Related papers: Two footnotes to the F. & M. Riesz theorem

In this note we study the problem of evaluating the trace of $f(T)-f(R)$, where $T$ and $R$ are contractions on Hilbert space with trace class difference, i.e., $T-R\in\boldsymbol{S}_1$ and $f$ is a function analytic in the unit disk ${\Bbb…

Let $\mathcal{A}$ denote the set of all analytic functions $f$ in the unit disk $\ID=\{z:\,|z|<1\}$ of the form $f(z)=z+\sum_{n=2}^{\infty}a_nz^n.$ Let $\mathcal{U}$ denote the set of all $f\in \mathcal{A}$, $f(z)/z\neq 0$ and satisfying…

We consider a class of non-conformal expanding maps on the $d$-dimensional torus. For an equilibrium measure of an H\"older potential, we prove an analogue of the Central Limit Theorem for the fluctuations of the logarithm of the measure of…

Given $E \subset \mathbb{R}^d$, $d \ge 2$, define ${\mathcal D}(E) \equiv {(x-y)/|x-y|: x,y \in E} \subset S^{d-1},$ the set of directions determined by $E$. We prove that if the Hausdorff dimension of $E$ is greater than $d-1$, then…

If $f$ is a real-valued function on $[-\pi,\pi]$ that is Henstock--Kurzweil integrable, let $u_r(\theta)$ be its Poisson integral. It is shown that $\|u_r\|_p=o(1/(1-r))$ as $r\to 1$ and this estimate is sharp for $1\leq p\leq\infty$. If…

\begin{abstract} Let $P\pm$ be the Riesz's projection operator and let $P_-= I - P_+$. We consider estimates of the expression $\|( |P_ + f | ^s + |P_- f |^s) ^{\frac{1}{s}}\|_{L^p (\mathbf{T})}$ in terms of Lebesgue $p$-norm of the…

Let $f$ be a holomorphic function on the unit disc, and $(S_{n_{k}})$ be a subsequence of its Taylor polynomials about $0$. It is shown that the nontangential limit of $f$ and lim$_{k\rightarrow \infty }S_{n_{k}}$ agree at almost all points…

Let $f = P[F]$ denote the Poisson integral of $F$ in the unit disk $\mathbb{D}$ with $F$ is an absolute continuous in the unit circle $\mathbb{T}$ and $\dot{F}\in L^p(\mathbb{T})$, where $\dot{F}(e^{it}) = \frac{d}{dt} F(e^{it})$ and $p \in…

The Ricci version of the Schur theorem is shown to hold for a wide class of Finsler metrics. What is more, let $F$ be any (positive definite) Finsler metric such that $\text{Ric} =\rho F^2$ with $\rho\colon M^n\rightarrow\mathbb{R}$ (i.e.,…

The Riesz-Markov theorem identifies any positive, finite, and regular Borel measure on the complex unit circle with a positive linear functional on the continuous functions. By the Weierstrass approximation theorem, the continuous functions…

Let $\mathbb{B}$ be the unit disc in $\mathbb{R}^2$, $\mathscr{H}$ be the completion of $C_0^\infty(\mathbb{B})$ under the norm $$\|u\|_{\mathscr{H}}=\left(\int_\mathbb{B}|\nabla…

In this paper we prove the following theorem: Suppose that $f_1,f_2\in H^\infty_\R(\D)$, with $\norm{f_1}_\infty,\norm{f_2}_{\infty}\leq 1$, with $$ \inf_{z\in\D}(\abs{f_1(z)}+\abs{f_2(z)})=\delta>0. $$ Assume for some $\epsilon>0$ and…

Let function $f$ be analytic in the unit disk ${\mathbb D}$ and be normalized so that $f(z)=z+a_2z^2+a_3z^3+\cdots$. In this paper we give sharp bounds of the modulus of its second, third and fourth coefficient, if $f$ satisfies \[…

We prove the following correction theorem: every function $f$ on the circumference $\mathbb{T}$ that is bounded by the $\alpha_1$-weight $w$ (this means that $Mw^2 \leq C w^2$) can be modified on a set $e$ with $\int\limits_{e} w \leq \eps$…

Let $\mathcal{A}$ denote the class of analytic functions in the unit disk $\mathbb{D}:=\{z\in\mathbb{C}:|z|<1\}$ satisfying $f(0)=0$ and $f'(0)=1$. Let $\mathcal{U}$ be the class of functions $f\in\mathcal{A}$ satisfying…

Let $D\in\mathbb{N}$, $q\in[2,\infty)$ and $(\mathbb{R}^D,|\cdot|,dx)$ be the Euclidean space equipped with the $D$-dimensional Lebesgue measure. In this article, via an auxiliary function space $\mathrm{WE}^{1,\,q}(\mathbb R^D)$ defined…

Let $(R,\mathfrak{m})$ be a Noetherian local ring of dimension $d\geq 2$. We prove that if $e(\widehat{R}_{red})>1$, then the classical Lech's inequality can be improved uniformly for all $\mathfrak{m}$-primary ideals, that is, there exists…

Let $f$ be a conformal (analytic and univalent) map defined on the open unit disk $\D$ of the complex plane $\IC$ that is continuous on the semi-circle $\partial \D^{+}=\{z\in\IC:|z|=1, {\rm{Im}}\,z>0\}$. The existence of a uniform upper…

We generalize a previous inequality related to a sharp version of the Littlewood conjecture on the minimal $L_1$-norm of $N$-term exponential sums $f$ on the unit circle. The new result concerns replacing the expression $\log(1+t|f|^2)$…

Two subanalytic subsets of $ \mathbb R^n$ are called $s$-equivalent at a common point $P$ if the Hausdorff distance between their intersections with the sphere centered at $P$ of radius $r$ vanishes to order $>s$ as $r$ tends to $0$. In…