I am working in the context of Gaussian white noise model (where we observe $n$ trajectories sampled via $d X(t)=f(t) d t+\sigma d W(t)$) and also in the non-parametric regression model (where we observe $n$ points, $X_i=f\left(x_i\right)+\sigma\varepsilon_i$, the $\varepsilon_i$ being i.i.d standard Gaussian variables)
I am wondering if, in those contexts, there exists non-trivial classes of functions where there exists a direct estimate of the sup norm of functions (i.e $\widehat{\|f\|_{\infty}}$) that achieves faster convergence rates (in expectation) than the sup norm of any estimator of the function (i.e $\|\hat{f}\|_{\infty}$), similarly to what we can get for the L2 norm for example.
References appreciated.