soc, 1977, 83: 569-645
[8] Dahlberg B E J, Kenig C E. A note on the almost everywhere behavior of solutions to the Schrödinger equation//Harmonic Analysis (Minneapolis, Minn, 1981). Lecture Notes in Math, 908. New York: Springer-Verlag, 1982: 205-209
[9] Du X, Guth L, Li X. A sharp Schrödinger maximal estimate in $\mathbb{R}^2$. Ann of Math, 2017, 186: 607-640
[10] Du X, Guth L, Li X, Zhang R. Pointwise convergence of Schrödinger solutions and multilinear refined Strichartz estimates. Forum Math Sigma, 2018, 6: e14
[11] Du X, Zhang R. Sharp $L^2$ estimates of the Schrödinger maximal function in higher dimensions. Ann of Math, 2019, 189: 837-861
[12] Liu H, Zeng H. Local estimate about Schrödinger maximal operator on H-type groups. Acta Math Sci, 2017, 37B(2): 527-538
[13] Marias M. $L^p$-boundedness of oscillating spectral multipliers on Riemannian manifolds. Annales Mathématiques, 2003, 10: 133-160
[14] Meaney C Müller D, Prestini E. A.e. convergence of spectral sums on Lie groups. Ann Inst Fourier (Grenoble), 2007, 57(5): 1509-1520
[15] Miao C, Yang J, Zheng J. An improved maximal inequality for 2D fractional order Schrödinger operators. Studia Mathematica, 2015, 230: 121-165
[16] Shubin M A. Special theory of the elliptic operators on noncompact manifolds. Astérisque, 1992, 207: 35-108
[17] Sjölin P. Regularity of solutions to the Schrödinger equation. Duke Math J, 1987, 55(3): 699-715
[18] Sjölin P. $L^p$ maximal estimates for solutions to the Schrödinger equation. Math Scand, 1997, 81(1): 35-68
[19] Vega L. Schrödinger equations: Pointwise convergence to the initial data. Proc Amer Math Soc, 1988, 102(4): 874-878
[20] Walther B G. Higher integrability for maximal oscillatory Fourier integrals. Ann Acad Sci Fenn Math, 2001, 26(1): 189-204
[21] Walther B G. Global range estimates for maximal oscillatory integrals with radial test functions. Illinois J Math, 2012, 56(2): 521-532
[22] Zhang C. Pointwise convergence of solutions to Schrödinger type equations. Nonlinear Anal, 2014, 109: 180-186
