  We study regularity properties of solutions to nonlinear and nonlocal
evolution problems driven by the so-called \emph{$0$-order fractional
$p-$Laplacian} type operators: $$ \partial_t u(x,t)=\mathcal{J}_p
u(x,t):=\int_{\mathbb{R}^n} J(x-y)|u(y,t)-u(x,t)|^{p-2}(u(y,t)-u(x,t))\,dy\,,
$$ where $n\ge 1$, $p>1$, $J\colon\mathbb{R}^n\to\mathbb{R}$ is a bounded
nonnegative function with compact support, $J(0)>0$ and normalized such that
$\|J\|_{\mathrm{L}^1(\mathbb{R}^n)}=1$, but not necessarily smooth. We deal
with Cauchy problems on the whole space, and with Dirichlet and Neumann
problems on bounded domains. Beside complementing the existing results about
existence and uniqueness theory, we focus on sharp regularity results in the
whole range $p\in (1,\infty)$. When $p>2$, we find an unexpected
$\mathrm{L}^q-\mathrm{L}^\infty$ regularization: the surprise comes from the
fact that this result is false in the linear case $p=2$.
  We show next that bounded solutions automatically gain higher time
regularity, more precisely that $u(x,\cdot)\in C^p_t$. We finally show that
solutions preserve the regularity of the initial datum up to certain order,
that we conjecture to be optimal ($p$-derivatives in space). When $p>1$ is
integer we can reach $C^\infty$ regularity (gained in time, preserved in space)
and even analyticity in time. The regularity estimates that we obtain are
quantitative and constructive (all computable constants), and have a local
character, allowing us to show further properties of the solutions: for
instance, initial singularities do not move with time. We also study the
asymptotic behavior for large times of solutions to Dirichlet and Neumann
problems. Our results are new also in the linear case and are sharp when $p$ is
integer. We expect them to be optimal for all $p>1$, supporting this claim with
some numerical simulations.
