Characterization of Carleson Measures via Spectral Estimates on Compact Manifolds with Boundary

Given a compact Riemannian manifold M with boundary of dimension \(m\geq 2\) , we study the space of functions \(E_L\) of \(L^2(M)\) generated by eigenfunctions of eigenvalues less than \(L\geq 1\) associated to Dirichlet Laplacian and Neumann Laplacian on M. The asymptotics of the reproducing kernel of the space \(E_L\) and a Bernstein type inequality for \(f\in E_L\) are discussed. Furthermore under suitable convexity assumptions on the boundary, the mean value inequalities of subharmonic functions associated to \(E_L\) in the scale \(\frac {1}{\sqrt {L}}\) 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 \(E_L\) , we show a characterization of the \(L^2\) -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 \(L^2\) -Carleson measures associated to Dirichlet Laplacian.