We consider observation of $N$ independent diffusion processes on a time interval $[0, 2T]$. Given a fixed time $t$, we construct a nonparametric estimator of $F_t (x) := \int_{-\infty}^{\infty} g(y) p_{t} (x, y) \, dy$, a transform of the transition density $p_t(x, y)$ with a known, real map $g$. The estimator is defined as the minimizer of a least-squares regression contrast over a finite-dimensional subspace of $\mathbb{L}^2(A, dx)$. We prove risk bounds in reference norms under various assumptions on $g$ and the estimation interval and study the bias- and variance terms to assess the rate of convergence in the standard $\mathbb{L}^2(A)$-norm. For this, we assume $F_t$ lies in a given regularity space, specifically the Sobolev(-Hermite) ellipsoids. We propose a model selection procedure and show that the least-square estimator achieves the bias-variance compromise. Estimating objects of shape $F_t$ has applications in financial mathematics. We conclude by illustrating the performance of our estimator for various diffusion processes and candidates for $g$, and finally Apply it to an option pricing scenario.