Let $K: [0,1]\times[0,1] \to \mathbb{R}$ be a symmetric positive definit and continuous function. It is known my Mercer's theorem that $$ [T_K \varphi](x) =\int_0^1 K(x,s) \varphi(s)\, ds $$ is compact and can be written as $$ K(s,t) = \sum_{j=1}^\infty \lambda_j \, e_j(s) \, e_j(t), $$ where $(e_j)_{j \in \mathbb{N}}$ is an orthonormal basis of $L^2([0,1])$.
My question is: What can be said about the uniform norm of the $e_i$, i.e. I am looking for a bound of the form $$ \|e_i\|_{L_\infty(0,1)} \leq C(\lambda_i). $$
Is there anything in this direction known? Thanks!