  We establish a local boundedness estimate for weak subsolutions to a doubly
nonlinear parabolic fractional $p$-Laplace equation. Our argument relies on
energy estimates and a parabolic nonlocal version of De Giorgi's method.
Furthermore, by means of a new algebraic inequality, we show that positive weak
supersolutions satisfy a reverse H\"older inequality. Finally, we also prove a
logarithmic decay estimate for positive supersolutions.
