A note on metric-measure spaces supporting Poincaré inequalities

Abstract

Dedicated to Professor Vladimir Maz'ya on the occasion of his 80th birthday.Abstract. Using a method of Korobenko, Maldonado and Rios we show a new characterization of doubling metric-measure spaces supporting Poincaré inequalities without assuming a priori that the measure is doubling.Non-smooth functions have played a key role in analysis since the nineteenth century. One fundamental development in this vein came with the introduction of Sobolev spaces, which turned out to be a key tool in studying nonlinear partial differential equations and calculus of variations. Although classically Sobolev functions themselves were not smooth, they were defined on smooth objects such as domains in the Euclidean space or, more generally, Riemannian manifolds. By the late 1970s it became well recognized that several results in real analysis required little structure from the underlying ambient space, and could be generalized to non-smooth settings, such as to the so-called spaces of homogeneous type. The latter spaces are (quasi)metric spaces equipped with a doubling Borel measure (see [9,10]). In fact, maximal functions, Hardy spaces, functions of bounded mean oscillation, and singular integrals of Calderón-Zygmund-type all continue to have a fruitful theory in the context of spaces of homogeneous type. However, this rich theory was, in a sense, only zeroth-order analysis given that no derivatives were involved. The study of first-order analysis with suitable generalizations of derivatives, a fundamental theorem of calculus, and Sobolev spaces, in the setting of spaces of homogeneous type, was initiated in the 1990s. This area, known as analysis on metric spaces, has since grown into a multifaceted theory which continues to play an important role in many areas of contemporary mathematics. For an introduction to the subject we recommend [2,3,4,5,8,12,13,19,20,23].One of the main objects of study in analysis on metric spaces are so called spaces supporting Poincaré inequalities introduced in [19]. To define this notion, recall that a metric-measure space (X, d, µ) is a metric space (X, d) with a Borel measure µ such that 0 < µ B(x, r) < ∞ for all x ∈ X and all r ∈ (0, ∞), where B(x, r) denotes the (open) metric ball with center x and radius r, i.e., B(x, r) := {y ∈ X : d(x, y) < r}. If the measure µ is doubling, that is, if there exists a finite constant C > 0 such that µ(2B) ≤ Cµ(B) for all balls B ⊆ X, then we call (X, d, µ) a doubling metric-measure space. The notation τ B denotes the dilation of a ball B by a factor τ ∈ (0, ∞), i.e., τ B := B(x, τ r). A Borel 2010 Mathematics Subject Classification. 30L99, 46E35.

Sobolev embedding implies regularity of measure in metric measure spaces

“…In this note, we prove the same result but without assuming the doubling condition, as conjectured in [3] and the proof of the same is inspired by [2] and [6].…”

Poincaré inequalities and $A_p$ weights on bow-ties

“…If q > p then the (q, p)-Poincaré inequality implies that µ is doubling, by Theorem 1 in Alvarado-Haj lasz [1]. Theorem 1.1 and Proposition 3.1 (and the fact that only connected spaces can support Poincaré inequalities) therefore give the following characterization of the (q, p)-Poincaré inequality, without presupposing that µ is doubling.…”

“…Let B = B(x 0 , r) be an arbitrary ball. If r ≤1 2 |x 0 |, then since w is approximately decreasing we get thatB w(x) dx w(|x 0 | − r) ≃ w(|x 0 | + r) ≃ ess inf B w.On the other hand, if r > 1 2 |x 0 |, then B ⊂ B 3r . Hence using Lemma 7.1 we get that α+n−1 φ(ρ) β dρ ≃ r α φ(3r) β .Moreover, ess infB w ≥ ess inf B3r w ≃ w(3r) ≃ r α φ(3r) β ,which shows that the A 1 -condition is satisfied for all balls.Using Proposition 7.2 we obtain the following result which is well known when β = 0, see Heinonen-Kilpeläinen-Martio[21, p. 10].…”

Admissibility versus Ap-Conditions on Regular Trees

“…Very recently, it was also shown in [5] that a measure on R is locally p-admissible if and only if it is given by a local Ap-weight. Moreover, on R n , p-admissible measures can be characterized by a stronger version of the Poincaré inequality, the (q, p)-Poincaré inequality with q > p. Under doubling, the ( , p)-Poincaré inequality improves to a (q, p)-Poincaré inequality with q > p by [10] and any measure satisfying (q, p)-Poincaré inequality with q > p is a doubling measure, see [1] and [17].…”

“…The following example from[1, Example 4] or[4, Example 6.2] gives a -regular tree with a nondoubling measure which satis es (4.10) for any < c < . Let X = (R+, dx, µ dµ) with µ(x) = min{ , x − }.…”