Related papers: A Sharp inequality for holomorphic functions on th…

We prove a contractive Hardy-Littlewood type inequality for functions from $H^p(\mathbb{T})$, $0 < p \le 2$ which is sharp in the first two Taylor coefficients and asymptotically at infinity.

It is a classical result that every subharmonic function, defined and ${\mathcal{L}}^p$-integrable for some $p$, $0<p<+\infty$, on the unit disk $\mathbb{D}$ of the complex plane ${\mathbb{C}}$ is for almost all $\theta$ of the form $o((1-|…

We present a natural family of Hilbert function spaces on the d-dimensional complex unit ball and classify which of them satisfy that subsets of the ball yield isometrically isomorphic subspaces if and only if there is an analytic…

In this study, we partially answer the question left open in Rudin's book "Function theory in polydiscs" on the structure of invariant subspaces of the Hardy space $H^2(U^n)$ on the polydisc $U^n$. We completely describe all invariant…

It is proved an inequality - integrated analogue of the Hardy inequality and as application simplified proof of the theorem of S. A. Vinogradov for the bounded Toeplitz operators on the space of functions analytic and bounded in the unit…

Let $\alpha>-1$ and assume that $f$ is $\alpha-$harmonic mapping defined in the unit disk that belongs to the Hardy class $h^p$ with $p\ge 1$. We obtain some sharp estimates of the type $|f(z)|\le g(|r|) \|f^\ast\|_p$ and $|Df(z)|\le…

The Patterson-Sullivan construction is proved almost surely to recover a Bergman function from its values on a random discrete subset sampled with the determinantal point process induced by the Bergman kernel on the unit ball $\mathbb{D}_d$…

Morrey's classical inequality implies the H\"older continuity of a function whose gradient is sufficiently integrable. Another consequence is the Hardy-type inequality $$ \lambda\biggl\|\frac{u}{d_\Omega^{1-n/p}}\biggr\|_{\infty}^p\le…

A new weighted Hardy-type inequality for functions from the Sobolev space $W_{p}^{1}$ is proved. It is assumed that functions vanish on small alternating pieces of the boundary. The proved inequality generalizes the classical known weighted…

Let $n$ be a positive integer. Let $\mathbf U$ be the unit disk, $p\ge 1$ and let $h^p(\mathbf U)$ be the Hardy space of harmonic functions. Kresin and Maz'ya in a recent paper found the representation for the function $H_{n,p}(z)$ in the…

In this paper we obtain some sharp Hardy inequalities with weight functions that may admit singularities on the unit sphere. In order to prove the main results of the paper we use some recent sharp inequalities for the lowest eigenvalue of…

We study Hardy classes on the disk associated to the equation $\bar\d w=\alpha\bar w$ for $\alpha\in L^r$ with $2\leq r<\infty$. The paper seems to be the first to deal with the case $r=2$. We prove an analog of the M.~Riesz theorem and a…

We~identify the standard weighted Bergman kernels of spaces of nearly holomorphic functions, in~the sense of Shimura, on~bounded symmetric domains. This also yields a description of the analogous kernels for spaces of…

This is the second in our series of papers concerning some reversed Hardy--Littlewood--Sobolev inequalities. In the present work, we establish the following sharp reversed Hardy--Littlewood--Sobolev inequality on the half space $\mathbb…

There are at least two directions concerning the extension of classical sharp Hardy-Littlewood-Sobolev inequality: (1) Extending the sharp inequality on general manifolds; (2) Extending it for the negative exponent $\lambda=n-\alpha$ (that…

We improve the classical discrete Hardy inequality for $ 1<p<\infty $ for functions on the natural numbers. For integer values of $ p $ the Hardy weight is an absolutely monotonic function.

n this article we consider functions meromorphic in the unit disk. We give an elementary proof for a condition that is sufficient for the univalence of such functions which also contains some known results. We include few open problems for…

In this paper we establish the reversed sharp Hardy-Littlewood-Sobolev (HLS for short) inequality on the upper half space and obtain a new HLS type integral inequality on the upper half space (extending an inequality found by Hang, Wang and…

We deal with weighted Hardy-Sobolev type inequalities for functions on $\mathbb{R}^d$, $d\geq 2$. The weights involved are anisotropic, given by products of powers of the distance to the origin and to a nontrivial subspace. We establish…

The definition of classical holomorphic function spaces such as the Hardy space or the Dirichlet space on the Hartogs triangle is not canonical. In this paper we introduce a natural family of holomorphic function spaces on the Hartogs…