Annales Fennici Mathematici

Assouad-type dimensions of overlapping self-affine sets

2024-01-11T10:27:37+02:00

We study the Assouad and quasi-Assoaud dimensions of dominated rectangular self-affine sets in the plane. In contrast to previous work on the dimension theory of self-affine sets, we assume that the sets satisfy certain separation conditions on the projection to the principal axis, but otherwise have arbitrary overlaps in the plane. We introduce and study regularity properties of a certain symbolic non-autonomous iterated function system corresponding to ``symbolic slices'' of the self-affine set. We then establish dimensional formulas for the self-affine sets in terms of the dimension of the projection along with the maximal dimension of slices orthogonal to the projection. Our results are new even in the case when the self-affine set satisfies the strong separation condition: in fact, as an application, we show that self-affine sets satisfying the strong separation condition can have distinct Assouad and quasi-Assouad dimensions, answering a question of the first named author.

Relative L^p-cohomology and application to Heintze groups

2024-01-31T10:34:30+02:00

We introduce the notion of relative \(L^p\)-cohomology as a quasi-isometry invariant defined for a Gromov-hyperbolic space and a point on its boundary at infinity and reproduce some basic properties of \(L^p\)-cohomology in this context. In the case of degree 1 we show a relation between the relative and the classical \(L^p\)-cohomology. As an application, we explicitly construct non-zero relative \(L^p\)-cohomology classes for a purely real Heintze group of the form \(\mathbb{R}^{n-1}\rtimes_\alpha\mathbb{R}\), which gives a way to prove that the eigenvalues of \(\alpha\), up to a scalar multiple, are invariant under quasi-isometries.

The range set of zero for harmonic mappings of the unit disk with sectorial boundary normalization

2024-02-04T19:25:15+02:00

Given a family \(\mathcal F\) of all complex-valued functions in a domain \(\Omega\subset\hat{\mathbb{C}}\), the authors introduce the range set \(RS_{\mathcal F}(A)\) of a set \(A\subset\Omega\) under the class in question, i.e. the set of all \(w\in\Bbb C\) such that \(w\in F(A)\) for a certain \(F\in\mathcal F\). Let \(T_1,T_2,T_3\) be closed arcs contained in the unit circle \(\Bbb T\) of the same length \(2\pi/3\) and covering \(\Bbb T\). The paper deals with the range set \(RS_{\mathcal F}(\{0\})\), where \(\mathcal F\) is the class of all complex-valued harmonic functions \(F\) of the unit disk \(\Bbb D\) into itself satisfying the following sectorial condition: For each \(k\in\{1,2,3\}\) and for almost every \(z\in T_k\) the radial limit \(F^*(z)\) of the function \(F\) at the point \(z\) belongs to the angular sector determined by the convex hull spanned by the origin and arc \(T_k\). In 2014 the authors proved that for any \(F\in\mathcal F\),

\(|F(z)|\le\frac{4}{3}-\frac{2}{\pi}\arctan\left(\frac{\sqrt{3}}{1+2|z|}\right), \quad z\in\Bbb D\),

by which \(|F(0)|\le 2/3\). This implies that \(RS_{\mathcal F}(\{0\})\) is a subset of the closed disk of radius 2/3 and centred at the origin. In the paper the range set \(RS_{\mathcal F}(\{0\})\) is precisely determined.

Exceptional projections of sets exhibiting almost dimension conservation

2024-02-14T10:38:59+02:00

We establish a packing dimension estimate on the exceptional sets of orthogonal projections of sets satisfying an almost dimension conservation law. In particular, the main result applies to homogeneous sets and to certain graph-directed sets. Connections are drawn to results of Rams and Orponen.

Parametrizability of infinitely generated attractors

2024-02-14T22:23:58+02:00

An infinite iterated function system (IIFS) is a countable collection of contraction maps on a compact metric space. In this paper we study the conditions under which the attractor of such a system admits a parameterization by a continuous or Hölder continuous map of the unit interval.

Minimal surfaces and the new main inequality

2024-03-01T12:23:14+02:00

We establish the new main inequality as a minimizing criterion for minimal maps into products of \(\mathbb{R}\)-trees, and the infinitesimal new main inequality as a stability criterion for minimal maps to \(\mathbb{R}^n\). Along the way, we develop a new perspective on destabilizing minimal surfaces in \(\mathbb{R}^n\), and as a consequence we reprove the instability of some classical minimal surfaces; for example, the Enneper surface.

Speeds of convergence for petals of semigroups of holomorphic functions

2024-03-01T15:32:20+02:00

We study the backward dynamics of one-parameter semigroups of holomorphic self-maps of the unit disk. More specifically, we introduce the speeds of convergence for petals of the semigroup, namely the total, orthogonal, and tangential speeds. These are analogous to speeds of convergence introduced by Bracci, yet profoundly different due to the nature of backward dynamics. Results are extracted on the asymptotic behavior of speeds of petals, depending on the type of the petal. We further discuss the asymptotic behavior of the hyperbolic distance along non-regular backward orbits.

Sobolev, BV and perimeter extensions in metric measure spaces

2024-03-12T14:19:44+02:00

We study extensions of sets and functions in general metric measure spaces. We show that an open set has the strong BV-extension property if and only if it has the strong extension property for sets of finite perimeter. We also prove several implications between the strong BV-extension property and extendability of two different non-equivalent versions of Sobolev \(W^{1,1}\)-spaces and show via examples that the remaining implications fail.

Oversampling and Donoho–Logan type theorems in model spaces

2024-03-14T13:12:57+02:00

The aim of this paper is to extend two results from the Paley–Wiener setting to more general model spaces. The first one is an analogue of the oversampling Shannon sampling formula. The second one is a version of the Donoho–Logan Large Sieve Theorem which is a quantitative estimate of the embedding of the Paley–Wiener space into an \(L^2(\mathbb{R},\mu)\) space.

On the Hausdorff dimension of radial slices

2024-03-14T13:46:53+02:00

Let \(t \in (1,2)\), and let \(B \subset \mathbb{R}^{2}\) be a Borel set with \(\dim_{\mathrm{H}} B > t\). I show that


\(\mathcal{H}^{1}(\{e \in S^{1} \colon \dim_{\mathrm{H}} (B \cap \ell_{x,e}) \geq t - 1\}) > 0\)

for all \(x \in \mathbb{R}^{2} \setminus E\), where \(\dim_{\mathrm{H}} E \leq 2 - t\). This is the sharp bound for \(\dim_{\mathrm{H}} E\). The main technical tool is an incidence inequality of the form


\(\mathcal{I}_{\delta}(\mu,\nu) \lesssim_{t} \delta \cdot \sqrt{I_{t}(\mu)I_{3 - t}(\nu)}\), \(t \in (1,2)\),

where \(\mu\) is a Borel measure on \(\mathbb{R}^{2}\), and \(\nu\) is a Borel measure on the set of lines in \(\mathbb{R}^{2}\), and \(\mathcal{I}_{\delta}(\mu,\nu)\) measures the \(\delta\)-incidences between \(\mu\) and the lines parametrised by \(\nu\). This inequality can be viewed as a \(\delta^{-\epsilon}\)-free version of a recent incidence theorem due to Fu and Ren. The proof in this paper avoids the high-low method, and the induction-on-scales scheme responsible for the \(\delta^{-\epsilon}\)-factor in Fu and Ren's work. Instead, the inequality is deduced from the classical smoothing properties of the \(X\)-ray transform.

On the (1/2,+)-caloric capacity of Cantor sets

2024-03-22T11:22:59+02:00

In the present paper we characterize the (1/2,+)-caloric capacity (associated with the 1/2-fractional heat equation) of the usual corner-like Cantor set of \(\mathbb{R}^{n+1}\). The results obtained for the latter are analogous to those found for Newtonian capacity. Moreover, we also characterize the BMO and Lip\(_\alpha\) variants (\(0<\alpha<1\)) of the 1/2-caloric capacity in terms of the Hausdorff contents \(H^n_\infty\) and \(H^{n+\alpha}_\infty\) respectively.

Mean value formulas on surfaces in Grushin spaces

2024-03-28T11:43:37+02:00

We prove (sub)mean value formulas at the point \(0\in\Sigma\) for (sub)harmonic functions on a hypersurface \(\Sigma\subset\mathbb{R}^{n+1}\) where the differentiable structure and the surface measure depend on the ambient Grushin structure.

Function theory off the complexified unit circle: Fréchet space structure and automorphisms

2024-04-10T10:04:59+03:00

Motivated by recent work on strict deformation quantization of the unit disk and the Riemann sphere, we study the Fréchet space structure of the set of holomorphic functions on the complement \(\Omega:=\{(z,w)\in \hat{\mathbb{C}}^2\colon z\cdot w\not=1\}\) of the complexified unit circle \(\{(z,w) \in \hat{\mathbb{C}}^2 \colon z\cdot w=1\}\). We also characterize the subgroup of all biholomorphic automorphisms of \(\Omega\) which leave the canonical Laplacian on \(\Omega\) invariant.

A geometric property of quadrilaterals

2024-04-16T09:46:23+03:00

Quadrilaterals in the complex plane play a significant part in the theory of planar quasiconformal mappings. Motivated by the geometric definition of quasiconformality, we prove that every quadrilateral with modulus in an interval \([1/K, K]\), where \(K>1\), contains a disk lying in its interior, of radius depending only on the internal distances between the pairs of opposite sides of the quadrilateral and on \(K\).