Related papers: A Counterexample to a Directed KKL Inequality
We investigate what we term "generalized sup-convolutions". We show that functional inequalities that enjoy an interpretation as sup-convolution inequalities can be deduced from the special case of indicator functions corresponding to a…
We establish a family of coercive Korn-type inequalities for generalised incompatible fields in the superlinear growth regime under sharp criteria. This extends and unifies several previously known inequalities that are pivotal to the…
In this paper, we show a weighted Hardy inequality in a limiting case for functions in weighted Sobolev spaces with respect to an invariant measure. We also prove that the constant in the left-hand side of the inequality is optimal. As…
We show that the following generalized version of Korn's second inequality with nonconstant measurable matrix valued coefficients P ||DuP+(DuP)^T||_q+||u||_q >= c ||Du||_q for u in W_0^{1,q}({\Omega};R^3), 1<q<{\infty} is in general false,…
We prove that the Harnack inequality fails for nonlocal kinetic equations. Such equations arise as linearized models for the Boltzmann equation without cutoff and are of hypoelliptic type. We provide a counterexample for the simplest…
It is disproved the Tokareva's conjecture that any balanced boolean function of appropriate degree is a derivative of some bent function. This result is based on new upper bounds for the numbers of bent and plateaued functions.
This is a continuation of our previous work 0712.4092. It is well known that various isoperimetric inequalities imply their functional ``counterparts'', but in general this is not an equivalence. We show that under certain convexity…
The hypercontractive inequality on the discrete cube plays a crucial role in many fundamental results in the Analysis of Boolean functions, such as the KKL theorem, Friedgut's junta theorem and the invariance principle. In these results the…
We prove that the isoperimetric inequalities in the euclidean and hyperbolic plane hold for all euclidean, respectively hyperbolic, cone-metrics on a disk with singularities of negative curvature. This is a discrete analog of the theorems…
Generalized trigonometric functions and generalized hyperbolic functions can be converted to each other by the duality formulas previously discovered by the authors. In this paper, we apply the duality formulas to prove dual pairs of…
Harris's correlation inequality states that any two monotone functions on the Boolean hypercube are positively correlated. Talagrand \cite{Talcorr} started a line of works in search of quantitative versions of this fact by providing a lower…
We extend the De Giorgi--Nash--Moser theory to a class of kinetic Fokker-Planck equations and deduce new results on the Landau-Coulomb equation. More precisely, we first study the H{\"o}lder regularity and establish a Harnack inequality for…
We study two generalizations of the Rudin-Keisler ordering to ultrafilters on complete Boolean algebras. To highlight the difference between them, we develop new techniques to construct incomparable ultrafilters in this setting.…
We prove a new inequality which improves on the classical Hardy inequality in the sense that a nonlinear integral quantity with super-quadratic growth, which is computed with respect to an inverse square weight, is controlled by the energy.…
We prove a sharp inequality conjectured by Bobkov on the measure of dilations of Borel sets in $\mathbb{R}^n$ by a $s$-concave probability. Our result gives a common generalization of an inequality of Nazarov, Sodin and Volberg and a…
In this paper we establish higher-order Sobolev and Rellich-type inequalities on non-compact Riemannian manifolds supporting an isoperimetric inequality. We highlight two notable settings: manifolds with non-negative Ricci curvature and…
Bohnenblust--Hille inequalities for Boolean cubes have been proven with dimension-free constants that grow subexponentially in the degree \cite{defant2019fourier}. Such inequalities have found great applications in learning low-degree…
It follows from a general theorem of Bonk and Eremenko that closed plane curves which are contractible in the complement to the integral lattice satisfy a linear isoperimetric inequality. We give an alternative proof of this fact. Our…
In a recent paper, we presented a new definition of influences in product spaces of continuous distributions, and showed that analogues of the most fundamental results on discrete influences, such as the KKL theorem, hold for the new…
In this comment, I point out a number of approximated derivations for the effective equation of motion, now been applied to d-wave superconductors by Kopnin and Volovik are invalid. The major error in those approximated derivations is the…