Representation formulas and weighted Poincaré inequalities for Hörmander vector fields

Sobolev spaces on metric-measure spaces

“…27 Because L p , 1 < p < ∞ is reflexive. 28 More generally, weak convergence xn x in a Banach space implies x ≤ lim infn→∞ xn . Indeed, for x * ∈ X * with x * = 1 we have x * (x) = limn→∞ x * (xn) ≤ lim infn→∞ xn and taking supremum over x * yields the claim (Hahn-Banach).…”

Subelliptic p -harmonic maps into spheres and the ghost of Hardy spaces

“…The statement of Theorem 2.6 below can be found for example in Franchi, Lu, and Wheeden [27], or in Capogna, Danielli, and Garofalo [8]. Various versions of this theorem appear also in Biroli …”

“…|Xu| , which holds for all x ∈ Ω 1 and all r ≤ 2R 0 , implies that u # 1−α,2R (x ) ≤ c M α,2R |Xu|(x ) for R ≤ R 0 . Then Lemma 3.3 gives (27). Now, Hölder inequality, the decay estimate (25), and the definition of the homogeneous dimension imply that for α = 1 − γ Q and any x ∈ Ω 1 we have …”

Weighted Trudinger-type inequalities

“…Weighted Poincaré-type inequalities have been extensively studied-see, for example, [6], [9], [11], [12], [13], [14], [15], [20], [21], [23]-but limiting cases of these inequalities have received scant attention (we know only of [4] and [26], both of which consider only special families of weights). Here, we shall investigate Trudinger-type inequalities for rather general pairs of weights (or measures) on fairly general domains.…”