Muckenhoupt Ap-properties of Distance Functions and Applications to Hardy–Sobolev -type Inequalities

Abstract

Abstract. Let X be a metric space equipped with a doubling measure. We consider weights w(x) = dist(x, E) −α , where E is a closed set in X and α ∈ R. We establish sharp conditions, based on the Assouad (co)dimension of E, for the inclusion of w in Muckenhoupt's A p classes of weights, 1 ≤ p < ∞. With the help of general A p -weighted embedding results, we then prove (global) Hardy-Sobolev inequalities and also fractional versions of such inequalities in the setting of metric spaces.

Fractional Hardy–Sobolev type inequalities for half spaces and John domains

“…Next we apply [3, Theorem 4.1], which yields two weight inequalities for the Riesz potentials, where the weights are powers of the distance function δ ∂D . In [3] the result is formulated in a general metric space, but it is straightforward to see that in R n the dimensional condition in [3, Theorem 4.1] coincides with (6). We remark that the proof of [3, Theorem 4.1] is based on the Muckenhoupt A p -properties of the powers of δ ∂D and general A p -weighted inequalities; the Euclidean space versions of the latter are originally due to Pérez [15].…”

“…We remark that the proof of [3, Theorem 4.1] is based on the Muckenhoupt A p -properties of the powers of δ ∂D and general A p -weighted inequalities; the Euclidean space versions of the latter are originally due to Pérez [15]. From [3,Theorem 4.1] it follows that…”

“…for functions u ∈ C ∞ 0 (D), where 0 < τ < 1 and D is an unbounded John domain satisfying the additional assumption that the Assouad dimension of the boundary ∂D is small enough; we use here the notation δ ∂D (x) = dist(x, ∂D). The proof of inequality (2) is based on the Riesz potential estimate from Section 2 and general two weight inequalities for Riesz potentials from [3]. An important feature in inequality (2) is that, due to the parameter 0 < τ < 1, the inner integral in the left-hand side is taken over a ball which is not too close to the boundary ∂D.…”

Traces of weighted function spaces: Dyadic norms and Whitney extensions

“…which is of the same form as the quantity (1) in the introduction. (8) and (9) in the setting of metric measure spaces were also considered in [43,Definition 5.1] in terms of a hyperbolic filling of R d . Another similar variant in the weighted Euclidean setting has been considered in [54].…”

“…First, it is well-known that for α > −1, the weight w α belongs to the Muckenhoupt class A r for all r > max(α + 1, 1), which implies that the measure µ α satisfies the doubling property with respect to the standard Euclidean metric (see e.g. [21,Chapter 15] or [9]). This in particular means that…”