arXiv subjects

Petar Melentijević

Publications and source records attributed to Petar Melentijević.

16 recordsLinked to original sources

arXiv2026

Stability of Wehrl-type Functionals and Concentration Estimates on Bergman Spaces of Log-Subharmonic Functions on the Unit Sphere

In this paper, we consider weighted Bergman classes $\mathcal{B}_{α,p}$ of real-analytic functions on $\mathbb{R}^{n}$, $n\geq 2$ whose moduli are log-subharmonic with respect to the spherical metric. Using the isoperimetric inequality for the spherical metric we prove a certain monotonicity property for super-level sets of $|f(x)|^p\mathcal{W}_n^α(x),$ where $f\in \mathcal{B}_{α,p}$ and $\mathcal{W}_n^α(x)$ is the Bergman weight. As a consequence, we obtain sharp concentration estimates and solve maximization problems for convex Wehrl-type functionals. We also establish quantitative stability estimates, including a bound for the distance of $|f|$ from spherical comparison profiles in terms of the concentration deficit.

math.CV↗

arXiv2026

Reverse Heinz type inequality, Mather beta function and coefficient estimates

In this paper, we prove that for two cyclically ordered sequences of complex numbers of unit modulus $(ξ_j)_{j=1}^{n}$ and $(ζ_j)_{j=1}^{n}$ the inequality: $$\bigg|\sum_{m=1}^{n}(ξ_{m+1}-ξ_m)ζ_m\bigg|^2+ \bigg|\sum_{m=1}^{n}(ξ_{m+1}-ξ_m)ζ_m^{-1}\bigg|^2\leqslant 4n^2\sin^2\fracπ{n}$$ holds for $n\geqslant 4$. As a consequence, for $n=4,$ we disprove Hall's conjecture on Heinz type inequality for harmonic self-mappings of the unit disk. The connection with the theory of elliptic billiards via the Mather beta function will also be given.

math.CV↗

arXiv2025

Sharp stability of convex functionals on weighted Bergman spaces with applications

Recently, Kulikov (\cite{Ku}) has shown that certain convex functionals on weighted Bergman spaces are maximized by reproducing kernels. We show a sharp quantitative stability of these estimates with the optimal norm and the exponent and an explicit constant asymptotically sharp in both directions ($α\rightarrow -1$ and $α\rightarrow +\infty$). Several applications of this result include recovering the appropriate result for Fock spaces, interpretation to Cauchy wavelets, and the Hardy space counterpart for functionals induced by increasing function. In addition, we prove a higher-dimensional analog of the main result assuming that all convex functionals on the weighted Bergman space $\mathcal{A}^2_α(\mathbb{B}_n)$ attain their extrema in reproducing kernels.

math.CA↗

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↗

arXiv2025

Best constants in reverse Riesz-type inequalities for analytic and co-analytic projections

Let $P_+$ be the Riesz's projection operator and let $P_-= I - P_+$. We consider the inequalities of the following form $$ \|f\|_{L^p(\mathbb{T})}\leq B_{p,s}\|( |P_ + f | ^s + |P_- f |^s) ^{\frac 1s}\|_{L^p (\mathbb{T})} $$ and prove them with sharp constant $B_{p,s}$ for $s \in [p',+\infty)$ and $1<p\leq 2$ and $p\geq 4,$ where $p':=\min\{p,\frac{p}{p-1}\}.$

math.CV↗

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↗

arXiv2023

$L^p$ norm of truncated Riesz transform and an improved dimension-free $L^p$ estimate for maximal Riesz transform

In this paper, we prove that the $L^p(\mathbb{R}^d)$ norm of the maximal truncated Riesz transform in terms of the $L^p(\mathbb{R}^d)$ norm of Riesz transform is dimension-free for any $2\leq p<\infty$, using integration by parts formula for radial Fourier multipliers. Moreover, we show that $$\|R_j^*f\|_{L^p}\leq \left({2+\frac{1}{\sqrt{2}}}\right)^{\frac{2}{p}}\|R_jf\|_{L^p},\ \ \mbox{for}\ \ p\geq2,\ \ d\geq2.$$ As by products of our calculations, we infer the $L^p$ norm contractivity of the truncated Riesz transforms $R^t_j$ in terms of $R_j$, and their accurate $L^p$ norms. More precisely, we prove: $$\|R^t_jf\|_{L^p}\leq\|R_jf\|_{L^p}$$ and $$\|R^t_j\|_{L^p}=\|R_j\|_{L^p},$$ for all $1 0.$

math.CA↗

arXiv2023

Hypercontractive inequalities for weighted Bergman spaces

\begin{abstract} We obtain sharp $L^p\rightarrow L^q$ hypercontractive inequalities for the weighted Bergman spaces on the unit disk $\mathbb{D}$ with the usual weights \\ $\frac{α-1}π(1-|z|^2)^{α-2},α>1$ for $q\geq 2,$ thus solving an interesting case of a problem from \cite{JANSON}. We also give some estimates for $0<q<2.$ \end{abstract}

math.CV↗

arXiv2023

Hollenbeck-Verbitsky conjecture on best constant inequalities for analytic and co-analytic projections

\begin{abstract} In this paper we address the problem of finding the best constants in inequalities of the form: $$ \|\big(|P_+f|^s+|P_-f|^s\big)^{\frac{1}{s}}\|_{L^p({\mathbb{T}})}\leq A_{p,s} \|f\|_{L^p({\mathbb{T}})},$$ where $P_+f$ and $P_-f$ denote analytic and co-analytic projection of a complex-valued function $f \in L^p({\mathbb{T}}),$ for $p \geq 2$ and all $s>0$, thus proving Hollenbeck-Verbitsky conjecture from \cite{HV.OTAA}. We also prove the same inequalities for\\ $1<p\leq\frac{4}{3}$ and $s\leq \sec^2\fracπ{2p}$ and confirm that $s=\sec^2\fracπ{2p}$ is the sharp cutoff for $s.$ The proof uses a method of plurisubharmonic minorants and an approach of proving the appropriate "elementary" inequalities that seems to be new in this topic. We show that this result implies best constants inequalities for the projections on the real-line and half-space multipliers on $\mathbb{R}^n$ and an analog for analytic martingales. A remark on an isoperimetric inequality for harmonic functions in the unit disk is also given. \end{abstract}

math.CV↗

arXiv2020

Best constants in inequalities involving analytic and co-analytic projections and Riesz theorem for various function spaces

\begin{abstract} Let $P\pm$ be the Riesz's projection operator and let $P_-= I - P_+$. We consider estimates of the expression $\|( |P_ + f | ^s + |P_- f |^s) ^{\frac{1}{s}}\|_{L^p (\mathbf{T})}$ in terms of Lebesgue $p$-norm of the function $f \in L^p(\mathbf{T})$. We find the accurate estimates for $p\geq 2$ and $0<s\leq p$, thus significantly improving results from \cite{KALAJ.TAMS} where it is considered for $s=2$ and $1<p<\infty$. Interestingly, for this range of $s$ there holds the appropriate vector-valued inequality with the same constant. Also, we obtain the right asymptotic of the constants for large $s$. This proves the conjecture of Hollenbeck and Verbitsky on the Riesz projection operator in some cases. As a consequence of inequalities we have proved in the paper we get Riesz-type theorems on conjugate harmonic functions for various function spaces. In particular, slightly general version of Stout's theorem for Lumer Hardy spaces is obtained by a new approach. \end{abstract}

math.FA↗

arXiv2020

Khavinson problem for hyperbolic harmonic mappings in Hardy space

\begin{abstract} In this paper, we partly solve the generalized Khavinson conjecture in the setting of hyperbolic harmonic mappings in Hardy space. Assume that $u=\mathcal{P}_Ω[ϕ]$ and $ϕ\in L^{p}(\partialΩ, \mathbb{R})$, where $p\in[1,\infty]$, $\mathcal{P}_Ω[ϕ]$ denotes the Poisson integral of $ϕ$ with respect to the hyperbolic Laplacian operator $Δ_{h}$ in $Ω$, and $Ω$ denotes the unit ball $\mathbb{B}^{n}$ or the half-space $\mathbb{H}^{n}$. For any $x\in Ω$ and $l\in \mathbb{S}^{n-1}$, let $\mathbf{C}_{Ω,q}(x)$ and $\mathbf{C}_{Ω,q}(x;l)$ denote the optimal numbers for the gradient estimate $$ |\nabla u(x)|\leq \mathbf{C}_{Ω,q}(x)\|ϕ\|_{ L^{p}(\partialΩ, \mathbb{R})} $$ and gradient estimate in the direction $l$ $$|\langle\nabla u(x),l\rangle|\leq \mathbf{C}_{Ω,q}(x;l)\|ϕ\|_{ L^{p}(\partialΩ, \mathbb{R})}, $$ respectively. Here $q$ is the conjugate of $p$. If $q=\infty$ or $q\in[\frac{2K_{0}-1}{n-1}+1,\frac{2K_{0}}{n-1}+1]\cap [1,\infty)$ with $K_{0}\in\mathbb{N}=\{0,1,2,\ldots\}$, then $\mathbf{C}_{\mathbb{B}^{n},q}(x)=\mathbf{C}_{\mathbb{B}^{n},q}(x;\pm\frac{x}{|x|})$ for any $x\in\mathbb{B}^{n}\backslash\{0\}$, and $\mathbf{C}_{\mathbb{H}^{n},q}(x)=\mathbf{C}_{\mathbb{H}^{n},q}(x;\pm e_{n})$ for any $x\in \mathbb{H}^{n}$, where $e_{n}=(0,\ldots,0,1)\in\mathbb{S}^{n-1}$. However, if $q\in(1,\frac{n}{n-1})$, then $\mathbf{C}_{\mathbb{B}^{n},q}(x)=\mathbf{C}_{\mathbb{B}^{n},q}(x;t_{x})$ for any $x\in\mathbb{B}^{n}\backslash\{0\}$, and $\mathbf{C}_{\mathbb{H}^{n},q}(x)=\mathbf{C}_{\mathbb{H}^{n},q}(x;t_{e_{n}})$ for any $x\in \mathbb{H}^{n}$. Here $t_{w}$ denotes any unit vector in $\mathbb{R}^{n}$ such that $\langle t_{w},w\rangle=0$ for $w\in \mathbb{R}^{n}\setminus\{0\}$. \end{abstract}

math.AP↗

arXiv2020

$L^p$-theory for Cauchy-transform on the unit disk

Let $\mathbb{D}$ be the unit disk and $φ\in L^p(\mathbb{D}, \mathrm{d}A)$, where $1\leq p\leq\infty$. For $z\in\mathbb{D}$, the Cauchy-transform on $\mathbb{D}$, denote by $\mathcal{P}$, is defined as follows: $$\mathcal{P}[φ](z)=-\int_{\mathbb{D}}\left(\frac{φ(w)}{w-z}+\frac{z\overline{φ(w)}}{1-\bar{w}z}\right)\mathrm{d}A(w).$$ The Beurling transform on $\mathbb{D}$, denote by $\mathcal{H}$, is now defined as the $z$-derivative of $\mathcal{P}$. In this paper, by using Hardy's type inequalities and Bessel functions, we show that $\|\mathcal{P}\|_{L^2\to L^2}=α\approx1.086$, where $α$ is a solution to the equation: $2J_0(2/α)-αJ_1(2/α)=0$, and $J_0$, $J_1$ are Bessel functions. Moreover, for $p>2$, by using Taylor expansion, Parseval's formula and hypergeometric functions, we also prove that $\|\mathcal{P}\|_{L^p\to L^{\infty}}=2(Γ(2-q)/Γ^2(2-\frac{q}{2}))^{1/q}$, where $q=p/(p-1)$ is the conjugate exponent of $p$, and $Γ$ is the Gamma function. Finally, applying the same techniques developed in this paper, we show that the Beurling transform $\mathcal{H}$ acts as an isometry of $L^2(\mathbb{D}, \mathrm{d}A)$.

math.CV↗

arXiv2020

Sharp Riesz-Fejér inequality for harmonic Hardy spaces

We prove sharp version of Riesz-Fejér inequality for functions in harmonic Hardy space $h^p(\mathbb{D})$ on the unit disk $\mathbb{D}$, for $p>1,$ thus extending the result from \cite{KPK} and resolving the posed conjecture.

math.FA↗

arXiv2019

A proof of the Khavinson conjecture

\begin{abstract} This paper deals with an extremal problem for bounded harmonic functions in the unit ball $\mathbb{B}^n.$ We solve the Khavinson conjecture in $\mathbb{R}^3,$ an intriguing open question since 1992 posed by D. Khavinson, later considered in a general context by Kresin and Maz'ya. Precisely, we obtain the following inequality: $$|\nabla u(x)|\leq \frac{1}{ρ^2}\bigg({\frac{(1+\frac{1}{3}ρ^2)^{\frac{3}{2}}}{1-ρ^2}-1}\bigg) \sup_{|y|<1} |u(y)|, $$ with $ρ=|x|,$ thus sharpening the previously known with $|\langle \nabla u(x),n_{x} \rangle |$ instead of $|\nabla u(x)|, $ where $n_{x}=\frac{x}{|x|}.$ \end{abstract}

math.AP↗

arXiv2018

Norm of the Bergman projection onto the Bloch space with $\mathcal{M}-$invariant gradient

The value of the operator norm of Bergman projections from $L^{\infty}(\mathbb{B}^n)$ to Bloch space is found in \cite{KalajMarkovic2014}. The authors of mentioned paper proposed the problem of calculating the norm of the same class of operators with different norm on the Bloch space - $\mathcal{M}$-invariant gradient. Here we prove that it has the conjectured value.

math.CV↗

arXiv2018

On the norm of the weighted Berezin transform

We consider a weighted Berezin transform: $$ B_α : L^{\infty} (\mathbb{B}^n) \to \ \mathcal{B},\quad α>-1,$$ defined, for $f \in L^{\infty} \left( \mathbb{B}^n \right)$ and $z \in \mathbb{B}^n$, by $$(B_αf) (z) = c_α\int_{\mathbb{B}^n} \frac{\left( 1-|z|^2 \right)^{n+1}}{|1 - \langle z, w \rangle|^{2n+2}} f(w) \left( 1-|w|^2 \right)^α\ d v(w),$$ where $c_α = \frac{Γ(α+n+1)}{Γ(α+1)π^n}$ , $v$ is the Lebesque measure and $\mathcal{B}$ is a Bloch-type space. We prove that $B_α$ is bounded iff $α>0$ and give the exact semi-norm of $ B_α$ for $0\leqα\leq 2n+3.$

math.CV↗