We consider the problem of estimating an unknown function $f$ in the heteroscedastic white noise setting under $\mathbb{L}^p$ risk. We show that the major connection which exists between Muckenhoupt theory and the geometrical properties of warped wavelet bases $\{\psi_{j,k}(G)\}$ allows us to consider spaces over which the minimax rate is stable for a wide class of variance functions $v$, contrarily to the usual wavelet approach. Adopting the maxiset point of view, we show that the hard thresholding procedure constructed on such a warped wavelet basis is close to the optimal over weighted Besov classes.