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Paata Ivanisvili

Paata Ivanisvili appears in the imported research catalog. Authorship, coauthor and topic links are available while profile ownership is still unclaimed.

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math.APmath.CAmath.FAmath.PRArtificial Intelligencemath.DGmath.OCMachine Learningmath.CO

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Published work

21 published item(s)

preprint2026arXiv

Almost-Orthogonality in Lp Spaces: A Case Study with Grok

Carbery proposed the following sharpened form of triangle inequality for many functions: for any $p\ge 2$ and any finite sequence $(f_j)_j\subset L^p$ we have \[ \Big\|\sum_j f_j\Big\|_p \ \le\ \left(\sup_{j} \sum_{k} α_{jk}^{\,c}\right)^{1/p'} \Big(\sum_j \|f_j\|_p^p\Big)^{1/p}, \] where $c=2$, $1/p+1/p'=1$, and $α_{jk}=\sqrt{\frac{\|f_{j}f_{k}\|_{p/2}}{\|f_{j}\|_{p}\|f_{k}\|_{p}}}$. In the first part of this paper we construct a counterexample showing that this inequality fails for every $p>2$. We then prove that if an estimate of the above form holds, the exponent must satisfy $c\le p'$. Finally, at the critical exponent $c=p'$, we establish the inequality for all integer values $p\ge 2$. In the second part of the paper we obtain a sharp three-function bound \[ \Big\|\sum_{j=1}^{3} f_j\Big\|_p \ \le\ \left(1+2Γ^{c(p)}\right)^{1/p'} \Big(\sum_{j=1}^{3} \|f_j\|_p^p\Big)^{1/p}, \] where $p \geq 3$, $c(p) = \frac{2\ln(2)}{(p-2)\ln(3)+2\ln(2)}$ and $Γ=Γ(f_1,f_2,f_3)\in[0,1]$ quantifies the degree of orthogonality among $f_1,f_2,f_3$. The exponent $c(p)$ is optimal, and improves upon the power $r(p) = \frac{6}{5p-4}$ obtained previously by Carlen, Frank, and Lieb. Some intermediate lemmas and inequalities appearing in this work were explored with the assistance of the large language model Grok.

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preprint2026arXiv

Grokability in five inequalities

In this note, we report five mathematical discoveries made in collaboration with Grok, all of which have been subsequently verified by the authors. These include an improved lower bound on the maximal Gaussian perimeter of convex sets in $\mathbb{R}^n$, sharper $L_2$-$L_1$ moment comparison inequalities on the Hamming cube $\{-1,1\}^n$, a strengthened autoconvolution inequality, improved asymptotic bounds on the size of the largest $g$-Sidon sets in $\{1,\dots,n\}$, and an optimal balanced Szarek's inequality.

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preprint2023arXiv

A new proof of the description of the convex hull of space curves with totally positive torsion

We give new proofs of the description convex hulls of space curves $γ: [a,b] \mapsto \mathbb{R}^{d}$ having totally positive torsion. These are curves such that all the leading principal minors of $d\times d$ matrix $(γ', γ'', \ldots, γ^{(d)})$ are positive. In particular, we recover parametric representation of the boundary of the convex hull, different formulas for its surface area and the volume of the convex hull, and the solution to a general moment problem corresponding to $γ$.

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preprint2022arXiv

Banach space valued Pisier and Riesz type inequalities on discrete cube

This is an attempt to build Banach space valued theory for certain singular integrals on Hamming cube. Of course all estimates below are dimension independent, and we tried to find ultimate sharp assumptions on the Banach space for a corresponding operators to be bounded. In certain cases we succeeded, although there are still many open questions, some of them are listed in the last Section. Using the approach of \cite{IVHV} and also quantum random variables approach of \cite{ELP} we generalize several theorems of Pisier \cite{P} and Hytönen-Naor \cite{HN}. We also improve the constant in $L^1$-Poincaré inequality on Hamming cube, the previous results are due to Talagrand and Ben Efraim--Lust-Piquard.

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preprint2020arXiv

Dimension independent Bernstein-Markov inequalities in Gauss space

We obtain the following dimension independent Bernstein-Markov inequality in Gauss space: for each $1\leq p<\infty$ there exists a constant $C_p>0$ such that for any $k\geq 1$ and all polynomials $P$ on $\mathbb{R}^{k}$ we have $$ \| \nabla P\|_{L^{p}(\mathbb{R}^{k}, \mathrm{d}γ_k)} \leq C_p (\mathrm{deg}\, P)^{\frac{1}{2}+\frac{1}π\arctan\left(\frac{|p-2|}{2\sqrt{p-1}}\right)}\|P\|_{L^{p}(\mathbb{R}^{k}, \mathrm{d}γ_k)}, $$ where $\mathrm{d}γ_k$ is the standard Gaussian measure on $\mathbb{R}^{k}$. We also show that under some mild growth assumptions on any function $B \in C^{2}((0,\infty))\cap C([0,\infty))$ with $B', B''>0$ we have $$ \int_{\mathbb{R}^{k}} B\left( |LP(x)|\right) \mathrm{d}γ_k(x) \leq \int_{\mathbb{R}^{k}} B\left( 10 (\mathrm{deg}P)^{α_{B}}|P(x)|\right)\mathrm{d}γ_k(x) $$ where $L=Δ-x\cdot \nabla $ is the generator of the Ornstein-Uhlenbeck semigroup and $$ α_{B} =1+\frac{2}π \arctan\left(\frac{1}{2}\sqrt{\sup_{s \in (0,\infty)}\left\{\frac{sB''(s)}{B'(s)}+\frac{B'(s)}{sB''(s)}\right\}-2}\right). $$

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preprint2020arXiv

Polynomial inequalities on the Hamming cube

Let $(X,\|\cdot\|_X)$ be a Banach space. The purpose of this article is to systematically investigate dimension independent properties of vector valued functions $f:\{-1,1\}^n\to X$ on the Hamming cube whose spectrum is bounded above or below. Our proofs exploit contractivity properties of the heat flow, induced by the geometry of the target space $(X,\|\cdot\|_X)$, combined with duality arguments and suitable tools from approximation theory and complex analysis. We obtain a series of improvements of various well-studied estimates for functions with bounded spectrum, including moment comparison results for low degree Walsh polynomials and Bernstein-Markov type inequalities, which constitute discrete vector valued analogues of Freud's inequality in Gauss space (1971). Many of these inequalities are new even for scalar valued functions. Furthermore, we provide a short proof of Mendel and Naor's heat smoothing theorem (2014) for functions on tail spaces with values in spaces of nontrivial type and we also prove a dual lower bound on the decay of the heat semigroup acting on functions with spectrum bounded from above. Finally, we improve the reverse Bernstein-Markov inequalities of Meyer (1984) and Mendel and Naor (2014) for functions with narrow enough spectrum and improve the bounds of Filmus, Hatami, Keller and Lifshitz (2016) on the $\ell_p$ sums of influences of bounded functions for $p\in\big(1,\frac{4}{3}\big)$.

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preprint2019arXiv

Sharpening the triangle inequality: envelopes between $L^{2}$ and $L^{p}$ spaces

Motivated by the inequality $\|f+g\|_{2}^{2} \leq \|f\|_{2}^{2}+2\|fg\|_{1}+\|g\|^{2}_{2}$, Carbery (2006) raised the question what is the "right" analogue of this estimate in $L^{p}$ for $p \neq 2$. Carlen, Frank, Ivanisvili and Lieb (2018) recently obtained an $L^{p}$ version of this inequality by providing upper bounds for $\|f+g\|_{p}^{p}$ in terms of the quantities $\|f\|_{p}^{p}, \|g\|_{p}^{p}$ and $\|fg\|_{p/2}^{p/2}$ when $p \in(0,1] \cup [2,\infty)$, and lower bounds when $p \in (-\infty,0) \cup (1,2)$, thereby proving (and improving) the suggested possible inequalities of Carbery. We continue investigation in this direction by refining the estimates of Carlen, Frank, Ivanisvili and Lieb. We obtain upper bounds for $\|f + g\|_p^p$ also when $p \in (-\infty,0) \cup (1,2)$ and lower bounds when $p \in (0,1] \cup [2,\infty)$. For $p \in [1,2]$ we extend our upper bounds to any finite number of functions. In addition, we show that all our upper and lower bounds of $\|f+g\|_{p}^{p}$ for $p \in \mathbb{R}$, $p\neq 0$, are the best possible in terms of the quantities $\|f\|_{p}^{p}, \|g\|_{p}^{p}$ and $\|fg\|_{p/2}^{p/2}$, and we characterize the equality cases.

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preprint2016arXiv

Isoperimetric functional inequalities via the maximum principle: the exterior differential systems approach

The goal of this note is to give the unified approach to the solutions of a class of isoperimetric problems by relating them to the exterior differential systems studied by R.~Bryant and P.~Griffiths. In this note we list several classical by now isopereimetric inequalities which can be proved in a unified way. This unified approach reduces them to the so-called exterior differential systems studied by Robert Bryant and Phillip Griffiths. To the best of our knowledge, this is the first article where this connection is used. After reviewing a list of classical inequalities (log-Sobolev inequality, Beckner's inequality, Bobkov's functional isoperimetric inequality and several other inequalities) we use our method to generate new isoperimetric inequalities, in particular, we found the sharpening of Beckner--Sobolev inequalities with Gaussian measure. Key words: log-Sobolev inequality, Poincaré inequality, Bobkov's inequality, Gaussian isoperimetry, semigroups, maximum principle, Monge--Ampère equation with drift, exterior differential systems, backwards heat equation, (B) theorem

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preprint2016arXiv

Lower bounds for uncentered maximal functions in any dimension

In this paper we address the following question: given $ p\in (1,\infty)$, $n \geq 1$, does there exists a constant $A(p,n)>1$ such that $\| M f\|_{L^{p}}\geq A(n,p) \| f\|_{L^{p}}$ for any nonnegative $f \in L^{p}(\mathbb{R}^{n})$, where $Mf$ is a maximal function operator defined over the family of shifts and dilates of a centrally symmetric convex body. The inequality fails in general for the centered maximal function operator, but nevertheless we give an affirmative answer to the question for the uncentered maximal function operator and the almost centered maximal function operator. In addition, we also present the Bellman function approach of Melas, Nikolidakis and Stavropoulos to maximal function operators defined over various types of families of sets, and in case of parallelepipeds we will show that $A(n,p)=\left(\frac{p}{p-1}\right)^{1/p}$.

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preprint2015arXiv

Bellman function for extremal problems in $\mathrm{BMO}$ II: evolution

In the paper "Bellman function for extremal problems in $\mathrm{BMO}$", the authors built the Bellman function for integral functionals on the $\mathrm{BMO}$ space. The present paper provides a development of the subject. We abandon the majority of unwanted restrictions on the function that generates the functional. It is the new evolutional approach that allows us to treat the problem in its natural setting. What is more, these new considerations lighten dynamical aspects of the Bellman function, in particular, evolution of its picture.

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preprint2015arXiv

Bellman partial differential equation and the hill property for classical isoperimetric problems

The goal of this note is to have a systematic approach to generating isoperimetric inequalities from two concrete type of PDEs. We call these PDEs Bellman type because a totally analogous equations happen to rule many sharp estimates for singular integrals in harmonic analysis, and such estimates were obtained with the use of Hamilton--Jacobi--Bellman PDE. We show how classical inequalities of Brascamp--Lieb, Prekopa--Leindler, Ehrhard are particular case of this scheme, which allows us to augment the stock of such inequalities. We approach the isoperimetric inequalities as a maximum (minimum) principle for special types of functions. These functions are compositions of "Bellman function" and an appropriate flow built on test functions. Then the existence of maximum (minimum) principle for such compositions can be reduced to the requirement that Bellman function satisfies a concrete class of nonlinear PDE (written down below). We are left to solve this nonlinear PDE (sometimes a possible task) to enjoy isoperimetric inequalities. The nonlinear PDE that we will describe in this article can be reduced sometimes to solving Laplacian eigenvalue problem, $\bar{\partial}$-equation of certain type or just the linear heat equation.

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preprint2014arXiv

Sharp estimates of integral functionals on classes of functions with small mean oscillation

We unify several Bellman function problems into one setting. For that purpose we define a class of functions that have, in a sense, small mean oscillation (this class depends on two convex sets in $\mathbb{R}^2$). We show how the unit ball in the $\mathrm{BMO}$ space, or a Muckenhoupt class, or a Gehring class can be described in such a fashion. Finally, we consider a Bellman function problem on these classes, discuss its solution and related questions.

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preprint2012arXiv

Bellman function for extremal problems in BMO

In this paper we develop the method of finding sharp estimates by using a Bellman function. In such a form the method appears in the proofs of the classical John--Nirenberg inequality and $L^p$ estimations of BMO functions. In the present paper we elaborate a method of solving the boundary value problem for the homogeneous Monge--Ampère equation in a parabolic strip for sufficiently smooth boundary conditions. In such a way we have obtained an algorithm of constructing an exact Bellman function for a large class of integral functionals in the BMO space.

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