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

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