Equivalence of critical and subcritical sharp Trudinger–Moser–Adams inequalities

Abstract

Sharp Trudinger–Moser inequalities on the first order Sobolev spaces and their analogous Adams inequalities on high order Sobolev spaces play an important role in geometric analysis, partial differential equations and other branches of modern mathematics. Such geometric inequalities have been studied extensively by many authors in recent years and there is a vast literature. There are two types of such optimal inequalities: critical and subcritical sharp inequalities, both are with best constants. Critical sharp inequalities are under the restriction of the full Sobolev norms for the functions under consideration, while the subcritical inequalities are under the restriction of the partial Sobolev norms for the functions under consideration. There are subtle differences between these two type of inequalities. Surprisingly, we prove in this paper that these critical and subcritical Trudinger–Moser and Adams inequalities are actually equivalent. Moreover, we also establish the asymptotic behavior of the supremum for the subcritical Trudinger–Moser and Adams inequalities on the entire Euclidean spaces, and provide a precise relationship between the suprema for the critical and subcritical Trudinger–Moser and Adams inequalities. This relationship of supremum is useful in establishing the existence and nonexistence of extremal functions for the Trudinger–Moser inequalities. Since the critical Trudinger–Moser and Adams inequalities can be easier to prove than subcritical ones in some occasions, and more difficult to establish in other occasions, our results and the method suggest a new approach to both the critical and subcritical Trudinger–Moser and Adams type inequalities.