On a class of functions with the graph box dimension .
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On a decomposition of non-negative Radon measures
Bérenger Akon Kpata (2019)
Archivum Mathematicum
We establish a decomposition of non-negative Radon measures on which extends that obtained by Strichartz [6] in the setting of -dimensional measures. As consequences, we deduce some well-known properties concerning the density of non-negative Radon measures. Furthermore, some properties of non-negative Radon measures having their Riesz potential in a Lebesgue space are obtained.
On a theorem of Beardon and Maskit.
Bishop, Christopher J. (1996)
Annales Academiae Scientiarum Fennicae. Mathematica
On distribution functions of the derivatives of weakly differentiable mappings
Victor J. Mizel (1994)
Annales de l'I.H.P. Analyse non linéaire
On entropy and Hausdorff dimension of measures defined through a non-homogeneous Markov process
Athanasios Batakis (2006)
Colloquium Mathematicae
We study the Hausdorff dimension of measures whose weight distribution satisfies a Markov non-homogeneous property. We prove, in particular, that the Hausdorff dimensions of this kind of measures coincide with their lower Rényi dimensions (entropy). Moreover, we show that the packing dimensions equal the upper Rényi dimensions. As an application we get a continuity property of the Hausdorff dimension of the measures, when viewed as a function of the distributed weights under the norm.
On fractals which are not so terrible
António M. Caetano (2002)
Fundamenta Mathematicae
The notion of NST domain and the closely related notion of ball condition, both topological in nature and quite useful within the theory of function spaces, are compared with each other (and with the older concept of porosity) and also with other notions of interest, like those of d-set and of interior regular domain, which have a measure-theoretical nature. Also, after extending the idea of NST (not so terrible) to a larger class of sets, the property is studied in the context of anisotropic self-affine...
On Nikodym-type sets in high dimensions
Themis Mitsis (2004)
Studia Mathematica
We prove that the complement of a higher-dimensional Nikodym set must have full Hausdorff dimension.
On random fractals with infinite branching: definition, measurability, dimensions
Artemi Berlinkov (2013)
Annales de l'I.H.P. Probabilités et statistiques
We investigate the definition and measurability questions of random fractals with infinite branching, and find, under certain conditions, a formula for the upper and lower Minkowski dimensions. For the case of a random self-similar set we obtain the packing dimension.
On selfsimilar Jordan curves on the plane.
Aseev, V.V., Tetenov, A.V., Kravchenko, A.S. (2003)
Sibirskij Matematicheskij Zhurnal
On some dimension problems for self-affine fractals.
Bernardi, M.P., Bondioli, C. (1999)
Zeitschrift für Analysis und ihre Anwendungen
On some properties of Hausdorff content related to instability.
Mattila, Pertti, Orobitg, Joan (1994)
Annales Academiae Scientiarum Fennicae. Series A I. Mathematica
On Some Properties of Separately Increasing Functions from [0,1]ⁿ into a Banach Space
Artur Michalak (2014)
Bulletin of the Polish Academy of Sciences. Mathematics
We say that a function f from [0,1] to a Banach space X is increasing with respect to E ⊂ X* if x* ∘ f is increasing for every x* ∈ E. A function is separately increasing if it is increasing in each variable separately. We show that if X is a Banach space that does not contain any isomorphic copy of c₀ or such that X* is separable, then for every separately increasing function with respect to any norming subset there exists a separately increasing function such that the sets of points of discontinuity...
On s-sets in spaces of homogeneous type
Marilina Carena, Marisa Toschi (2015)
Colloquium Mathematicae
Let (X,d,μ) be a space of homogeneous type. We study the relationship between two types of s-sets: relative to a distance and relative to a measure. We find a condition on a closed subset F of X under which F is an s-set relative to the measure μ if and only if F is an s-set relative to δ. Here δ denotes the quasi-distance defined by Macías and Segovia such that (X,δ,μ) is a normal space. In order to prove this result, we prove a covering type lemma and a type of Hausdorff measure based criterion...
On subseries.
Tibor Salát (1964)
Mathematische Zeitschrift
On summability of measures with thin spectra
Maria Roginskaya, Michaël Wojciechowski (2004)
Annales de l’institut Fourier
We study different conditions on the set of roots of the Fourier transform of a measure on the Euclidean space, which yield that the measure is absolutely continuous with respect to the Lebesgue measure. We construct a monotone sequence in the real line with this property. We construct a closed subset of which contains a lot of lines of some fixed direction, with the property that every measure with spectrum contained in this set is absolutely continuous. We also give examples of sets with such property...
On the 1/2 Problem of Besicovitch: quasi-arcs do not contain sharp saw-teeth.
Hany M. Farag (2002)
Revista Matemática Iberoamericana
In this paper we give an alternative proof of our recent result that totally unrectifiable 1-sets which satisfy a measure-theoretic flatness condition at almost every point and sufficiently small scales, satisfy Besicovitch's 1/2-Conjecture which states that the lower spherical density for totally unrectifiable 1-sets should be bounded above by 1/2 at almost every point. This is in contrast to rectifiable 1-sets which actually possess a density equal to unity at almost every point. Our present method...
On the Absolute Continuity of a Surface Representation
Hans Martin Reimann (1971)
Commentarii mathematici Helvetici
On the analytic capacity and curvature of some Cantor sets with non-σ-finite length.
Pertti Mattila (1996)
Publicacions Matemàtiques
We show that if a Cantor set E as considered by Garnett in [G2] has positive Hausdorff h-measure for a non-decreasing function h satisfying ∫01 r−3 h(r)2 dr < ∞, then the analytic capacity of E is positive. Our tool will be the Menger three-point curvature and Melnikov’s identity relating it to the Cauchy kernel. We shall also prove some related more general results.
On the complexification of the Weierstrass non-differentiable function.
Barański, Krzysztof (2002)
Annales Academiae Scientiarum Fennicae. Mathematica
On the conformal gauge of a compact metric space
Matias Carrasco Piaggio (2013)
Annales scientifiques de l'École Normale Supérieure
In this article we study the Ahlfors regular conformal gauge of a compact metric space , and its conformal dimension . Using a sequence of finite coverings of , we construct distances in its Ahlfors regular conformal gauge of controlled Hausdorff dimension. We obtain in this way a combinatorial description, up to bi-Lipschitz homeomorphisms, of all the metrics in the gauge. We show how to compute using the critical exponent associated to the combinatorial modulus.
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