Weighted Hardy inequalities and the size of the boundary

Lehrbäck, Juha

doi:10.1007/s00229-008-0208-5

Cited by 32 publications

(59 citation statements)

References 13 publications

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“…holds whenever x ∈ E and 0 < r < d (E). This condition was used by Aikawa [1] in connection to the so-called quasiadditivity of capacity, which has subsequently turned out to be intimately related to Hardy inequalities, we refer to [22,23,25].…”

Section: Concepts Of Dimension and The P (S)-propertymentioning

confidence: 99%

“…Suppose that E = ∅ is a compact and porous set in R n . Let β < 0 and 1 ≤ p < q < np/(n − p) < ∞ be such that the Hardy-Sobolev inequality (23) holds for every f ∈ C ∞ 0 (R n ). Then…”

Section: Necessary Conditions For Global Inequalitiesmentioning

confidence: 99%

“…Such an inequality holds, for instance, in a Lipschitz domain G for 1 < p < ∞ if (and only if) and β < p − 1, as was shown by Nečas [30]. On the other hand, if, roughly speaking, ∂G contains an isolated part of dimension n − p + β, then the (p, β)-Hardy inequality can not be valid in G ⊂ R n ; we refer to [20,23].…”

Section: Introductionmentioning

confidence: 99%

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In between the inequalities of Sobolev and Hardy

Lehrbäck

Vähäkangas

2016

Journal of Functional Analysis

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Abstract. We establish both sufficient and necessary conditions for the validity of the so-called Hardy-Sobolev inequalities on open sets of the Euclidean space. These inequalities form a natural interpolating scale between the (weighted) Sobolev inequalities and the (weighted) Hardy inequalities. The Assouad dimension of the complement of the open set turns out to play an important role in both sufficient and necessary conditions.

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Section: Concepts Of Dimension and The P (S)-propertymentioning

confidence: 99%

Section: Necessary Conditions For Global Inequalitiesmentioning

confidence: 99%

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In between the inequalities of Sobolev and Hardy

Lehrbäck

Vähäkangas

2016

Journal of Functional Analysis

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“…holds if 1 < p < n and ∂G is 'thin', i.e., if dim A (∂G) < n − p. Indeed, as it is observed in [17], this result is implicitly contained in [16]. On the other hand, it is well known that inequality (1.1) holds if R n \ G is (1, p) uniformly fat with 1 < p ≤ n, we refer to [20].…”

Section: Introductionmentioning

confidence: 66%

Hardy inequalities in Triebel–Lizorkin spaces II. Aikawa dimension

Ihnatsyeva

Vähäkangas

2013

Annali di Matematica

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Abstract. We prove inequalities of Hardy type for functions in Triebel-Lizorkin spaces F s pq (G) on a domain G ⊂ R n , whose boundary has the Aikawa dimension strictly less than n − sp. IntroductionIn this paper, we study Hardy-type inequalities for functions in Triebel-Lizorkin spaces F s pq (G); see [26] for the case of bounded smooth domains G. We assume that the boundary ∂G of a domain G is 'thin', in the sense that its Aikawa dimension is strictly smaller than n − sp. The notion of the Aikawa dimension appears in connection with the quasiadditivity of Riesz capacity, [2]; subsequently, it has turned out to be useful in other questions in the theory of function spaces, see e.g. [10,23]. In particular, it is known that, for every f ∈ C ∞ 0 (G), a 'classical' Hardy inequalityholds if 1 < p < n and ∂G is 'thin', i.e., if dim A (∂G) < n − p. Indeed, as it is observed in [17], this result is implicitly contained in [16]. On the other hand, it is well known that inequality (1.1) holds if R n \ G is (1, p) uniformly fat with 1 < p ≤ n, we refer to [20]. These two last results exhibit a dichotomy between 'thin' and 'fat' sets which manifests in Hardy-type inequalities.Though our main result is Theorem 1.5, we also formulate and prove the following illustrative theorem under an additional assumption that G is a John domain.1.2. Theorem. Let n ≥ 2, 1 < p < ∞, and 0 < s < min{1, n/p}. Suppose that G is a Johnwhere a constant C depends on parameters n, s, p, and G.Recall that bounded Lipschitz domains, and bounded domains with the interior cone condition, are John domains. Also, the Koch snowflake G is a John domain with dim A (∂G) = log 4/ log 3. For bounded Lipschitz domains G in the 'fat' case of sp > 1, inequality (1.3) holds for every f ∈ C ∞ 0 (G) without the L p -term f L p (G) on the right hand side; furthermore, the L p -term cannot2000 Mathematics Subject Classification. 46E35, 26D15.

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“…For simplicity, we give the construction only for n D 2, but the essential ideas for higher dimensional examples are exactly the same as in the planar case. We would like to mention that a more general (and more detailed) treatment of domains having properties similar to those in the following construction is given in [8].…”

Section: The Case Q Pmentioning

confidence: 98%