TY - GEN
T1 - Characterization of Carleson Measures via Spectral Estimates on Compact Manifolds with Boundary
AU - Xu, Xiangjin
N1 - Publisher Copyright: © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024.
PY - 2024
Y1 - 2024
N2 - Given a compact Riemannian manifold M with boundary of dimension m≥2, we study the space of functions EL of L2(M) generated by eigenfunctions of eigenvalues less than L≥1 associated to Dirichlet Laplacian and Neumann Laplacian on M. The asymptotics of the reproducing kernel of the space EL and a Bernstein type inequality for f∈EL are discussed. Furthermore under suitable convexity assumptions on the boundary, the mean value inequalities of subharmonic functions associated to EL in the scale 1L are achieved on the metric ball with possible nonempty intersection with the boundary, which generalizes the classical mean value inequality on the interior geodesic ball by Li, Schoen, and Yau. Applying the asymptotic estimates, Bernstein type inequality and mean value inequality on these spaces EL, we show a characterization of the L2-Carleson measures associated to Neumann Laplacian with the interior rolling R-ball condition on the boundary, and give a counterexample to invalid the characterization of the L2-Carleson measures associated to Dirichlet Laplacian.
AB - Given a compact Riemannian manifold M with boundary of dimension m≥2, we study the space of functions EL of L2(M) generated by eigenfunctions of eigenvalues less than L≥1 associated to Dirichlet Laplacian and Neumann Laplacian on M. The asymptotics of the reproducing kernel of the space EL and a Bernstein type inequality for f∈EL are discussed. Furthermore under suitable convexity assumptions on the boundary, the mean value inequalities of subharmonic functions associated to EL in the scale 1L are achieved on the metric ball with possible nonempty intersection with the boundary, which generalizes the classical mean value inequality on the interior geodesic ball by Li, Schoen, and Yau. Applying the asymptotic estimates, Bernstein type inequality and mean value inequality on these spaces EL, we show a characterization of the L2-Carleson measures associated to Neumann Laplacian with the interior rolling R-ball condition on the boundary, and give a counterexample to invalid the characterization of the L2-Carleson measures associated to Dirichlet Laplacian.
KW - Asymptotics of spectral function
KW - Bernstein type inequality
KW - Carleson measure
KW - Mean value inequality
UR - https://www.scopus.com/pages/publications/85210321309
U2 - 10.1007/978-3-031-69706-7_1
DO - 10.1007/978-3-031-69706-7_1
M3 - Conference contribution
SN - 9783031697050
T3 - Springer Proceedings in Mathematics and Statistics
SP - 1
EP - 23
BT - Applied Mathematical Analysis and Computations I - 1st SGMC
A2 - Wanduku, Divine
A2 - Zheng, Shijun
A2 - Chen, Zhan
A2 - Sills, Andrew
A2 - Zhou, Haomin
A2 - Agyingi, Ephraim
PB - Springer
T2 - 1st Southern Georgia Mathematics Conference, SGMC 2021
Y2 - 2 April 2021 through 3 April 2021
ER -