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We derive sharp Adams inequalities for the Riesz and other potentials offunctions with arbitrary compact support in R^n. Up to now such results wereonly known for a class of functions whose supports have uniformly boundedmeasure. We obtain several sharp Moser-Trudinger inequalities for the criticalSobolev space W^{d,n/d} on R^n and on the hyperbolic space H^n The only knownresults so far are for d= 1, both on R^n and H^n, and for d = 2 on R^n. Othersharp inequalities are obtained for general elliptic operators with constantcoefficients and for trace type Borel measures. We introduce critical potentialspaces on which our results can be extended to noninteger values of d.

Sharp Adams and Moser-Trudinger inequalities on R^n and other spaces of infinite measure