An estimate in the spirit of Poincaré's inequality

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Regularity estimates for elliptic nonlocal operators (2015, background): “hen there exists a constant c = c(d,D), which is independent of α, and an extension operator E : Hα/2(D) → Hα/2(Rd) with norm kEk ≤ c. Furthermore, we will need the following Poincaré inequality, cf. [28]. Fact 3.2 (Poincaré I). Let D ⊂ Rd be a bounded Lipschitz domain, and let 0 < α0 ≤ α < 2.”
On weighted Poincaré inequalities (2012, methodology): “(4) ˆ B1 |u(x)− uφB1 | φ(x) dx ≤ M ˆ 1”
On weighted Poincar\'e inequalities (2012, background): “Then for M = 8 |B1| |B1/2| Φ(0) Φ(1/2) (4) ˆ”
Characterization of Sobolev and BV Spaces (2011, background): “We finally remark that there is a large body of work on related non-local functionals, including papers addressing compactness (see [6], [33]), applications to problems and further questions (see [10], [11]), extended looks at non-radial mollifiers (see [32], [33]), Γ-convergence of non-local functi”
Regularity results and Harnack inequalities for minimizers and solutions of nonlocal problems: A unified approach via fractional De Giorgi classes (2016, background): “2] reveals, together with the sharp Poincaré-Wirtingertype estimate of [7, 53] and the extension inequality (4.”