Sobolev inequalities in disguise

Abstract

We present a simple and direct proof of the equivalence of various functional inequalities such as Sobolev or Nash inequalities. This proof applies in the context of Riemannian or sub-elliptic geometry, as well as on graphs and to certain non-local Sobolev norms. It only uses elementary cut-off arguments. This method has interesting consequences concerning Trudinger type inequalities.

Geometric Implications of the Poincaré Inequality

“…Or we may replace the upper gradient by the local Lipschitz constant, see [14]. For more information about the Poincaré inequality, see also for example [2], [6] and [9]. Let Γ be a family of curves in X and let 1 ≤ p < ∞.…”

Hypercontractivity, Nash inequalities and subordination for classes of nonlinear semigroups

“…Moreover, generalizations of functional inequalities of Nash and Gagliardo-Nirenberg type have been considered in [2], and in particular it is shown there that, under suitable assumptions, the validity of a single Gagliardo-Nirenberg inequality implies the validity of a whole class of them.…”

Sobolev Inequalities in Familiar and Unfamiliar Settings

“…The paper [4] shows that a great number of other interesting Sobolev type inequalities follow as a corollary of the above result. …”

“…Many results in the spirit of these equivalences can be found in [75] in the context of Euclidean domains. A discussion in a very general setting is in [4] (see also [87,Chapt. 3]).…”

“…is how many different equivalent form it takes (hence the title "Sobolev inequalities in disguise" of [4]). Despite the equivalence of this different forms, some appear "stronger" than other.…”