  Consider nonparametric function estimation under $L^p$-loss. The minimax rate
for estimation of the regression function over a H\"older ball with smoothness
index $\beta$ is $n^{-\beta/(2\beta+1)}$ if $1\leq p<\infty$ and $(n/\log
n)^{-\beta/(2\beta+1)}$ if $p=\infty.$ There are many known procedures that
either attain this rate for $p=\infty$ but are suboptimal by a $\log n$ factor
in the case $p<\infty$ or the other way around. In this article, we construct
an estimator that simultaneously achieves the optimal rates under $L^p$-risk
for all $1\leq p\leq \infty$ without prior knowledge of $\beta.$ In contrast to
classical wavelet thresholding methods that kill small empirical wavelet
coefficients and keep large ones, it is essential for simultaneous adaptation
that on each resolution level, the largest empirical wavelet coefficients are
truncated. This leads to a completely different point of view on wavelet
thresholding. The crucial part in the construction of the estimator is the size
of the truncation level which is linked to the unknown smoothness index.
Although estimation of the smoothness index is known to be a difficult task,
there is a data-driven choice of the truncation level that is sufficiently
precise for our purpose.
