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

Adams inequalities for Riesz subcritical potentials