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Consider nonparametric function estimation under $L^p$-loss. The minimax ratefor estimation of the regression function over a H\"older ball with smoothnessindex $\beta$ is $n^{-\beta/(2\beta+1)}$ if $1\leq p<\infty$ and $(n/\logn)^{-\beta/(2\beta+1)}$ if $p=\infty.$ There are many known procedures thateither attain this rate for $p=\infty$ but are suboptimal by a $\log n$ factorin the case $p<\infty$ or the other way around. In this article, we constructan estimator that simultaneously achieves the optimal rates under $L^p$-riskfor all $1\leq p\leq \infty$ without prior knowledge of $\beta.$ In contrast toclassical wavelet thresholding methods that kill small empirical waveletcoefficients and keep large ones, it is essential for simultaneous adaptationthat on each resolution level, the largest empirical wavelet coefficients aretruncated. This leads to a completely different point of view on waveletthresholding. The crucial part in the construction of the estimator is the sizeof 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 sufficientlyprecise for our purpose.

On an estimator achieving the adaptive rate in nonparametric regression under $L^p$-loss for all $1\leq p \leq \infty$