  For a scalar conservation law with strictly convex flux, by Oleinik's
estimates the total variation of a solution with initial data $\overline{u}\in
\bf{L}^\infty(\mathbb R)$ decays like $t^{-1}$. This paper introduces a class
of intermediate domains $\mathcal P_\alpha$, $0<\alpha<1$, such that for
$\overline u\in \mathcal P_\alpha$ a faster decay rate is achieved:
$\mathrm{Tot.Var.}\bigl\{ u(t,\cdot)\bigr\}\sim t^{\alpha-1}$. A key ingredient
of the analysis is a ``Fourier-type" decomposition of $\overline u$ into
components which oscillate more and more rapidly. The results aim at extending
the theory of fractional domains for analytic semigroups to an entirely
nonlinear setting.
