  We prove local boundedness of variational solutions to the double phase
equation
  \begin{align*}
  \partial_t u +&
P.V.\int_{\mathbb{R}^N}\frac{|u(x,t)-u(y,t)|^{p-2}(u(x,t)-u(y,t))}{|x-y|^{N+ps}}\\
&+a(x,y)\frac{|u(x,t)-u(y,t)|^{q-2}(u(x,t)-u(y,t))}{|x-y|^{N+qs'}} \,dy = 0,
  \end{align*}
  under the restrictions $s,s'\in (0,1),\, 1 < p \leq q \leq p\,\frac{2s+N}{N}$
and the non-negative function $(x,y)\mapsto a(x,y)$ is assumed to be measurable
and bounded.
