Adams inequalities for Riesz subcritical potentials

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.