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Marijan Markovic

Publications and source records attributed to Marijan Markovic.

At least 19 recordsLinked to original sources
arXiv2025

The distance function and Lipschitz classes of mappings between metric spaces

We investigate when the local Lipschitz property of the real-valued function $g(z) = d_Y (f(z),A)$ implies the global Lipschitz property of the mapping $f:X\to Y$ between the metric spaces $(X,d_X)$ and $(Y,d_Y)$. Here, $d_Y(y,A)$ denotes the distance of $y\in Y$ from the non-empty set $\subseteq Y$. As a consequence, we find that an analytic function on a uniform domain of a normed space belongs to the Lipschitz class if and only if its modulus satisfies the same condition; in the case of the unit disk this result is proved by K. Dyakonov. We use the recently established version of a classical theorem by Hardy and Littlewood for mappings between metric spaces. This paper is a continuation of the recent article by the author [14].

math.CV↗
arXiv2025

A condition equivalent to the H\"{o}lder continuity of harmonic functions on unbounded Lipschitz domains

Our main result concerns the behavior of bounded harmonic functions on a domain in $\mathbb{R}^N$ which may be represented as a strict epigraph of a Lipschitz function on $\mathbb{R}^{N-1}$. Generally speaking, the result says that the H\"{o}lder continuity of a harmonic function on such a domain is equivalent to the uniform H\"{o}lder continuity along the straight lines determined by the vector $\mathbf{e}_N$, where $\mathbf{e}_1,\mathbf{e}_2,\dots,\mathbf {e}_N$ is the base of standard vectors in $\mathbb{R}^N$. More precisely, let $\Psi$ be a Lipschitz function on $\mathbb {R}^{N-1}$, and $U$ be a real-valued bounded harmonic function on $E_\Psi=\{(x',x_N): x'\in\mathbb{R}^{N-1}, x_N>\Psi(x')\}$. We show that for $\alpha\in(0,1)$ the following two conditions on $U$ are equivalent: (a) There exists a constant $C$ such that \begin{equation*} | U(x',x_N) - U(x',y_N)|\le C |x_N - y_N|^\alpha,\quad x'\in \mathbb {R}^{N-1}, x_N, y_N > \Psi (x'); \end{equation*} (b) There exists a constant $\tilde {C}$ such that \begin{equation*} |U(x) - U (y)|\le \tilde{C} |x-y|^\alpha,\quad x, y\in E_\Psi. \end{equation*} Moreover, the constant $\tilde {C}$ depends linearly on $C$. The result holds as well for vector-valued harmonic functions and, therefore, for analytic mappings.

math.CV↗
arXiv2025

Lipschitz conditions on bounded harmonic functions on the upper half-space

This work is devoted to Lipschitz conditions on bounded harmonic functions on the upper half-space in $\mathbb {R}^n$. Among other results we prove the following one. Let $U(x',x_n)$ be a real-valued bounded harmonic function on the upper half-space $\mathbb {R}^n_+ = \{(x',x_n):x'\in \mathbb{R}^{n-1}, x_n\in (0,\infty)\}$, which is continuous on the closure of this domain. Assume that for $\alpha\in (0,1)$ there exists a constant $C$ such that for every $x'\in \mathbb{R}^{n-1}$ we have $| |U|(x',x_n) - |U|(x',0)|\le Cx_n^\alpha,\, x_n\in (0,\infty)$. Then there exists a constant $\tilde {C}$ such that $|U(x) - U (y)| \le \tilde{C} |x-y|^\alpha,\, x,y\in \mathbb{R}^{n}_+$.

math.CV↗
arXiv2024

Conformally Natural Families of Probability Distributions on Hyperbolic Disc with a View on Geometric Deep Learning

We introduce the novel family of probability distributions on hyperbolic disc. The distinctive property of the proposed family is invariance under the actions of the group of disc-preserving conformal mappings. The group-invariance property renders it a convenient and tractable model for encoding uncertainties in hyperbolic data. Potential applications in Geometric Deep Learning and bioinformatics are numerous, some of them are briefly discussed. We also emphasize analogies with hyperbolic coherent states in quantum physics.

cs.LG↗
arXiv2024

Regularly oscillating mappings between metric spaces and a theorem of Hardy and Littlewood

This paper is motivated by the classical theorem due to Hardy and Littlewood which concerns analytic mappings on the unit disk and relates the growth of the derivative with the H\"{o}lder continuity. We obtain a version of this result in a very general setting -- for regularly oscillating mappings on a metric space equipped with a weight, which is a continuous and positive function, with values in another metric space. As a consequence, we derive the Hardy and Littlewood theorem for analytic mappings on the unit ball of a normed space.

math.CV↗
arXiv2021

A criterion for normality of analytic mappings

In this paper we give a generalization and improvement of the Pavlović result on the characterization of continuously differentiable functions in the Bloch space on the unit ball in $\mathbb{R}^m$. Then we derive a Holland--Walsh type theorem for analytic normal mappings on the unit disk.

math.CV↗
arXiv2021

On holomorphic functions on negatively curved manifolds

Based on a well known Sh.-T. Yau theorem we obtain that the real part of a holomorphic function on a Kähler manifold with the Ricci curvature bounded from below by $-1$ is contractive with respect to the distance on the manifold and the hyperbolic distance on $(-1,1)$ inhered from the domain $(-1,1)\times\mathbb{R}$. Moreover, in the case of bounded holomorphic functions we prove that the modulus is contractive with respect to the distance on the manifold and the hyperbolic distance on the unit disk.

math.CV↗
arXiv2021

Representations for the Bloch type semi-norm of Frechet differentiable mappings

In this paper we give some results concerning Frechet differentiable mappings between domains in normed spaces with controlled growth. The results are mainly motivated by Pavlovic's equality for the Bloch semi-norm of continuously differentiable mappings in the Bloch class on the unit ball of the Euclidean space as well as the very recent Jocic's generalization of this result.

math.CV↗
arXiv2020

Riesz's Theorem for Lumer's Hardy Spaces

In this note we obtain a version of the well-known Riesz's theorem on conjugate harmonic functions for Lumer's Hardy spaces $(Lh)^2(Ω)$ on arbitrary domains $Ω$: If a real-valued harmonic function $U\in (Lh)^2(Ω)$ has a harmonic conjugate $V$ on $Ω$ (i.e., a real-valued harmonic function such that $U+ iV$ is analytic on $Ω$), then $U+iV$ also belongs to $(Lh)^2(Ω)$, and for the normalized conjugate we have the norm estimate $\|U+iV\|_{(Lh)^2(Ω)}\le\sqrt{2} \|U\|_{(Lh)^2(Ω)}$, with the best possible constant.

math.CV↗
arXiv2020

Hardy-Littlewood theorems and the Bergman distance

We obtain non-Euclidean versions of classical theorems due to Hardy and Littlewood concerning smoothness of the boundary function of an analytic mapping on the unit disk with an appropriate growth condition.

math.CV↗
arXiv2016

Solution to the Khavinson problem near the boundary of the unit ball

This paper deals with an extremal problem for harmonic functions in the unit ball of $\mathbf{R}^n$. We are concerned with the pointwise sharp estimates for the gradient of real--valued bounded harmonic functions. Our main result may be formulated as follows. The sharp constants in the estimates for the absolute value of the radial derivative and the modulus of the gradient of a bounded harmonic function coincide near the boundary of the unit ball. This result partially confirms a conjecture posed by D. Khavinson.

math.CV↗
arXiv2015

Semi-norms of the Bergman projection

It is known that the Bergman projection operator maps the space of essentially bounded functions in the unit ball in the d-dimensional complex vector space onto the Bloch space of the unit ball. This paper deals with the various semi-norms of the Bergman projection. We improve some recent results.

math.CV↗
arXiv2015

On the Forelli-Rudin projection theorem

Motivated by the Forelli--Rudin projection theorem we give in this paper a criterion for boundedness of an integral operator on weighted Lebesgue spaces in the interval $(0,1)$. We also calculate the precise norm of this integral operator. This is the content of the first part of the paper. In the second part, as applications, we give some results concerning the Bergman projection and the Berezin transform. We derive a generalization of the Dostanić result on the norm of the Berezin transform acting on Lebesgue spaces over the unit ball in $\mathbf{C}^n$.

math.CV↗
arXiv2015

A sharp constant for the Bergman projection

For the Bergman projection operator $P$ we prove that $ \|P\|_{L^1(B,dλ)\rightarrow B_1}= \frac {(2n+1)!}{n!}.$ Here $λ$ stands for the invariant metric in the unit ball $B$ of $\mathbf{C}^n$, and $B_1$ denotes the Besov space with an adequate semi--norm. We also consider a generalization of this result. This generalizes some recent results due to Perälä.

math.CV↗
arXiv2015

Sharp inequalities over the unit polydisc

Motivated by some results due to Burbea we prove that if a certain sharp integral inequality holds for functions in the unit polydisc which belong to concrete Hardy spaces, then it also holds, in an appropriate form, in the case of functions from arbitrary Hardy spaces. We also examine the equality case. We present an application of this main result to a Burbea inequality which includes an isoperimetric type inequality as a special case.

math.CV↗
arXiv2014

Normality and boundary behavior of arbitrary and meromorphic functions along simple curves and applications

We establish the theorems that give necessary and sufficient conditions for an arbitrary function defined in the unit disk of complex plane in order to has boundary values along classes of equivalencies of simple curves. Our results generalize the well--known theorems on asymptotic and angular boundary behavior of meromorphic functions (Lindolf, Lehto--Virtanen, and Seidel--Walsh type theorems). The results are applied to the study of boundary behavior of meromorphic functions along curves using $P-$sequences, as well as in the proof of the uniqueness theorem similar to Saginjan's one. Constructed examples of functions show that the results cannot be improved.

math.CV↗
arXiv2014

On harmonic functions and the hyperbolic metric

Motivated by some recent results of Kalaj and Vuorinen (Proc. Amer. Math. Soc., 2012), we prove that positive harmonic functions defined in the upper half--plane are contractions w.r.t. hyperbolic metrics of half--plane and positive part of the real line, respectively

math.CV↗