We consider the problem of estimating an unknown function $f$ in the heteroscedastic white noise setting under $\mathbb{L}^p$ risk. We exhibit the minimax rate of convergence over usual Besov spaces and over weighted Besov spaces for a wide class of variance functions. We show via the maxiset approach that the natural hard thresholding procedure constructed on warped wavelet bases is close to the optimal over weighted Besov spaces.