  We derive Adams inequalities for potentials on general measure spaces,
extending and improving previous results obtained by the authors. The integral
operators involved, which we call "Riesz subcritical", have kernels whose
decreasing rearrangements are not worse than that of the Riesz kernel on
${\mathbb R}^n$, where the kernel is large, but they behave better where the
kernel is small. The new element is a "critical integrability" condition on the
kernel at infinity. Typical examples of such kernels are fundamental solutions
of nonhomogeneous differential, or pseudo-differential, operators. Another
example is the Riesz kernel itself restricted to suitable measurable sets,
which we name "Riesz subcritical domains". Such domains are characterized in
terms of their growth at infinity. As a consequence of the general results we
obtain several new sharp Adams and Moser-Trudinger inequalities on ${\mathbb
R}^n$, on the hyperbolic space, on Riesz subcritical domains, and on domains
where the Poincar\'e inequality holds.
