Related papers: On bilinear Hilbert transform along two polynomial…

We address some questions concerning indecomposable polynomials and their behaviour under specialization. For instance we give a bound on a prime $p$ for the reduction modulo $p$ of an indecomposable polynomial $P(x)\in \Zz[x]$ to remain…

We establish tight inapproximability bounds for max-LINSAT, the problem of maximizing the number of satisfied linear constraints over the finite field $\mathbb{F}_q$, where each constraint accepts $r$ values. Specifically, we prove by a…

We give tight bounds on the degree $\ell$ homogenous parts $f_\ell$ of a bounded function $f$ on the cube. We show that if $f: \{\pm 1\}^n \rightarrow [-1,1]$ has degree $d$, then $\| f_\ell \|_\infty$ is bounded by $d^\ell/\ell!$, and $\|…

Given a polynomial \( H(x) \) over \(\mathbb{F}_{q^n}\), we study permutation polynomials of the form \( x + \gamma \mathrm{Tr}(H(x)) \) over \(\mathbb{F}_{q^n}\). Let \[P_H=\{\gamma\in \mathbb{F}_{q^n} : x+\gamma \mathrm{Tr}(H(x))~\text{is…

In this paper we study the $L^p$-$L^q$ boundedness of the Fourier multipliers in the setting where the underlying Fourier analysis is introduced with respect to the eigenfunctions of an anharmonic oscillator $A$. Using the notion of a…

We define a discrete version of the bilinear spherical maximal function, and show bilinear $l^{p}(\mathbb{Z}^d)\times l^{q}(\mathbb{Z}^d) \to l^{r}(\mathbb{Z}^d)$ bounds for $d \geq 3$, $\frac{1}{p} + \frac{1}{q} \geq \frac{1}{r}$,…

Commutators of a large class of bilinear operators and multiplication by functions in a certain subspace of the space of functions of bounded mean oscillations are shown to be jointly compact. Under a similar commutation, fractional…

Let $n\ge 1$ and $\mathfrak{T}_{m}$ be the bilinear square Fourier multiplier operator associated with a symbol $m$, which is defined by $$ \mathfrak{T}_{m}(f_1,f_2)(x) = \biggl( \int_{0}^\infty\Big|\int_{(\mathbb{R}^n)^2} e^{2\pi ix\cdot…

We introduce a new class of holomorphic polynomials extending the classical Gould--Hopper to two complex variables. The considered polynomials include the $1$-D and $2$-D holomorphic and polyanalytic It\^o--Hermite polynomials as particular…

We derive upper bounds for probabilities of the form $P(g(\mathbf{X})\geq t)$ using the southwest boundary (recently introduced in our previous work) $\partial_{\mathrm{SW}} Q(g^{-1}[t,\infty))$, where $Q$ is a reflection to the first…

For exponents $p,q\in (1,\infty),$ we study the $L^p$-to-$L^q$ boundedness and compactness of the commutator $[b,H_{\gamma}] = bH_{\gamma} - H_{\gamma}b,$ where $H_{\gamma}$ is the Hilbert transform along the monomial curve $\gamma$ and the…

We present a new form and a short full proof of explicit two-sided estimates for the distribution function F_{n,p}(x) of the binomial law from the paper published by D.Alfers and H.Dinges in 1984. These inequalities are universal (valid for…

Let $q$ be an odd prime power and let $X(m,q)$ be the set of symmetric bilinear forms on an $m$-dimensional vector space over $\mathbb{F}_q$. The partition of $X(m,q)$ induced by the action of the general linear group gives rise to a…

We present a new effective Nullstellensatz with bounds for the degrees which depend not only on the number of variables and on the degrees of the input polynomials but also on an additional parameter called the {\it geometric degree of the…

Let $f$ be a diagonal hypersurface in $A_p=\mathbb{F}_p[[x_1,\dots,x_n]]$. We study the behavior of the function $\phi_{f,p}({a}/{p^e})=p^{-ne}\dim_{\mathbb{F}_p}\big(A_p/(x_1^{p^e},\dots,x_n^{p^e},f^a)\big)$ which encodes information about…

For the bi-orthogonal polynomials with the third degree polynomial potential functions, the 3 x 3 matrix Riemann-Hilbert problem is explicitly constructed. The developed approach admits an extension to the bi-orthogonal polynomials with…

A result of Gilbert shows that every completely bounded multiplier $f$ of the Fourier algebra $A(G)$ arises from a pair of bounded continuous maps $\alpha,\beta:G \rightarrow K$, where $K$ is a Hilbert space, and $f(s^{-1}t) =…

We show that if the dyadic Hilbert transform with values in a Banach space is $L^p$ bounded, then so is the Hilbert transform, with a linear relation of the bounds. This result is the counterpart of [arXiv:2212.00090] where the opposite…

We complete the $L^p$ boundedness theory of commutators of Hilbert transforms along monomial curves by providing the previously missing lower bounds. This optimal result now covers all monomial curves while previous results had significant…

In this paper, we use the unified transform method to consider the initial-boundary value problem for the coupled Fokas-Lenells equations on the half-line, assuming that the solution $\{q(x,t),r(x,t)\}$ of the coupled Fokas-Lenells…