Suppose we are on a closed Riemannian manifold $M$. Any function $f\in C^\infty(M)$ may be decomposed as $$f = \sum_{j = 0}^\infty f_j\phi_j,$$ where $\phi_j\in C^\infty(M)$ are the Laplace eigenfunctions with associated eigenvalues $\lambda_j$ ($||{\phi_j}||_{L^2} = 1$) and $f_j:=\langle f,\phi_j\rangle_{L^2}$.

I was wondering, is there a way to prove that for any $\epsilon>0$ there exists an $C_\epsilon > 0$ such that $$||\phi_j||_{L^\infty}\leq C_\epsilon \lambda_j^{n/4+\epsilon}$$ for all $j\geq0$. Moreover, I'd like to conclude something like $$\forall f\in C^\infty(M),\forall K>0,\exists C_K>0:|f_j|\leq \frac{C_K}{\lambda_j^K}.$$

Is this possible/straightforward?

Suppose we are on a closed Riemannian manifold $M$. Any function $f\in C^\infty(M)$ may be decomposed as $$f = \sum_{j = 0}^\infty f_j\phi_j,$$ where $\phi_j\in C^\infty(M)$ are the Laplace eigenfunctions with associated eigenvalues $\lambda_j$ ($||{\phi_j}||_{L^2} = 1$) and $f_j:=\langle f,\phi_j\rangle_{L^2}$.

I was wondering, is there a way to prove that for any $\epsilon>0$ there exists an $C_\epsilon > 0$ such that $$||\phi_j||_{L^\infty}\leq C_\epsilon \lambda_j^{n/4+\epsilon}$$ for all $j\geq0$. Moreover, I'd like to conclude something like $$\forall f\in C^\infty(M),\forall K>0,\exists C_K>0:|f_j|\leq \frac{C_K}{\lambda_j^K}.$$

How is this possible/straightforward?