  The aim of this paper is to obtain new fine properties of entropy solutions
of nonlinear scalar conservation laws. For this purpose, we study some
"fractional $BV$ spaces" denoted $BV^s$, for $0 < s \leq 1$, introduced by Love
and Young in 1937. The $BV^s(\R)$ spaces are very closed to the critical
Sobolev space $W^{s,1/s}(\R)$. We investigate these spaces in relation with
one-dimensional scalar conservation laws. $BV^s$ spaces allow to work with less
regular functions than BV functions and appear to be more natural in this
context. We obtain a stability result for entropy solutions with $BV^s$ initial
data. Furthermore, for the first time we get the maximal $W^{s,p}$ smoothing
effect conjectured by P.-L. Lions, B. Perthame and E. Tadmor for all nonlinear
degenerate convex fluxes.
