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

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