  We show a strong maximum principle and an Alexandrov-Bakelman-Pucci estimate
for the weak solutions of a Cauchy problem featuring Caputo time-derivatives
and non-local operators in space variables given in terms of Bernstein
functions of the Laplacian. To achieve this, first we propose a suitable
meaning of a weak solution, show their existence and uniqueness, and establish
a probabilistic representation in terms of time-changed Brownian motion. As an
application, we also discuss an inverse source problem.
