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We develop the regularity theory for solutions to space-time nonlocal equations driven by fractional powers of the heat operator $$(\partial_t-\Delta)^su(t,x)=f(t,x),\quad\hbox{for}~0<s<1.$$ This equation is of order $s$ in time and $2s$ in space. It plays for space-time nonlocal equations like the generalized master equation the same role as the fractional Laplacian for nonlocal in space equations. We obtain a pointwise integro-differential formula for $(\partial_t-\Delta)^su(t,x)$ as well as parabolic maximum principles. An extension problem to characterize this nonlocal equation with a local degenerate parabolic equation is proved. Then we show parabolic interior and boundary Harnack inequalities. H\"older and Schauder estimates for the space-time Poisson problem are deduced via a novel characterization of parabolic H\"older spaces. Some crucial tools in the analysis are the language of semigroups and the Cauchy Integral Theorem.

Regularity Theory and Extension Problem for Fractional Nonlocal Parabolic Equations and the Master Equation