  We study the regularity of a porous medium equation with nonlocal diffusion
effects given by an inverse fractional Laplacian operator. The precise model is
$u_t=\nabla\cdot(u\nabla (-\Delta)^{-1/2}u).$ For definiteness, the problem is
posed in $\{x\in\mathbb{R}^N, t\in \mathbb{R}\}$ with nonnegative initial data
$u(x,0)$ that are integrable and decay at infinity. Previous papers have
established the existence of mass-preserving, nonnegative weak solutions
satisfying energy estimates and finite propagation, as well as the boundedness
of nonnegative solutions with $L^1$ data, for the more general family of
equations $u_t=\nabla\cdot(u\nabla (-\Delta)^{-s}u)$, $0<s<1$.
  Here we establish the $C^\alpha$ regularity of such weak solutions in the
difficult fractional exponent case $s=1/2$. For the other fractional exponents
$s\in (0,1)$ this H\"older regularity has been proved in $[5]$. The method
combines delicate De Giorgi type estimates with iterated geometric corrections
that are needed to avoid the divergence of some essential energy integrals due
to fractional long-range effects.
