e to prove results for $\|\hat f(t;h_{1,n},  h_{2,n})-f(t)\|_\infty$, however controlling the bias part of this error, $|E\hat f(t;h_{1,n},  h_{2,n})-f(t)|$, seems to require that $f(t)$ be bounded away from zero, so, we will consider instead the supremum of the estimation error on the ‘ideal’ regions $$\label{region0}
D_r=D_r(f):=\{t: f(t)>r, |t|<1/r\},\ \ r>0,$$ and will eventually replace $D_r$ by a region that depends on the data only and that can be made arbitrarily close to the positivity set of $f$. (We will not display the argument $f$ in $D_r(f)$ unless confusion is possible.) It is known (Hall, Hu and Marron (1995), Novak (1999)) that the bias reduction does hold for $f$ and $K$ four times differentiable and that then one has a bias of the order of $h_{2,n}^4$. This leads almost immediately in the case of the ideal estimator, and with some relatively hard work in the case of the real estimator, to an a.s. rate of convergence of $\hat f(t;h_{1,n},  h_{2,n})-f(t)$ (fixed $t$) of the order of $n^{-4/9}$ if we take $h_{2,n}\simeq n^{-1/9}$ and $h_{1,n}\simeq n^{-2/9}$ or of a smaller order, and this is best possible ($n^{-4/9}$ is the minimax rate in the $L_2(P)$ norm for estimating $f(t)$ four times differentiable with continuity). The minimax rate for the sup norm in this case is $(n/\log n)^{-4/9}$ and we show in this article that this rate is achieved by the estimator (\[realest0\]) uniformly in $D_r$ and in a similar data-dependent region. (See e.g. Efromovich (1999) for minimax rates.) Concretely, we prove the following theorem, in fact, as explained below, a [*uniform* ]{}version of it.

\[main0\] Assume the density $f$ and its first four derivatives are uniformly continuous and bounded, that the same is true for the kernel $K$, which, moreover is non-negative, has support contained in $[-T,T]$, $T<\infty$, integrates to 1 and is symmetric about zero. Set $h_{2,n}= ((\log n)/n)^{1/9}$ and $h_{1,n}=n^{-2/9}$ (or $h_{1,n}=n^{-(2+\eta)/9}$ for some $0\le\eta<1$), $n\in\mathbb N$. Then, for all $r>0$