Asymptotic behavior of $L^2$-normalized eigenfunctions of the Laplace-Beltrami operator on a closed Riemannian manifold

Let $e(x,y,\l)$ be the spectral function and ${\chi}_\l$ the unit band spectral projection operator, with respect to the Laplace-Beltrami operator $\D_M$ on a closed Riemannian manifold $M$. We firstly review the one-term asymptotic formula of $e(x,x,\l)$ as $\l\to\infty$ by H{\" o}rmander (1968) and the one of $\p^\al_x\p^\bt_y e(x,y,\l)|_{x=y}$ as $\l\to\infty$ in a geodesic normal coordinate chart by the author (2004) and the sharp asymptotic estimates from above of the mapping norm $\|\chi_\l\|_{L_2\to L_p}$ ($2\leq p\leq\infty$) by Sogge (1988 $&$ 1989) and of the mapping norm $\|\chi_\l\|_{L_2\to {\rm Sobolev} L_p}$ by the author (2004). In the paper we show the one term asymptotic formula for $e(x,y,\l)$ as $\l\to\infty$, provided that the Riemannian distance between $x$ and $y$ is ${\rm O}(1/\l)$. As a consequence, we obtain the sharp estimate of the mapping norm $\|\chi_\l\|_{L_2\to C^\d}$ ($0<\d<1$), where $C^\d(M)$ is the space of H{\" o}lder continuous functions with exponent $\d$ on $M$. Moreover, we show a geometric property of the eigenfunction $e_\l$: $\D_M e_\l+\l^2 e_\l=0$, which says that $1/\l$ is comparable to the distance between the nodal set of $e_\l$ (where $e_\l$ vanishes) and the concentrating set of $e_\l$ (where $e_\l$ attains its maximum or minimum) as $\l\to\infty$.