On the axiomatic approach to Harnackʼs inequality in doubling quasi-metric spaces

A visual formalism for weights satisfying reverse inequalities

“…Then, it is proved that if for any given ball B ∈ B Ω and any scalar λ with λ − u > 0 in B we have λ − u ∈ K Ω , then u ∈ RC weak (p, ∞) for every p > 0. See, for instance [24,Section 3]. Therefore, putting these two results together, if u is a positive subsolution (Lu ≥ 0), then L(λ − u) = −Lu ≤ 0, so that λ − u ∈ K Ω and, consequently, u ∈ RC weak (p, ∞) for every p > 0.…”

“…Finally, by choosing the p in Figure 40 equal to the p 0 in Figure 41 and using the fact that u p 0 is doubling, since it is A 2 , so that dashed lines in Figure 40 Later on, L. Caffarelli's seminal work [4] on fully non-linear elliptic equations (see also [5,Section 4.2]) greatly enriched and simplified Krylov-Safonov's theory rendering it quite flexible and still manageable. This, in fact, paved the way for the axiomatizations of Krylov-Safonov's theory in doubling quasi-metric spaces carried out in [1,13,24]. All these axiomatic approaches involve, implicitly or explicitly, the so-called critical density and power-like decay properties.…”

“…The next key step in Krylov-Safonov's approach consists in showing that the class of positive supersolutions possesses the power-like decay property, see for instance Theorem 2.1.3 on page 36 of [19] and Lemma 4.6 on page 33 of [5]. Now, the fact that the class of supersolutions has the power-like decay property (and since it is closed under multiplication by positive constants) amounts to the existence of constants 0 < δ < 1 ≤ C, depending only on , N , Λ 2 /Λ 1 , and dimension n, such that every supersolution u satisfies the reverse inequality RC(−∞, δ, C) (see [24,Remark 2]); namely, In other words, u δ ∈ A 1 (Ω). Inequality (9.22) is usually referred to as the weak Harnack inequality for u and it is here illustrated in Figure 43.…”

Potential Theory Results for a Class of PDOs Admitting a Global Fundamental Solution

“…It is also well known that a natural framework for the Harnack inequality is that of doubling metric spaces (where a doubling condition and a Poincaré inequality are available): Aimar, Forzani and Toledano [3]; Barlow and Bass [5]; Di Fazio, Gutiérrez and Lanconelli [36]; Grigor'yan and Saloff-Coste [51]; Gutiérrez and Lanconelli [53]; Hebisch and Saloff-Coste [54]; Indratno, Maldonado and Silwal [57]; Kinnunen, Marola, Miranda and Paronetto [59]; Mohammed [64]; Saloff-Coste [70].…”

The Strong Maximum Principle and the Harnack inequality for a class of hypoelliptic non-Hörmander operators

“…Nowadays, it is known that the natural framework for Harnack-type theorems is the setting of doubling metric spaces: see e.g., Aimar, Forzani and Toledano [2]; Barlow and Bass [4]; Di Fazio, Gutiérrez and Lanconelli [19]; Grigor'yan and Saloff-Coste [34]; Gutiérrez and Lanconelli [36]; Hebisch and Saloff-Coste [37]; Indratno, Maldonado and Silwal [40]; Kinnunen, Marola, Miranda and Paronetto [43]; Mohammed [52]; Saloff-Coste [62]. In this framework it appears that the Harnack Inequality holds true whenever some axiomatic assumptions are satisfied (roughly speaking, a doubling condition and a Poincaré inequality): this fact has a strong theoretical impact, even if it is not always simple to verify whether these axiomatic assumptions are satisfied for a given PDO.…”