In this paper we consider a smooth bounded domain Ω⊂RN and a
parametric family of radially symmetric kernels Kϵ:RN→R+
such that, for each ϵ∈(0,1), its L1−norm is finite but it blows
up as ϵ→0. Our aim is to establish an ϵ independent
modulus of continuity in Ω, for the solution uϵ of the
homogeneous Dirichlet problem \begin{equation*} \left \{ \begin{array}{rcll} -
\I_\epsilon [u] \&=\& f \& \mbox{in} \ \Omega. \\ u \&=\& 0 \& \mbox{in} \
\Omega^c, \end{array} \right . \end{equation*} where f∈C(Ωˉ)
and the operator \I_\epsilon has the form \begin{equation*} \I_\epsilon[u](x)
= \frac12\int \limits_{\R^N} [u(x + z) + u(x - z) - 2u(x)]K_\epsilon(z)dz
\end{equation*} and it approaches the fractional Laplacian as ϵ→0.
The modulus of continuity is obtained combining the comparison principle with
the translation invariance of \I_\epsilon, constructing suitable barriers
that allow to manage the discontinuities that the solution uϵ may
have on ∂Ω. Extensions of this result to fully non-linear
elliptic and parabolic operators are also discussed