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David Kalaj

Publications and source records attributed to David Kalaj.

At least 37 records · Page 2Linked to original sources
arXiv2025

On a sharp form of curvature conjecture for minimal graphs

Recently, the author and Melentijević resolved the longstanding Gaussian curvature problem by proving the sharp inequality \[ |\mathcal{K}| < c_0 = \frac{π^2}{2} \] for minimal graphs over the unit disk, evaluated at the point of the graph lying directly above the origin. The constant \( c_0 \) is known as the \emph{Heinz constant}. Building on this result, we obtain an improved estimate for the Hopf constant \( c_1 \). In addition, we show that for any prescribed unit normal vector \( \mathbf{n} \), there exists a minimal graph over the unit disk -- bending in the coordinate directions -- whose Gaussian curvature at the point above the origin is strictly smaller, yet arbitrarily close to, the curvature of the associated Scherk-type surface with the same normal, situated above a bicentric quadrilateral. This sharp inequality strengthens the classical result of Finn and Osserman, which applies in the special case when the unit normal is \( (0,0,1) \).

math.DG↗
arXiv2025

Gaussian curvature conjecture for minimal graphs

In this paper, we solve the longstanding Gaussian curvature conjecture of a minimal graph $S$ over the unit disk. The conjecture asserts that for any minimal graph above the unit disk, the Gaussian curvature at the point directly above the origin satisfies the sharp inequality \( |\mathcal{K}| < \frac{π^2}{2} \). We first reduce the conjecture to the problem of estimating the Gaussian curvature of certain Scherk-type minimal surfaces defined over bicentric quadrilaterals inscribed in the unit disk, containing the origin. We then provide a sharp estimate for the Gaussian curvature of these minimal surfaces at the point above the origin. Our proof employs complex-analytic methods, as the minimal surfaces in question allow a conformal harmonic parameterization.

math.DG↗
arXiv2025

Zygmund theorem for harmonic quasiregular mappings

Let $K\ge 1$. We prove Zygmund theorem for $K-$quasiregular harmonic mappings in the unit disk $\mathbb{D}$ in the complex plane by providing a constant $C(K)$ in the inequality $$\|f\|_{1}\le C(K)(1+\|\mathrm{Re}\,(f)\log^+ |\mathrm{Re}\, f|\|_1),$$ provided that $\mathrm{Im}\,f(0)=0$. Moreover for a quasiregular harmonic mapping $f=(f_1,\dots, f_n)$ defined in the unit ball $\mathbb{B}\subset \mathbb{R}^n$, we prove the asymptotically sharp inequality $$\|f\|_{1}-|f(0)|\le (n-1)K^2(\|f_1\log f_1\|_1- f_1(0)\log f_1(0)),$$ when $K\to 1$, provided that $f_1$ is positive.

math.CV↗
arXiv2025

Weighted contractivity for derivatives of functions in the Bergman space on the unit disk

In a recent paper, Ramos and Tilli proved certain sharp inequality for analytic functions in subdomains of the unit disk. We will generalize their main inequality for derivatives of functions from Bergman space with respect to two diferent measures. Some connections with an analog for the Fock spaces, earlier investigated in Kalaj, will also be discussed.

math.CV↗
arXiv2024

Conformal and holomorphic barycenters in hyperbolic balls

We introduce the notions of \textit{conformal barycenter} and \textit{holomorphic barycenter} of a measurable set $D$ in the hyperbolic ball. The two barycenters coincide in the disk, but they differ in multidimensional balls $\mathbb{C}^m \cong \mathbb{R}^{2m}$. These notions are counterparts of barycenters of measures on spheres, introduced by Douady and Earle in 1986.

math.DG↗
arXiv2024

Uniform stability of concentration inequalities and applications

We prove a sharp quantitative version of recent Faber-Krahn inequalities for the continuous Wavelet transforms associated to a certain family of Cauchy wavelet windows . Our results are uniform on the parameters of the family of Cauchy wavelets, and asymptotically sharp in both directions. As a corollary of our results, we are able to recover not only the original result for the short-time Fourier transform as a limiting procedure, but also a new concentration result for functions in Hardy spaces. This is a completely novel result about optimal concentration of Poisson extensions, and our proof automatically comes with a sharp stability version of that inequality. Our techniques highlight the intertwining of geometric and complex-analytic arguments involved in the context of concentration inequalities. In particular, in the process of deriving uniform results, we obtain a refinement over the proof of a previous result by the first and fourth authors together with A. Guerra and P. Tilli, further improving the current understanding of the geometry of near extremals in all contexts under consideration.

math.FA↗
arXiv2024

Radiall symmetry of minimizers to the weighted $p-$Dirichlet energy

Let $\mathbb{A}=\{z: r< |z| 1$ then such diffeomorphism exist always. If $p=1$, then the conformal modulus of $\A^\ast$ must not be greater or equal to $π/2$. This curious phenomenon is opposite to the Nitsche type phenomenon known for the standard Dirichlet energy.

math.AP↗
arXiv2024

A sharp estimate of area for sublevel-set of Blaschke products

Let $\mathbb{D}$ be the unit disk in the complex plane. Among other results, we prove the following curious result for a finite Blaschke product: $$B(z)=e ^{is}\prod_{k=1}^d \frac{z-a_k}{1-z \overline{a_k}}.$$ The Lebesgue measure of the sublevel set of $B$ satisfies the following sharp inequality for $t \in [0,1]$: $$|\{z\in \mathbb{D}:|B(z)|<t\}|\le πt^{2/d},$$ with equality at a single point $t\in(0,1)$ if and only if $a_k=0$ for every $k$. In that case the equality is attained for every $t$.

math.CV↗
arXiv2024

Contraction property of Fock type space of log-subharmonic functions in $\mathbb{R}^m$

We prove a contraction property of Fock type spaces $\mathcal{L}_α^p$ of log-subharmonic functions in $\mathbb{R}^n$. To prove the result, we demonstrate a certain monotonic property of measures of the superlevel set of the function $u(x) = |f(x)|^p e^{-\fracα{2} p |x|^2}$, provided that $f$ is a certain log-subharmonic function in $\mathbb{R}^m$. The result recover a contraction property of holomorphic functions in the Fock space $\mathcal{F}_α^p$ proved by Carlen in \cite{carlen}.

math.CV↗
arXiv2024

Minimization of Dirichlet energy of $j-$degree mappings between annuli

Let $\mathbb{A}$ and $\mathbb{B}$ be circular annuli in the complex plane and consider the Dirichlet energy integral of $j-$degree mappings between $\mathbb{A}$ and $\mathbb{B}$. Then we minimize this energy integral. The minimizer is a $j-$degree harmonic mapping between annuli $\mathbb{A}$ and $\mathbb{B}$ provided it exits. If such a harmonic mapping does not exist, then the minimizer is still a $j-$degree mapping which is harmonic in $\mathbb{A}'\subset \mathbb{A}$ and it is a squeezing mapping in its complementary annulus $\mathbb{A}''=\mathbb{A}\setminus \mathbb{A}$. Such a result is an extension of the certain result of Astala, Iwaniec and Martin \cite{astala2010}.

math.CV↗
arXiv2024

Sharp pointwise estimate of $α-$harmonic functions

Let $α>-1$ and assume that $f$ is $α-$harmonic mapping defined in the unit disk that belongs to the Hardy class $h^p$ with $p\ge 1$. We obtain some sharp estimates of the type $|f(z)|\le g(|r|) \|f^\ast\|_p$ and $|Df(z)|\le h(|r|)\|f^\ast\|_p$. We also prove a Schwarz type lemma for the class of $α-$harmonic mappings of the unit disk onto itself fixing the origin.

math.CV↗
arXiv2023

Riesz and Kolmogorov inequality for harmonic quasiregular mappings

Let $K\ge 1$ and $p\in(1,2]$. We obtain asymptotically sharp constant $c(K,p)$, when $K\to 1$ in the inequality $$\|\Im f\|_{p}\le c(K,p)\|\Re(f)\|_p$$ where $f\in \mathbf{h}^p$ is a $K-$quasiregular harmonic mapping in the unit disk belonging to the Hardy space $\mathbf{h}^p$, under the conditions $\arg(f(0))\in (-π/(2p),π/(2p))$ and $f(\mathbb{D})\cap(-\infty,0)=\emptyset$. The paper improves a recent result by Liu and Zhu in \cite{aimzhu}. We also extend this result for the quasiregular harmonic mappings in the unit ball in $\mathbb{R}^n$. We also extend Kolmogorov theorem for quasiregular harmonic mappings in the plane.

math.CV↗
arXiv2023

Harmonic quasiconformal mappings between $\mathscr{C}^1$ smooth Jordan domains

We prove the following result. If $f$ is a harmonic quasiconformal mapping between two Jordan domains $D$ and $Ω$ having $\mathscr{C}^1$ boundaries, then the function $f$ is globally Hölder continuous for every $α<1$ but it is not Lipschitz in general. This extends and improves a classical theorem of S. Warschawski for conformal mappings.

math.CV↗
arXiv2023

On M. Riesz conjugate function theorem for harmonic functions

Let $L^p(\mathbf{T})$ be the Lesbegue space of complex-valued functions defined in the unit circle $\mathbf{T}=\{z: |z|=1\}\subseteq \mathbb{C}$. In this paper, we address the problem of finding the best constant in the inequality of the form: $$\|f\|_{L^p(\mathbf{T})}\le A_{p,b} \|(|P_+ f|^2+b| P_{-} f|^2)^{1/2}\|_{L^p(\mathbf{T})}.$$ Here $p\in[1,2]$, $b>0$, and by $P_{-} f$ and $ P_+ f$ are denoted co-analytic and analytic projection of a function $f\in L^p(\mathbf{T})$. The equality is "attained" for a quasiconformal harmonic mapping. The result extends a sharp version of M. Riesz conjugate function theorem of Pichorides and Verbitsky and some well-known estimates for holomorphic functions.

math.CV↗
arXiv2023

Schwarz lemma for real harmonic functions onto surfaces with non-negative Gaussian curvature

Assume that $f$ is a real $ρ$-harmonic function of the unit disk $\mathbb{D}$ onto the interval $(-1,1)$, where $ρ(u,v)=R(u)$ is a metric defined in the infinite strip $(-1,1)\times \mathbb{R}$. Then we prove that $|\nabla f(z)|(1-|z|^2)\le \frac{4}π(1-f(z)^2)$ for all $z\in\mathbb{D}$, provided that $ρ$ has a non-negative Gaussian curvature. This extends several results in the field and answers to a conjecture proposed by the first author in 2014. Such an inequality is not true for negatively curved metrics.

math.CV↗
arXiv2023

A Faber-Krahn type inequality for log-subharmonic functions in the hyperbolic ball

Assume that $Δ_h$ is the hyperbolic Laplacian in the unit ball $\mathbb{B}$ and assume that $Φ_n$ is the unique radial solution of Poisson equation $Δ_h \log Φ_n =-4 (n-1)^2$ satisfying the condition $Φ_n(0)=1$ and $Φ_n(ζ)=0$ for $ζ\in \partial\mathbb{B}$. We explicitly solve the question of maximizing $$ R_n(f,Ω)= \frac{\int_Ω|f(x)|^2 Φ_n^α(|x|) \, dτ(x)}{\|f\|^2_{\mathbf{B}^2_α}}, $$ over all $f \in\mathbf{B}^2_α$ and $Ω\subset \mathbb{B}$ with $τ(Ω) = s,$ where $dτ$ denotes the invariant measure on $\mathbb{B},$ and $\|f\|_{{B}^2_α}^2 = \int_\mathbb{B} |f(x)|^2 Φ_n^α(|x|) dτ(x) < \infty.$ This result extends the main result of Tilli and the second author \cite{ramostilli} to a higher-dimensional context. Our proof relies on a version of the techniques used for the two-dimensional case, with several additional technical difficulties arising from the definition of the weights $Φ_n$ through hypergeometric functions. Additionally, we show that an immediate relationship between a concentration result for log-sunharmonic functions and one for the Wavelet transform is only available in dimension one.

math.AP↗
arXiv2022

Contraction property of differential operator on Fock space

In the recent paper, \cite{tilli} Nicola and Tilli proved the Faber-Krahn inequality, which for $p=2$, states the following. If $f\in\mathcal{F}_α^2$ is an entire function from the corresponding Fock space, then $$\frac{1}π\int_Ω |f(z)|^2 e^{-π|z|^2} dx dy \le (1-e^{-|Ω|}) \|f\|^2_{2,π}.$$ Here $Ω$ is a domain in the complex plane and $|Ω|$ is its Lebesgue measure. This inequality is sharp and equality can be attained. We prove the following sharp inequality $$\int_Ω \frac{|f^{(n)}(z)|^2e^{-π|z|^2}}{π^n n ! L_n(-π|z|^2)}dxdy \le (1-e^{-(n+1)|Ω|})\|f\|^2_{2,π},$$ where $L_n$ is Laguerre polynomial, and $n\in\{0,1,2,3,4\} $. For $n=0$ it coincides with the result of Nicola and Tilli.

math.CV↗