arXiv subjects

Antti Rasila

Publications and source records attributed to Antti Rasila.

At least 19 recordsLinked to original sources

arXiv2026

Hölder continuity and composition operators in pluriharmonic Bloch spaces

We show a Hölder estimate of order $1/n$ for pluriharmonic Bloch mappings in the unit ball $\mathbb{B}^n \subset \mathbb{C}^n$ with respect to the Bergman metric. We apply this to obtain a sufficient condition for the composition operator on the pluriharmonic Bloch space to be bounded below. As a partial converse, we also give a necessary condition for the boundedness below of the composition operator on the Bloch space of holomorphic mappings in $\mathbb{B}^n$.

math.CV↗

arXiv2026

Isoperimetric-type inequalities for pluriharmonic functions on the polydisc

We prove isoperimetric-type inequalities for complex-valued pluriharmonic functions in the unit polydisc $\mathbb{U}^n\subset\mathbb{C}^n$. Denote by $h^p(\mathbb{U}^n)$ and $b^p_{\mathbf q}(\mathbb{U}^n)$, respectively, the pluriharmonic Hardy space and the pluriharmonic weighted Bergman space in $\mathbb{U}^n$, where $\mathbf q=(q_1,\ldots,q_n)>-\mathbf1$. For $m\in\mathbb{N}$, $m\geq2$, write $\mathbf{m-2}=(m-2,\ldots,m-2)$ and let \[ dμ_{\mathbf{m-2}}(z) =\frac{(m-1)^n}{π^n} \prod_{k=1}^n \left[(1-|z_k|^2)^{m-2}\,dx_kdy_k\right], \qquad z_k=x_k+iy_k. \] We prove that if $1 2$, we refine this in the form \[ \|f\|_{b^{mp}_{\mathbf{m-2}}(\mathbb{U}^n)} \leq \left[ \sqrt2\cos\left(\fracπ{4m}\right) \right]^{2n/p} \|f\|_{h^p(\mathbb{U}^n)}. \] Consequently, the obtained constant in the diagonal inclusion tends to $1$ as $p\to\infty$, for fixed $m$ and $n$. When $m=2$ and $n=1$, the latter estimate coincides with best-known planar estimate. Explicit lower bounds at $p=2$, together with the dimension-free upper estimate, show that the optimal diagonal constants converge to $\sqrt2$ as $m\to\infty$, uniformly in the dimension.

math.CV↗

arXiv2026

Walk-on-Cubes Monte Carlo Simulation for nonisotropic fractional Laplace, Helmholtz, and Yukawa equations

We study the nonisotropic fractional analogs of Laplace, Helmholtz and Yukawa equations. We provide a Duffin correspondence for the Yukawa equation and a Feynman--Kac reconstruction for the Helmholtz equation. The foundation of our analysis is the fact that the nonisotropic fractional Laplace equation is related to a symmetric $α$-stable Lévy process with independent identically distributed components. By using this relation we provide a Walk-on-Cubes algorithm that simulates the solutions of Helmholtz and Yukawa equations.

math.NA↗

arXiv2026

Zygmund type spaces of harmonic functions on the real unit ball

In this paper, we obtain several characterizations of harmonic Zygmund type spaces on the real unit ball $\mathbb{B}$ of $\mathbb{R}^n$. First, we characterize the harmonic Zygmund space $\mathcal{HZ}^α$ ($0<α\leq2$) in terms of Zygmund type conditions. Our result extends Theorem 3.4 of [M. Aljuaid and F. Colonna, On the harmonic Zygmund spaces, Bull. Aust. Math. Soc. 101 (2020), 466--476.] to the setting of the real unit ball $\mathbb{B}$. Second, we establish an analogue of the Holland-Walsh characterization of the harmonic Zygmund space $\mathcal{HZ}$. Finally, an integral characterization of $\mathcal{HZ}^α$ is also discussed. All obtained results can be viewed as counterparts of the known results for Bloch spaces.

math.CV↗

arXiv2026

Shortest paths in planar domains with hyperbolic type metrics

We study planar domains $G$ equipped with a hyperbolic type metric and approximate geodesics that join two points $x,y \in G$ and their lengths. We present an algorithm that enables one to approximate the shortest distance in polygonal domains taken with respect to the quasihyperbolic metric. The method is based on Dijkstra's algorithm, and we give several examples demonstrating how the algorithm works and analyze its accuracy. We experimentally demonstrate several previously theoretically observed features of geodesics, such as the relationship between hyperbolic and quasihyperbolic distance in the unit disk. We also investigate bifurcation of geodesics and the connection of this phenomenon to the medial axis of the domain.

math.MG↗

arXiv2026

Properties for ($α,β$)-harmonic functions

We investigate properties of ($α,β$)-harmonic functions. First, we discuss the coefficient estimates for ($α,β$)-harmonic functions. In particular, we obtain Heinz's inequality for ($α,β$)-harmonic functions, propose a coefficient bound for normalized univalent ($α,β$)-harmonic functions and prove that this holds for the subclass that consists of starlike functions. Furthermore, by utilizing the relationship between ($α,β$)-harmonic functions and harmonic functions, we obtain Radó's theorem, Koebe type covering theorems and an area theorem. Finally, we show growth estimates and distortion estimates for ($α,β$)-harmonic functions by using the $L^p$ norms of the boundary functions.

math.CV↗

arXiv2026

Harmonic K-quasiconformal Koebe functions: construction and application to Pavlovic's problem

We first construct the harmonic K-quasiconformal Koebe functions, filling a long-standing foundational gap in geometric function theory. This construction provides a unified parametric candidate extremal function framework for conformal mappings, quasiconformal mappings, and harmonic mappings, and we formulate related conjectures for the extremal theory of harmonic K-quasiconformal mappings. By combining this construction with Astala and Koskela's Hp-theory for quasiconformal mappings, we establish a sharp result concerning the optimal order of harmonic K-quasiconformal mappings with bounded Schwarzian norm in harmonic Hardy spaces. Motivated by the work of Chuaqui, Hernandez, and Martin [Math. Ann. 367, 1099-1122, 2017], this result gives a partial solution to Pavlovic's 2014 open problem on the embeddings of harmonic quasiconformal mappings into Hardy spaces, and outlines a path toward its complete solution.

math.CV↗

arXiv2026

Applications of Reproducing Kernels in composition operators

In this paper, we illustrate the effectiveness of reproducing kernel Hilbert space techniques in the study of composition operators. For weighted Hardy spaces on the unit disk, we characterize the composition operators whose adjoint is again a composition operator. Using reproducing kernel methods, we obtain a classification of bounded weighted composition operators acting between reproducing kernel Hilbert spaces. We also show that the reproducing kernel techniques yield simpler proofs of several known results, highlighting the role of reproducing kernels as a unifying structural tool in the analysis of composition operators.

math.FA↗

arXiv2026

On Koebe-type functions for harmonic quasiconformal mappings

This paper studies a class of Koebe-type harmonic quasiconformal functions. It is motivated by the shear construction of Clunie and Sheil-Small [Ann. Acad. Sci. Fenn. Ser. A I Math. 9: 3--25, 1984] and the harmonic quasiconformal Koebe function. Equivalent univalence conditions, pre-Schwarzian and Schwarzian norms, coefficient inequalities, as well as growth and area theorems for this family of functions are established. These findings improve several previously known results.

math.CV↗

arXiv2025

Hardy spaces of harmonic quasiconformal mappings and Baernstein's theorem

Let $\mathcal{S}_H^0(K)$, $K\ge 1$, be the class of normalized $K$-quasiconformal harmonic mappings in the unit disk. We obtain Baernstein type extremal results for the analytic and co-analytic parts of functions in the geometric subclasses of $\mathcal{S}_H^0(K)$. We then apply these results to obtain integral means estimates for the respective classes. Furthermore, we find the range of $p>0$ such that these geometric classes of harmonic quasiconformal mappings are contained in the Hardy space $h^p$, thereby refining some earlier results of Nowak.

math.CV↗

arXiv2025

On the average scale-invariant Cassinian metric

We establish geometric relationships between the average scale-invariant Cassinian metric and other hyperbolic type metrics. In addition, we study the local convexity properties of the scale-invariant metric balls in Euclidean once punctured spaces.

math.MG↗

arXiv2025

On harmonic quasiregular mappings in Bergman spaces

A classical result of Hardy and Littlewood says that if $f=u+iv$ is analytic in the unit disk $\mathbb{D}$ and $u$ is in the harmonic Bergman space $a^p$ ($0 0$ such that every univalent harmonic function $f$ (and the partial derivatives $f_θ,\, rf_r$) is of class $a^p$. This result extends nicely to harmonic quasiconformal mappings in $\mathbb{D}$.

math.CV↗

arXiv2025

A class of linear operators on Bergman spaces

We study the boundedness of the linear operator $S$ on $L^{p}_{a}(dA_α)$ $(0<p<\infty)$. In particular, we obtain a sufficient and necessary condition for the compactness of the linear operator $S$ on $L^{p}_{a}(dA_α)$ $(1<p<\infty)$. Our results weaken the assumptions of earlier results of J. Miao and D. Zheng in a certain sense.

math.FA↗

arXiv2025

Note on real and imaginary parts of harmonic quasiregular mappings

If $f=u+iv$ is analytic in the unit disk $\mathbb{D}$, it is known that the integral means $M_p(r,u)$ and $M_p(r,v)$ have the same order of growth. This is false if $f$ is a (complex-valued) harmonic function. However, we prove that the same principle holds if we assume, in addition, that $f$ is $K$-quasiregular in $\mathbb{D}$. The case $0<p<1$ is particularly interesting, and is an extension of the recent Riesz type theorems for harmonic quasiregular mappings by several authors. Further, we proceed to show that the real and imaginary parts of a harmonic quasiregular mapping have the same degree of smoothness on the boundary.

math.CV↗

arXiv2025

Geometric characterizations of inner uniformity through Gromov hyperbolicity

In this paper, we study the characterization of inner uniformity of bounded domains $G$ in $\IR^n$, and prove that the following three conditions are equivalent: $(1)$ $G$ is inner uniform; $(2)$ $G$ is Gromov hyperbolic and its inner metric boundary is naturally quasisymmetrically equivalent to the Gromov boundary; $(3)$ $G$ is Gromov hyperbolic and linearly locally connected with respect to the inner metric. The equivalence between the conditions $(1)$ and $(2)$, and the implication from $(2)$ to $(3)$ affirmatively answer three questions raised by Bonk, Heinonen, and Koskela in 2001.

math.CV↗

arXiv2025

Zygmund's theorem for harmonic quasiregular mappings

Given an analytic function $f=u+iv$ in the unit disk $\mathbb{D}$, Zygmund's theorem gives the minimal growth restriction on $u$ which ensures that $v$ is in the Hardy space $h^1$. This need not be true if $f$ is a complex-valued harmonic function. However, we prove that Zygmund's theorem holds if $f$ is a harmonic $K$-quasiregular mapping in $\ID$. Our work makes further progress on the recent Riesz-type theorem of Liu and Zhu (Adv. Math., 2023), and the Kolmogorov-type theorem of Kalaj (J. Math. Anal. Appl., 2025), for harmonic quasiregular mappings. We also obtain a partial converse, thus showing that the proposed growth condition is the best possible. Furthermore, as an application of the classical conjugate function theorems, we establish a harmonic analogue of a well-known result of Hardy and Littlewood.

math.CV↗

arXiv2024

Positive solutions of the $\mathcal{A}$-Laplace equation with a potential

In this paper, we study positive solutions of the quasilinear elliptic equation $$Q'_{p,\mathcal{A},V}[u]\triangleq-\mathrm{div}{\mathcal{A}(x,\nabla u)}+V(x)|u|^{p-2}u=0,$$ in a domain $Ω\subseteq \mathbb{R}^n$, where $n\geq 2$, $1<p<\infty$, the divergence of $\mathcal{A}$ is the well known $\mathcal{A}$-Laplace operator considered in the influential book of Heinonen, Kilpeläinen, and Martio, and the potential $V$ belongs to a certain local Morrey space. The main aim of the paper is to extend criticality theory to the operator $Q'_{p,\mathcal{A},V}$. In particular, we prove an Agmon-Allegretto-Piepenbrink (AAP) type theorem, establish the uniqueness and simplicity of the principal eigenvalue of $Q'_{p,\mathcal{A},V}$ in a domain $ω\SubsetΩ$, and give various characterizations of criticality. Furthermore, we also study positive solutions of the equation $Q'_{p,\mathcal{A},V}[u]=0$ of minimal growth at infinity in $Ω$, the existence of a minimal positive Green function, and the minimal decay at infinity of Hardy-weights.

math.AP↗

arXiv2024

Efficient simulation of mixed boundary value problems and conformal mappings

In this paper, we present a stochastic method for the simulation of Laplace's equation with a mixed boundary condition in planar domains that are polygonal or bounded by circular arcs. We call this method the Reflected Walk-on-Spheres algorithm. The method combines a traditional Walk-on-Spheres algorithm with use of reflections at the Neumann boundaries. We apply our algorithm to simulate numerical conformal mappings from certain quadrilaterals to the corresponding canonical domains, and to compute their conformal moduli. Finally, we give examples of the method on three dimensional polyhedral domains, and use it to simulate the heat flow on an L-shaped insulated polyhedron.

math.NA↗