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Tomasz Adamowicz

Publications and source records attributed to Tomasz Adamowicz.

At least 19 recordsLinked to original sources
arXiv2026

Hardy spaces and quasiregular mappings: averaged derivatives and the $\mathbb{BMO}$ case

We study the Hardy spaces $\mathcal{H}^p$, $0<p<\infty$ of quasiregular mappings on the unit ball $\mathbb{B}^n$ in ${\mathbb{R}}^n$ under the appropriate growth and multiplicity conditions. Our focus is on the averaged derivatives of maps and their Harnack and quantitative Harnack estimates. The averaged derivatives are employed to study the non-tangential limit functions and non-tangential maximal functions of quasiregular mappings and to characterize $\mathcal{H}^p$ in the case of finite multiplicity of $f$. Moreover, we study relations between quasiregular mappings, averaged derivatives, BMO spaces and Carleson measures on $\mathbb{B}^n$ and the role of the multiplicity of a map. We also apply our results to the second order elliptic PDEs and $\mathcal{A}$-harmonic equations. Our paper extends results by Astala and Koskela [AK] and Nolder [No1] to the setting of quasiregular maps.

math.CV↗
arXiv2024

$\epsilon$-Approximability and Quantitative Fatou Property on Lipschitz-graph domains for a class of non-harmonic functions

We study the class of functions on Lipschitz-graph domains satisfying a differential-oscillation condition and show that such functions are $\epsilon$-approximable. As a consequence we obtain the quantitative Fatou theorem in the spirit of works e.g. by Garnett and Bortz-Hofmann. Such a class contains harmonic functions, as well as non-harmonic ones, for example nonnegative subharmonic functions, as illustrated by our discussion.

math.AP↗
arXiv2024

Carleson measures on domains in Heisenberg groups

We study the Carleson measures on NTA and ADP domains in the Heisenberg groups $\mathbb{H}^n$ and provide two characterizations of such measures: (1) in terms of the level sets of subelliptic harmonic functions and (2) via the $1$-quasiconformal family of mappings on the Kor\'anyi--Reimann unit ball. Moreover, we establish the $L^2$-bounds for the square function $S_{\alpha}$ of a subelliptic harmonic function and the Carleson measure estimates for the BMO boundary data, both on NTA domains in $\mathbb{H}^n$. Finally, we prove a Fatou-type theorem on $(\epsilon, \delta)$-domains in $\mathbb{H}^n$.

math.AP↗
arXiv2024

Harmonic Curves From Euclidean Domains to Heisenberg Group H1

We define and study the harmonic curves on domains in $\mathbb{R}^n$ into the first Heisenberg group $\mathbb{H}^1$. These are the $C^2$-regular mappings which are critical points of the second Dirichlet energy and satisfy the weak isotropicity condition. We investigate the geometry of such curves including the comparison and maximum principles, the Harnack inequalities, the Liouville theorems, the existence results, the Phragm\`en-Lindel\"of theorem, as well as the three spheres theorem.

math.AP↗
arXiv2023

Hardy spaces and quasiregular mappings

We study Hardy spaces $\mathcal{H}^p$, $0<p<\infty$ for quasiregular mappings on the unit ball $B$ in $\mathbb{R}^n$ which satisfy appropriate growth and multiplicity conditions. Under these conditions we recover several classical results for analytic functions and quasiconformal mappings in $\mathcal{H}^p$. In particular, we characterize $\mathcal{H}^p$ in terms of non-tangential limit functions and non-tangential maximal functions of quasiregular mappings. Among applications we show that every quasiregular map in our class belongs to $\mathcal{H}^p$ for some $p=p(n,K)$. Moreover, we provide characterization of Carleson measures on $B$ via integral inequalities for quasiregular mappings on $B$. We also discuss the Bergman spaces of quasiregular mappings and their relations to $\mathcal{H}^p$ spaces and analyze correspondence between results for $\mathcal{H}^p$ spaces and $\mathcal{A}$-harmonic functions. A key difference between the previously known results for quasiconformal mappings in $\mathbb{R}^n$ and our setting is the role of multiplicity conditions and the growth of mappings that need not be injective. Our paper extends results by Astala and Koskela, Jerison and Weitsman, Jones, Nolder, and Zinsmeister.

math.CV↗
arXiv2023

Isoperimetric inequalities and regularity of $A$-harmonic functions on surfaces

We investigate the logarithmic and power-type convexity of the length of the level curves for $a$-harmonic functions on smooth surfaces and related isoperimetric inequalities. In particular, our analysis covers the $p$-harmonic and the minimal surface equations. As an auxiliary result, we obtain higher Sobolev regularity properties of the solutions, including the $W^{2,2}$ regularity. The results are complemented by a number of estimates for the derivatives $L'$ and $L''$ of the length of the level curve function $L$, as well as by examples illustrating the presentation. Our work generalizes results due to Alessandrini, Longinetti, Talenti and Lewis in the Euclidean setting, as well as a recent article of ours devoted to the harmonic case on surfaces.

math.AP↗
arXiv2023

Asymptotically mean value harmonic functions in sub-Riemannian and RCD settings

We consider weakly and strongly asymptotically mean value harmonic (amv-harmonic) functions on subriemannian and RCD settings. We demonstrate that, in non-collapsed RCD-spaces with vanishing metric measure boundary, Cheeger harmonic functions are weakly amv-harmonic and that, in Carnot groups, weak amv-harmonicity equivalently characterizes harmonicity in the sense of the sub-Laplacian. In homogeneous Carnot groups of step $2$, we prove a Blaschke-Privaloff-Zaremba type theorem. Similar results are discussed in the settings of Riemannian manifolds and for Alexandrov surfaces.

math.DG↗
arXiv2022

The Liouville theorem for discrete symmetric averaging operators

We introduce averaging operators on lattices $\mathbb{Z}^d$ and study the Liouville property for functions satisfying mean value properties associated to such operators. This framework encloses discrete harmonic, $p$-harmonic, $\infty$-harmonic and the so-called game $p$-harmonic functions. Our approach provides an elementary alternative proof of the Liouville Theorem for positive $p$-harmonic functions on $\mathbb{Z}^d$.

math.AP↗
arXiv2022

Hardy spaces and quasiconformal maps in the Heisenberg group

We define Hardy spaces $H^p$, $0 0$ such that every $K$-quasiconformal map $f:B \to f(B) \subset \mathbb{H}^1$ belongs to $H^p$ for all $0<p<p_0(K)$. Second, we give two equivalent conditions for the $H^p$ membership of a quasiconformal map $f$, one in terms of the radial limits of $f$, and one using a nontangential maximal function of $f$. As an application, we characterize Carleson measures on $B$ via integral inequalities for quasiconformal mappings on $B$ and their radial limits. Our paper thus extends results by Astala and Koskela, Jerison and Weitsman, Nolder, and Zinsmeister, from $\mathbb{R}^n$ to $\mathbb{H}^1$. A crucial difference between the proofs in $\mathbb{R}^n$ and $\mathbb{H}^1$ is caused by the nonisotropic nature of the Korányi unit sphere with its two characteristic points.

math.CV↗
arXiv2021

Schwarzians on the Heisenberg group

We study various notions of the Schwarzian derivative for contact mappings in the Heisenberg group $\mathbb{H}_1$ and introduce two definitions: (1) the CR Schwarzian derivative based on the conformal connection approach studied by Osgood and Stowe and, recently, by Son; (2) the classical type Schwarzian refering to the well-known complex analytic definition. In particular, we take into consideration the effect of conformal rigidity and the limitations it imposes. Moreover, we study the kernels of both Schwarzians and the cocycle conditions. Our auxiliary results include a characterization of the contact conformal vector fields. Inspired by ideas of Chuaqui--Duren--Osgood, Hernández, Martín and Venegas, we introduce the Preschwarzian for mappings in $\mathbb{H}_1$. Furthermore, we study results in the theory of subelliptic PDEs for the horizontal Jacobian and related differential expressions for harmonic mappings and the gradient harmonic mappings, the latter notion introduced here in the setting of $\mathbb{H}_1$.

math.AP↗
arXiv2021

Isoperimetric inequalities and geometry of level curves of harmonic functions on smooth and singular surfaces

We investigate the logarithmic convexity of the length of the level curves for harmonic functions on surfaces and related isoperimetric type inequalities. The results deal with smooth surfaces, as well as with singular Alexandrov surfaces (also called surfaces with bounded integral curvature), a class which includes for instance surfaces with conical singularities and surfaces of CAT(0) type. Moreover, we study the geodesic curvature of the level curves and of the steepest descent for harmonic functions on surfaces with non-necessarily constant Gaussian curvature $K$. Such geodesic curvature functions turn out to satisfy certain Laplace-type equations and inequalities, from which we infer various maximum and minimum principles. The results are complemented by a number of growth estimates for the derivatives $L'$ and $L''$ of the length of the level curve function $L$, as well as by examples illustrating the presentation. Our work generalizes some results due to Alessandrini, Longinetti, Talenti, Ma--Zhang and Wang--Wang.

math.AP↗
arXiv2020

Hardy spaces for quasiregular mappings and composition operators

We define Hardy spaces $\mathcal{H}^p$ for quasiregular mappings in the plane, and show that for a particular class of these mappings many of the classical properties that hold in the classical setting of analytic mappings still hold. This particular class of quasiregular mappings can be characterised in terms of composition operators when the symbol is quasiconformal. Relations between Carleson measures and Hardy spaces play an important role in the discussion. This program was initiated and developed for Hardy spaces of quasiconformal mappings by Astala and Koskela in 2011 in their paper $\mathcal{H}^p$-theory for Quasiconformal Mappings.

math.CV↗
arXiv2020

Asymptotically mean value harmonic functions in doubling metric measure spaces

We consider functions with an asymptotic mean value property, known to characterize harmonicity in Riemannian manifolds, in doubling metric measure spaces. We show that the strongly amv-harmonic functions are H\"older continuous for any exponent below one. More generally, we define the class of functions with finite amv-norm and show that functions in this class belong to a fractional Hajlasz-Sobolev space and their blow-ups satisfy the mean-value property. Furthermore, in the weighted Euclidean setting we find an elliptic PDE satisfied by amv-harmonic functions.

math.AP↗
arXiv2019

A Koebe distortion theorem for quasiconformal mappings in the Heisenberg group

We prove a Koebe distortion theorem for the average derivative of a quasiconformal mapping between domains in the sub-Riemannian Heisenberg group $\mathbb{H}_1$. Several auxiliary properties of quasiconformal mappings between subdomains of $\mathbb{H}_1$ are proven, including distortion of balls estimates and local BMO-estimates for the logarithm of the Jacobian of a quasiconformal mapping. Applications of the Koebe theorem include diameter bounds for images of curves, comparison of integrals of the average derivative and the operator norm of the horizontal differential, as well as the study of quasiconformal densities and metrics in domains in $\mathbb{H}_1$. The theorems are discussed for the sub-Riemannian and the Korányi distances. This extends results due to Astala--Gehring, Astala--Koskela, Koskela and Bonk--Koskela--Rohde.

math.MG↗
arXiv2018

The prime end capacity of inaccessible prime ends, resolutivity, and the Kellogg property

Prime end boundaries $\partial_PΩ$ of domains $Ω$ are studied in the setting of complete doubling metric measure spaces supporting a $p$-Poincaré inequality. Notions of rectifiably (in)accessible- and (in)finitely far away prime ends are introduced and employed in classification of prime ends. We show that, for a given domain, the prime end capacity of the collection of all rectifiably inaccessible prime ends together will all non-singleton prime ends is zero. We show the resolutivity of continouous functions on $\partial_PΩ$ which are Lipschitz continuous with respect to the Mazurkiewicz metric when restricted to the collection $\partial_{SP}Ω$ of all accessible prime ends. Furthermore, bounded perturbations of such functions in $\partial_PΩ\setminus\partial_{SP}Ω$ yield the same Perron solution. In the final part of the paper, we demonstrate the (resolutive) Kellogg property with respect to the prime end boundary of bounded domains in the metric space. Notions given in this paper are illustrated by a number of examples.

math.MG↗
arXiv2018

The Rad\'o-Kneser-Choquet theorem for $p$-harmonic mappings between Riemannian surfaces

In the planar setting the Rad\'o-Kneser-Choquet theorem states that a harmonic map from the unit disk onto a Jordan domain bounded by a convex curve is a diffeomorphism provided that the boundary mapping is a homeomorphism. We prove the injectivity criterion of Rad\'o-Kneser-Choquet for $p$-harmonic mappings between Riemannian surfaces. In our proof of the injecticity criterion we approximate the $p$-harmonic map with auxiliary mappings that solve uniformly elliptic systems. We prove that each auxiliary mapping has a positive Jacobian by a homotopy argument. We keep the maps injective all the way through the homotopy with the help of the minimum principle for a certain subharmonic expression that is related to the Jacobian.

math.AP↗
arXiv2017

Mean value property and harmonicity on Carnot-Carathéodory groups

We study strongly harmonic functions in Carnot-Carathéodory groups defined via the mean value property with respect to the Lebesgue measure. For such functions we show their Sobolev regularity and smoothness. Moreover, we prove that strongly harmonic functions satisfy the sub-Laplace equation for the appropriate gauge norm and that the inclusion is sharp. We observe that spherical harmonic polynomials in $\mathbb{H}_1$ are both strongly harmonic and satisfy the sub-Laplace equation. Our presentation is illustrated by examples.

math.AP↗