Related papers: Conjugate Transforms on Dyadic Group

Let $n \in \mathbb{Z}_{\geq 3}$ be given. We prove Lebesgue-almost everywhere pointwise inversion formulae for the Siegel transforms in the geometry of numbers. These inversion formulae are quite general; for instance, they are valid for…

Let $K_n(x)$ denote the Fej\'er kernel given by $$K_n(x)=\sum_{j=-n}^n\left(1-\frac{|j|}{n+1}\right)e^{-ijx}$$ and let $\sigma_nf(x)=(K_n\ast f)(x)$, where as usual $f\ast g$ denotes the convolution of $f$ and $g$. Let the sequence…

Let $E$ be an arbitrary closed set on the unit circle $\partial \mathbb{D}$, u be a harmonic function on the unit disk $\mathbb{D}$ satisfying $|u(z)|\lesssim (1-|z|)^\gamma \rho^{-q}(z)$ where $\rho(z)= \mathop{\rm dist}(z, E)$, $\gamma$,…

We study fluctuations of ergodic averages generated by actions of amenable groups. In the setting of an abstract ergodic theorem for locally compact second countable amenable groups acting on uniformly convex Banach spaces, we deduce a…

We establish sharp upper bounds for shifted moments of quadratic Dirichlet $L$-function under the generalized Riemann hypothesis. Our result is then used to prove bounds for moments of quadratic Dirichlet character sums.

We prove the following higher-order Szego theorems: if a measure on the unit circle has absolutely continuous part $w(\theta)$ and Verblunsky coefficients $\alpha$ with square-summable variation, then for any positive integer $m$, $\int…

We derive explicit bounds for the computation of normalizing constants $Z$ for log-concave densities $\pi = \exp(-U)/Z$ with respect to the Lebesgue measure on $\mathbb{R}^d$. Our approach relies on a Gaussian annealing combined with recent…

An embedding theorem for Sobolev spaces built upon general Musielak-Orlicz norms is offered. These norms are defined in terms of generalized Young functions which also depend on the $x$ variable. Under minimal conditions on the latter…

We investigate the extreme values of the Riemann zeta function $\zeta(s)$. On the 1-line, we obtain a lower bound evaluation $$\max_{t\in[1,T]}|\zeta(1+\i t)|\ge {\rm e}^\gamma(\log_2T+\log_3T+c),$$ with an effective constant $c$ which…

We consider ergodic families of Verblunsky coefficients generated by minimal aperiodic subshifts. Simon conjectured that the associated probability measures on the unit circle have essential support of zero Lebesgue measure. We prove this…

The finite Hilbert transform $T$, when acting in the classical Zygmund space $\logl$ (over $(-1,1)$), was intensively studied in \cite{curbera-okada-ricker-log}. In this note an integral representation of $T$ is established via the…

For a Schwartz function $f$ on the plane and a non-zero $v\in\ZR^2$ define the Hilbert transform of $f$ in the direction $v$ to be $$ H_vf(x)=\text{p.v.}\int_\ZR f(x-vy) \frac{dy}y $$ Let $\zeta$ be a Schwartz function with frequency…

We first prove that the Legendre transform is the only continuous and $\mathrm{SL}(n)$ contravariant valuation that behaves as a conjugation of two important translations on super-coercive, lower semi-continuous, and convex functions. Then…

We study weighted Walsh--Carleson maximal operators arising from dyadic martingale transforms associated with Walsh--Fourier partial sums. For weights satisfying a uniform dyadic variation condition and a uniform bound at the top dyadic…

Let $\alpha \in (0,2)$ and let $\beta>0$. Fix $-\pi<\varphi\leq \pi$ such that $|\varphi|>\alpha \pi/2$. We obtain asymptotic upper bounds on the Fourier transform of the radially symmetric tempered distribution \begin{equation*}…

Let $f_{\bf c}(r)=\sum_{n=0}^\infty e^{c_n}r^n$ be an analytic function; ${\bf c}=(c_n)\in l_\infty$. We assume that $r$ is some logarithmically convex and lower semicontinuous functional on a locally convex topological space $L$. In this…

We explore the optimality of the constants making valid the recently established Little Grothendieck inequality for JB$^*$-triples and JB$^*$-algebras. In our main result we prove that for each bounded linear operator $T$ from a…

We give an improvement of sharp Berezin type bounds on the Riesz means $\sum_k(\Lambda-\lambda_k)_+^\sigma$ of the eigenvalues $\lambda_k$ of the Dirichlet Laplacian in a domain if $\sigma\geq 3/2$. It contains a correction term of the…

Let $\Omega$ be a Polish space with Borel $\sigma$-field $\mathcal{F}$ and countably generated sub $\sigma$-field $\mathcal{G}\subset\mathcal{F}$. Denote by $\mathcal{L}(\mathcal{F})$ the set of all bounded $\mathcal{F}$-upper semianalytic…

This work proves pointwise convergence of the truncated Fourier double integral of non-Lebesgue integrable bounded variation functions. This leads to the Dirichlet-Jordan theorem proof for non-Lebesgue integrable functions, which has not…