  We prove some regularity estimates for viscosity solutions to a class of
possible degenerate and singular integro-differential equations whose leading
operator switches between two different types of fractional elliptic phases,
according to the zero set of a modulating coefficient $a=a(\cdot,\cdot)$. The
model case is driven by the following nonlocal double phase operator, $$ \int
\!\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{n+sp}}\, {\rm d}y + \int
\!a(x,y)\frac{|u(x)-u(y)|^{q-2}(u(x)-u(y))}{|x-y|^{n+tq}}\, {\rm d}y, $$ where
$q\geq p$ and $a(\cdot,\cdot)\geqq 0$. Our results do also apply for
inhomogeneous equations, for very general classes of measurable kernels. By
simply assuming the boundedness of the modulating coefficient, we are able to
prove that the solutions are H\"older continuous, whereas similar sharp results
for the classical local case do require $a$ to be H\"older continuous.
  To our knowledge, this is the first (regularity) result for nonlocal double
phase problems.
