  In this paper, we develop a functional analytical theory for establishing
that mild solutions of first-order Cauchy problems involving homogeneous
operators of order zero are strong solutions; in particular, the first-order
time derivative satisfies a global regularity estimate depending only on the
initial value and the positive time. We apply those results to the Cauchy
problem associated with the total variational flow operator and the nonlocal
fractional 1-Laplace operator.
