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Paata Ivanisvili

Publications and source records attributed to Paata Ivanisvili.

At least 19 recordsLinked to original sources
arXiv2026

The missing two-point inequality in Weissler's conjecture

Weissler's conjectured characterization of complex hypercontractivity on the Hamming cube remained open in the ranges $2<p\le q<3$ and $\frac32<p\le q<2$. Ivanisvili--Nazarov established the diagonal cases. We prove the remaining strict off-diagonal cases through the corresponding two-point inequality. Our argument gives a self-contained proof for every $2<p\le q<\infty$ and, by duality, for every $1<p\le q<2$, including the previously known diagonal cases. The proof is based on a flow argument.

math.FA↗
arXiv2026

Polynomial growth of Bohnenblust--Hille constants on the Hamming cube

We prove that the Bohnenblust--Hille constants for Walsh polynomials on the Hamming cube grow at most polynomially in the degree. More precisely, there is an absolute constant $K$ such that every $f :\{-1,1\}^{n} \to \mathbb{C}$ of degree at most $m$ satisfies $$ \left(\sum_{|S|\le m}|\widehat f(S)|^{2m/(m+1)}\right)^{(m+1)/(2m)} \le Km^{27}\|f\|_\infty. $$ The estimate is uniform in the dimension. The exponent $27$ is not optimized.

math.FA↗
arXiv2026

Sharp isoperimetric inequalities on the Hamming cube near the critical exponent

An isoperimetric inequality on the Hamming cube for exponents $β\ge 0.50057$ is proved, achieving equality on any subcube. This was previously known for $β\ge \log_2(3/2)\approx 0.585$. Improved bounds are also obtained at the critical exponent $β=0.5$, including a bound that is asymptotically sharp for small subsets. A key ingredient is a new Bellman-type function involving the Gaussian isoperimetric profile which appears to be a good approximation of the true envelope function. Verification uses computer-assisted proofs and interval arithmetic. Applications include progress towards a conjecture of Kahn and Park as well as sharp Poincaré inequalities for Boolean-valued functions near $L^1$.

math.CA↗
arXiv2026

The Hayman--Wu constant is $π^2$

We show that for every conformal map $ϕ:\mathbb{D}\toΩ\subsetneq\mathbb{C}$ from the unit disk onto a simply connected proper domain $Ω$, the length of $ϕ^{-1}(Ω\cap L)$ is at most $π^2$ for every line $L$.

math.CV↗
arXiv2026

A counterexample to the Kato conjecture for positive commutators

We disprove the conjectural converse to Kato's positivity criterion for commutators of functions of the canonical position and momentum operators $Q$ and $P$ by showing that the operator \[ i\,[\,\arctan(P),\,\arctan(Q)\,] \] is nonnegative and nonzero.

math.FA↗
arXiv2026

The Winding Number at the Critical Hölder Exponent 1/3: Failure of Universal Fourier Summation

The degree (or winding number) of a sufficiently regular map $f:\mathbb{T} \to \mathbb {S^1}$ is given in terms of its Fourier coefficients by $$ \operatorname{deg} f = \sum_{n\in\mathbb{Z}} n\,|\widehat{f}(n)|^2. $$ At lower regularity the series may diverge, but, as shown by Kahane, when $f$ is $α$-Hölder continuous with $α>1/3$, then the degree can be recovered by a universal linear summation process. We show that no summation process satisfying Brezis's natural axioms can recover the degree universally for $α=1/3$, thereby resolving an open problem by Brezis.

math.CA↗
arXiv2026

A Beckmann boundary form of Talagrand's conjecture on the discrete cube

We introduce the Beckmann boundary of a Boolean function \[ \mathsf{B}(f)=\inf_{\operatorname{div} V=Lf}\mathbb E\|V(x)\|_2. \] Here \[ L=\sum_iD_i,\qquad D_i f(x)=\frac{f(x)-f(x^{\oplus i})}{2}, \] and $\operatorname{div} V(x)=\sum_i (V_{i}(x)-V_{i}(x^{\oplus i}))$. This nonlocal quantity is no larger than the usual two-sided, one-sided, colored, optimized colored, or optimized fractional colored boundaries. Nevertheless, every nonconstant Boolean $f$ satisfies \[ \mathsf{B}(f)\gtrsim \operatorname{Var}(f) \sqrt{\log\!\left(1+\frac{1}{\sum_i\operatorname{Inf}_i(f)^2}\right)}. \] We also prove strong one-sided fractional spectral estimates. If $A\subset\{-1,1\}^n$ and \[ h_{A}(x)=\#\{i:x\in A,\ x^{\oplus i}\notin A\}, \] then, for $0<α<1$, \[ \sum_{S\ne\varnothing}|S|^α\widehat{\mathbf 1_{A}}(S)^2 \lesssim_α\mathbb Eω_α(h_{A}), \] where $ω_α(m)=\sqrt m$ for $α<1/2$, $ω_{1/2}(m)=\sqrt m\log(e+m)$, and $ω_α(m)=m^α$ for $α>1/2$. These profiles are sharp, up to $α$-dependent constants, for majority. We also show that the comparison is genuinely nonreversible: an explicit quotient-cube family makes the optimized fractional, and hence optimized colored, boundary exceed $\mathsf{B}$ by a factor $\gtrsim\sqrt{\log n}$. We further obtain a driftless Bernstein-multiplier inequality.

math.CA↗
arXiv2026

Minimizers for Coulomb gases constrained to a halfspace

We consider a family of optimization problems, based on a mean-field description of particles interacting through Coulomb forces in a quadratic trap. In addition, the particles are constrained to lie in a halfspace and we are interested in the way the particle distribution changes as the halfspace varies. In particular, we can prove the existence of a phase transition, thereby settling a recent conjecture by Byun, Forrester, Majumdar and Schehr.

math.AP↗
arXiv2026

The distance between homotopy classes of Sobolev maps on spheres

We consider self-maps of a sphere in the critical Sobolev space with a given Brouwer degree. Our main result is that the (directed) distance between maps of different degrees is equal to an explicit constant times the difference in degrees. In the case of the 2-sphere this resolves an open problem by Brezis.

math.AP↗
arXiv2026

Geometric Block Exponents and a Uniform Mixed-Alphabet Sumset Inequality

We study sharp exponents in inequalities for pairs of finite geometric blocks. We characterize exactly when the endpoint $t=1$ determines the optimal exponent and compute the exponent that is uniform in the length of one block. For a two-term first block the answer is $p_0=\log 4/\log 6$. This yields a uniform two-slice max-convolution inequality and, for every $m,d\ge1$, the dimension-free mixed-alphabet sumset bound \[ |A+B|\ge (|A||B|)^{p_0}, \qquad A\subset\{0,1\}^d,\quad B\subset\{0,1,\ldots,m\}^d. \] For every $m\ge2$, the exponent $p_0$ is best possible; for $m=1$, a larger exponent is available.

math.CO↗
arXiv2026

Dimension-free bounds for Riesz transforms on the Hamming cube via a Bellman function

We give a Bellman-function proof of the dimension-free estimate \[ \Big\| \vec{R} f \Big\|_{L^p(Ω;\,\ell^2)} \lesssim (p-1) \,\|f\|_{L^p(Ω)}, \qquad 2\le p<\infty, \] for the vector of Riesz transforms associated with the Walsh number operator on the Hamming cube $Ω=\{-1,1\}^n$, as well as for locally compact abelian groups, in particular $Ω=\mathbb{Z}^n$. The argument is based on a Poisson semigroup representation, symmetrized estimates along edges of $Ω$, and a two-point inequality. This is the first non noncommutative proof of this result, after the seminal papers of Lust-Piquard and later Junge-Mei-Parcet. According to an example of Lamberton, for $1<p<2$ such a dimension-free bound is known to be false.

math.FA↗
arXiv2026

The sharp extension norm for a planar sector

We compute explicitly the infimum of the norms of $W^{1,2}$-extension operators for planar sectors and exhibit an extension operator attaining this infimum. This solves an open problem posed by Maz'ya.

math.FA↗
arXiv2026

Carbery's inequality in the Schatten--von Neumann classes

Carbery posed a question of sharpened triangle inequalities for families of operators in the Schatten--von Neumann classes $S_p$, $p\geq 2$. He established a weaker form of the desired estimate for even integer values of $p$. In the commutative setting the corresponding sharp inequality (with optimal exponent $p'=\frac{p}{p-1}$) was recently obtained for all integer $p\geq 2$. In the present work we resolve Carbery's question completely in the non-commutative setting: we prove the sharp inequality $$ \Big\|\sum_{j} T_{j}\Big\|_{S_{p}}\leq \|(α_{ij}^{p'})\|^{1/p'}_{\ell_{2}\to \ell_{2}} \Big( \sum_{j} \|T_{j}\|^{p}_{S_{p}}\Big)^{1/p} $$ for all $p\geq 2$ and all countable sequences of operators $(T_{j}) \subset S_{p}$, where $α_{ij}$ are almost orthogonality coefficients. The proof is based on a block-operator reduction and a complex interpolation of the polar parts of the blocks.

math.FA↗
arXiv2026

The sharp log Sobolev inequality on finite cycles

We settle the problem of finding the sharp constant in the log Sobolev inequality on the $n$-cycle for all $n\ge 4$, by showing that it is equal to half of the spectral gap. We deduce this result from an optimal cubic Sobolev inequality.

math.AP↗
arXiv2026

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.

math.CA↗