A note on the dimensions of Assouad and Aikawa

Lehrbäck, Juha; Tuominen, Heli

doi:10.2969/jmsj/06520343

Cited by 37 publications

(54 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Our study continues and combines e.g. the works of [1,2,3,5,17,24,25,27,32,40]. A common theme to most of our results is that we explore means to obtain knowledge on the internal structure of a closed set E ⊂ X when only external information is available.…”

Section: Introductionmentioning

confidence: 80%

“…The first equation in (3.11) was also proven in [24]. On the other hand, it is not hard to give examples where µ is doubling and any given inequality in (3.8) is strict for a set E ⊂ X; compare to [24,Example 4.3].…”

Section: Lemma 34 If µ Is a Doubling Measure On X And Ementioning

confidence: 99%

“…A concept of dimension defined via integrals as in (3.7) was first used by Aikawa in [1] for subsets of R d (see also [2]). Thus, in [24], where the interest originates from such integral estimates, the lower Assouad codimension is called the Aikawa codimension. However, the resemblance between the conditions in (3.5) and (3.6) and the definitions of upper and lower Assouad dimensions justifies the present terminology; compare also (3.4) to Lemma 3.4 below.…”

Section: Dimensions In Metric Measure Spacesmentioning

confidence: 99%

See 2 more Smart Citations

Dimensions, Whitney covers, and tubular neighborhoods

Käenmäki¹,

Lehrbäck²,

Вуоринен³

2013

Indiana Univ. Math. J.

View full text Add to dashboard Cite

Abstract. Working in doubling metric spaces, we examine the connections between different dimensions, Whitney covers, and geometrical properties of tubular neighborhoods. In the Euclidean space, we relate these concepts to the behavior of the surface area of the boundaries of parallel sets. In particular, we give characterizations for the Minkowski and the spherical dimensions by means of the Whitney ball count.

show abstract

Section: Introductionmentioning

confidence: 80%

Section: Lemma 34 If µ Is a Doubling Measure On X And Ementioning

confidence: 99%

Section: Dimensions In Metric Measure Spacesmentioning

confidence: 99%

See 1 more Smart Citation

Dimensions, Whitney covers, and tubular neighborhoods

Käenmäki¹,

Lehrbäck²,

Вуоринен³

2013

Indiana Univ. Math. J.

View full text Add to dashboard Cite

show abstract

“…It has been recently shown in [18] that the dimensions of Assouad and Aikawa of a given set S ⊂ R n coincide.…”

Section: Remark a Set S Is Porous In Rmentioning

confidence: 99%

Hardy inequalities in Triebel–Lizorkin spaces II. Aikawa dimension

Ihnatsyeva

Vähäkangas

2013

Annali di Matematica

View full text Add to dashboard Cite

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.

show abstract

“…(1) the boundary ∂G of a John domain is porous in R n ; (2) the Assouad dimension of a porous set in R n is strictly less than n, [14]; (3) the Assouad dimension of ∂G coincides with the Aikawa dimension of ∂G; we refer to a recent paper [13]. Indeed, by (1)-(3), the Aikawa dimension of ∂G is strictly less than n, and inequality (2.4) follows.…”

Section: Notation and Preliminariesmentioning

confidence: 99%

Aspects of local-to-global results

Hurri-Syrjänen¹,

Marola²,

Vähäkangas³

2014

Bulletin of the London Mathematical Society

View full text Add to dashboard Cite

Abstract. We establish local to global results for a function space which is larger than the well known BMO space, and was also introduced by John and Nirenberg. IntroductionThe space of functions of bounded mean oscillation, abbreviated to BMO, is introduced by John and Nirenberg [12]. In the same paper, John and Nirenberg introduced a larger space of functions. As opposed to any BMO function, that has exponentially decaying distribution function, a function in this larger space is known to belong to a weak L p -space, [12, Lemma 3]; the inclusion being strict, see [1, Example 3.5]. We extend this weak-type inequality to the case of John domains. The equivalence of local and global BMO norms is a rather well-known result, due to Reimann and Rychener [17]. We obtain the corresponding local to global result for the mentioned larger space of functions.Let G be a proper open subset of R n , n ≥ 1. The following condition was introduced in [12]: Let f : G → R be a function in L 1 (G) and let us assume that there exists 1 < p < ∞ such thatwhere the supremum is taken over all partitions P(G) of G into cubes such that Q ⊂ G for each Q ∈ P(G), the interiors of these cubes are pairwise disjoint, and G = Q∈P(G) Q. We call such partitions admissible. It is shown in [12, Lemma 3] that a function satisfying (1.1), with G being a cube Q in R n , belongs to a weak L p (Q)-space. More precisely, there exists a positive constant C, depending only on n and p, so that for all f ∈ L 1 (Q),We refer to [7,19,1] for other proofs of this result.We mention papers [5,6,15,16] where a related discrete summability condition is studied, and a recent paper [2] where its relation to condition (1.1) is discussed. In [5], in particular, the authors prove a local to global result in connection with this discrete summability condition.

show abstract

A note on the dimensions of Assouad and Aikawa

Cited by 37 publications

References 11 publications

Dimensions, Whitney covers, and tubular neighborhoods

Dimensions, Whitney covers, and tubular neighborhoods

Hardy inequalities in Triebel–Lizorkin spaces II. Aikawa dimension

Aspects of local-to-global results

Contact Info

Product

Resources

About