arXiv subjects

Efstathios Konstantinos Chrontsios Garitsis

Publications and source records attributed to Efstathios Konstantinos Chrontsios Garitsis.

15 recordsLinked to original sources

arXiv2026

A purely metric characterization of supercritical Sobolev spaces

In this paper we provide a purely metric and derivative-free characterization of continuous mappings in the local supercritical Sobolev space. More specifically, we show that a continuous mapping $f\colon\mathbb{R}^n\to\mathbb{R}^m$ lies in the local Sobolev space $W^{1,p}_\text{loc}(\mathbb{R}^n\colon\mathbb{R}^m)$ if and only if $f$ is a $(p,1-n/p)$-compactly Hölder mapping, where $p>n$. In fact, we prove the following more general result: whenever $X$ is a complete $Q$-Ahlfors regular metric space supporting a $p$-Poincaré inequality with $p>Q$, and $V$ is any Banach space, the compactly Hölder class $CH^{p,1-Q/p}(X\colon V)$ consists precisely of the continuous representatives of the local Hajłasz and Newtonian Sobolev spaces $M^{1,p}_\text{loc} (X \colon V)$ and $N^{1,p}_\text{loc}(X\colon V)$. We also establish quantitative comparisons between the corresponding seminorms.

math.FA↗

arXiv2026

On the Assouad dimension of Weierstrass function graphs

The class of Weierstrass functions $W_{a,b}$ is one of the first class of examples of continuous and nowhere differentiable real functions. A challenging line of research has been to determine the various dimensions of graphs $G(W_{a,b})$ of such functions. For instance, the Hausdorff dimension of $G(W_{a,b})$ was only recently determined by Shen in 2018, after a long series of partial results by many different authors. While the Assouad dimension of $G(W_{a,b})$ remains an open problem, also posed as a question by J. M. Fraser, there have been many indications that it might be equal to $2$. Such indications include the graph of Wiener processes, the graphs of almost all Hölder functions in the Baire category sense, and the graphs of Weierstrass functions after a series of countably many reflections all having Assouad dimension equal to $2$. In this paper we show that this is not the case, providing a quantitative upper bound on the Assouad dimension of $G(W_{a,b})$ that is strictly less than $2$. In particular, we show that such a bound is true for a class of generalized Weierstrass functions $W_{a,b}^ϕ(x) = \sum_{j=0}^\infty a^jϕ(b^j x)$, which includes $W_{a,b}$ and the class of Takagi functions. The latter fact is used to also answer in the negative a conjecture of H. Yu on the Assouad dimension of graphs of Takagi functions for parameters $a\in (0,1)$, $b\in (1/a, \infty)\cap \mathbb{Z}$.

math.DS↗

arXiv2026

Sobolev spaces on snowtrees

We introduce a discrete-energy Sobolev space $\mathcal{W}^{1,p}_{\mathscr V}(T)$ on Ahlfors regular snowtrees, a class of metric trees where every arc is a snowflake of the same type. Our main result shows that, for every partition $\mathscr V$ and every $1<p<\infty$, this discrete space coincides quantitatively with the Korevaar--Schoen space on $T$. This fact and the independence of the space on the particular partition used to define $\mathcal{W}^{1,p}_{\mathscr V}(T)$ are both novel even for the class of geodesic trees. We also determine the critical Korevaar-Schoen exponent for Ahlfors regular snowtrees and prove capacity attainment and upper estimates, which reveal the appropriate walk dimension needed for the corresponding probabilistic profile on these trees.

math.MG↗

arXiv2026

Universal quasiconformal trees

A quasiconformal tree is a doubling (compact) metric tree in which the diameter of each arc is comparable to the distance of its endpoints. We show that for each integer $n\geq 2$, the class of all quasiconformal trees with uniform branch separation and valence at most $n$, contains a quasisymmetrically ''universal'' element, that is, an element of this class into which every other element can be embedded quasisymmetrically. We also show that every quasiconformal tree with uniform branch separation quasisymmetrically embeds into $\mathbb{R}^2$. Our results answer two questions of Bonk and Meyer from 2022, in higher generality, and partially answer one question of Bonk and Meyer from 2020.

math.MG↗

arXiv2025

On the dimension distortion under fractionally smooth mappings

We determine the extent to which certain classes of fractionally `smooth' continuous mappings between metric spaces distort various dimensions, including the Hausdorff, upper Minkowski (box-counting), and upper intermediate dimensions. Our intermediate and Minkowski dimension distortion results are new even for continuous (fractional) Sobolev and, more generally, Triebel--Lizorkin and Besov mappings between Euclidean spaces, complementing the work of Hencl-Honzík (2015) and Huynh (2022). Moreover, our results also extend the aforementioned work, as well as the work of Kaufman (2000) and Fraser-Tyson (2025) to certain weighted Euclidean spaces and, more generally, to doubling metric measure spaces. As an application of our main result, we quantify the corresponding dimension distortion properties of quasisymmetric mappings for non-Ahlfors regular subsets of metric measure spaces, strengthening a result of Bishop-Hakobyan-Williams (2016).

math.CA↗

arXiv2025

Dimensions and metric dyadic cubes

In this note, we provide equivalent definitions for fractal geometric dimensions through dyadic cube constructions. Given a metric space $X$ with finite Assouad dimension, i.e., satisfying the doubling property, we show that the construction of systems of dyadic cubes by Hytönen-Kairema is compatible with many dimensions. In particular, the Hausdorff, Minkowski, and Assouad dimensions can be equivalently expressed solely using dyadic cubes in the aforementioned system. The same is true for the Assouad spectrum, a collection of dimensions introduced by Fraser-Yu.

math.MG↗

arXiv2025

Hölder spiral arcs

We establish a quantitative necessary and sufficient condition for a spiral arc to be a Hölder arc. The class of spiral arcs contains the polynomial spirals studied by Fraser, and the elliptical spirals studied by Burrell-Falconer-Fraser. As an application, we recover the sharp result on the Hölder winding problem for polynomial spirals. Moreover, we provide a sharp exponent estimate for the Hölder classification of polynomial spirals, which coincides with the corresponding quasiconformal classification estimate, and improve certain exponent bounds of Burrell-Falconer-Fraser on the Hölder classification of elliptical spirals.

math.CA↗

arXiv2025

On the Assouad spectrum of Hölder and Sobolev graphs

We provide upper bounds for the Assouad spectrum $\dim_A^θ(\text{Gr}(f))$ of the graph of a real-valued Hölder or Sobolev function $f$ defined on an interval $I \subset \mathbb{R}$. We demonstrate via examples that all of our bounds are sharp. In the setting of Hölder graphs, we further provide a geometric algorithm which takes as input the graph of an $α$-Hölder continuous function satisfying a matching lower oscillation condition with exponent $α$ and returns the graph of a new $α$-Hölder continuous function for which the Assouad $θ$-spectrum realizes the stated upper bound for all $θ\in (0,1)$. Examples of functions to which this algorithm applies include the continuous nowhere differentiable functions of Weierstrass and Takagi.

math.CA↗

arXiv2025

Hölder continuity and dimensions of fractal Fourier series

Motivated by applications in number theory, analysis, and fractal geometry, we consider regularity properties and dimensions of graphs associated with Fourier series of the form $F(t)=\sum_{n=1}^\infty f(n)e^{2πi nt}/n$, for a large class of coefficient functions $f$. Our main result states that if, for some constants $C$ and $α$ with $0<α<1$, we have $|\sum_{1\le n\le x}f(n)e^{2πi nt}|\le C x^α$ uniformly in $x\ge 1$ and $t\in \mathbb{R}$, then the series $F(t)$ is Hölder continuous with exponent $1-α$, and the graph of $|F(t)|$ on the interval $[0,1]$ has box-counting dimension $\leq 1+α$. As applications we recover the best-possible uniform Hölder exponents for the Weierstrass functions $\sum_{k=1}^\infty a^k\cos(2πb^k t)$ and the Riemann function $\sum_{n=1}^\infty \sin(πn^2 t)/n^2$. Moreoever, under the assumption of the Generalized Riemann Hypothesis, we obtain nontrivial bounds for Hölder exponents and dimensions associated with series of the form $\sum_{n=1}^\infty μ(n)e^{2πi n^kt}/n^k$, where $μ$ is the Möbius function.

math.CA↗

arXiv2025

On concentric fractal spheres and spiral shells

We investigate dimension-theoretic properties of concentric topological spheres, which are fractal sets emerging both in pure and applied mathematics. We calculate the box dimension and Assouad spectrum of such collections, and use them to prove that fractal spheres cannot be shrunk into a point at a polynomial rate. We also apply these dimension estimates to quasiconformally classify certain spiral shells, a generalization of planar spirals in higher dimensions. This classification also provides a bi-Hölder map between shells, and constitutes an addition to a general programme of research proposed by J. Fraser.

math.DS↗

arXiv2024

Quasiregular distortion of dimensions

We investigate the distortion of the Assouad dimension and (regularized) spectrum of sets under planar quasiregular maps. While the respective results for the Hausdorff and upper box-counting dimension follow immediately from their quasiconformal counterparts by employing elementary properties of these dimension notions (e.g. countable stability and Lipschitz stability), the Assouad dimension and spectrum do not share such properties. We obtain upper bounds on the Assouad dimension and spectrum of images of compact sets under planar quasiregular maps by studying their behavior around their critical points. As an application the invariance of porosity of compact subsets of the plane under quasiregular maps is established.

math.CV↗

arXiv2024

Minkowski weak embedding theorem

A well-known theorem of Assouad states that metric spaces satisfying the doubling property can be snowflaked and bi-Lipschitz embedded into Euclidean spaces. Due to the invariance of many geometric properties under bi-Lipschitz maps, this result greatly facilitates the study of such spaces. We prove a non-injective analog of this embedding theorem for spaces of finite Minkowski dimension. This allows for non-doubling spaces to be weakly embedded and studied in the usual Euclidean setting. Such spaces often arise in the context of random geometry and mathematical physics with the Brownian continuum tree and Liouville quantum gravity metrics being prominent examples.

math.MG↗

arXiv2024

Sobolev mappings on metric spaces and Minkowski dimension

We introduce the class of compactly Hölder mappings between metric spaces and determine the extent to which they distort the Minkowski dimension of a given set. These mappings are defined purely with metric notions and can be seen as a generalization of Sobolev mappings, without the requirement for a measure on the source space. In fact, we show that if $f:X\rightarrow Y$ is a continuous mapping lying in some super-critical Newtonian-Sobolev space $N^{1,p}(X,μ)$, under standard assumptions on the metric measure space $(X,d,μ)$, it is then a compactly Hölder mapping. The dimension distortion result we obtain is new even for Sobolev mappings between weighted Euclidean spaces and generalizes previous results of Kaufman and Bishop-Hakobyan-Williams.

math.DS↗

arXiv2024

A geometric property of quadrilaterals

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 $K$.

math.CV↗

arXiv2022

Quasiconformal distortion of the Assouad spectrum and classification of polynomial spirals

We investigate the distortion of Assouad dimension and the Assouad spectrum under Euclidean quasiconformal maps. Our results complement existing conclusions for Hausdorff and box-counting dimension due to Gehring--Väisälä and others. As an application, we classify polynomial spirals $S_a:=\{x^{-a}e^{\mathbf{i} x}:x>0\}$ up to quasiconformal equivalence, up to the level of the dilatation. Specifically, for $a>b>0$ we show that there exists a quasiconformal map $f$ of $\mathbb{C}$ with dilatation $K_f$ and $f(S_a)=S_b$ if and only if $K_f \ge \tfrac{a}{b}$.

math.CV↗