2018
DOI: 10.1007/s11118-017-9674-2
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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.

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Cited by 41 publications
(34 citation statements)
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“…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].…”
Section: Weighted Fractional Hardy-sobolev Inequalitiesmentioning
confidence: 99%
See 2 more Smart Citations
“…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].…”
Section: Weighted Fractional Hardy-sobolev Inequalitiesmentioning
confidence: 99%
“…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…”
Section: Weighted Fractional Hardy-sobolev Inequalitiesmentioning
confidence: 99%
See 1 more Smart Citation
“…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].…”
Section: Definitions and Preliminariesmentioning
confidence: 99%
“…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…”
Section: Definitions and Preliminariesmentioning
confidence: 99%