  We consider Smoluchowski's equation with a homogeneous kernel of the form
$a(x,y) = x^\alpha y ^\beta + x^\beta y^\alpha$ with $-1 < \alpha \leq \beta <
1$ and $\lambda := \alpha + \beta \in (-1,1)$. We first show that self-similar
solutions of this equation are infinitely differentiable and prove sharp
results on the behavior of self-similar profiles at $y = 0$ in the case $\alpha
< 0$. We also give some partial uniqueness results for self-similar profiles:
in the case $\alpha = 0$ we prove that two profiles with the same mass and
moment of order $\lambda$ are necessarily equal, while in the case $\alpha < 0$
we prove that two profiles with the same moments of order $\alpha$ and $\beta$,
and which are asymptotic at $y=0$, are equal. Our methods include a new
representation of the coagulation operator, and estimates of its regularity
using derivatives of fractional order.
