  We consider a smooth one-parameter family $t \to f_t$ of diffeomorphisms with
compact transitive Axiom A attractors. Our first result (corrected) is that for
any function $G$ in the Sobolev space $H^r_p$, with $p>1$ and $0<r<1/p$, the
map $R(t)$ sending $t$ to the average of $G$ with respect to the SRB measure of
$f_t$ is $\alpha$-H\"older continuous for all $\alpha <r- |log \mathcal
J|/(p|log \nu_s|$) where $\mathcal J\le 1$ is the strongest volume contraction
and $\nu_s<1$ is the weakest contraction. This applies to
$\theta(x)=h(x)\Theta(g(x)-a)$ (for all $\alpha <1- |log \mathcal J|/|log
\nu_s|$) for $h$ and $g$ smooth and $\Theta$ the Heaviside function, if $a$ is
not a critical value of $g$. Our second result says that for any such function
so that, in addition, the intersection of the set of points $x$ so that
$g(x)=a$ with the support of $h$ is foliated by "admissible stable leaves" of
$f_t$, the map $R(t)$ is differentiable. (We provide distributional linear
response and fluctuation-dissipation formulas for the derivative.) Obtaining
linear response or fractional response for such observables is motivated by
extreme-value theory. --- Second version, following the referee's comments: We
explain better the cone choices around (2.4). Appendix A contains information
on the Banach spaces. We added the paragraph containing (2.6) in the proof of
Theorem 2.1. In the proof of Theorem 3.3, we do not need to introduce
mollifiers. However, the new argument around (2.6) is not available here, so we
must replace the pair $(u-1, |s-1|)$ by $(u-2, |s-2|)$. This is why we now
assume that $h$ is $C^3$ and that $g$ and the foliations are $C^4$. --- Third
version: We have added a corrigendum modifying the first result (Theorem 2.1).
