Weighted Norm Inequalities for the Hardy Operator

In this paper.a characterizationis,obtained for those pairs of weight funetions on(0=∞) for which the Hardy operator Pf(x)=f(t)dt is bounded from (μ) to ,0相似文献   

Asymptotic approximations to the Hardy–Littlewood function

The function Q(x):=_∑n≥1(1/n)sin(x/n)$Q\left(x\right):={\sum }_{n\ge 1}(1/n)sin(x/n)$ was introduced by Hardy and Littlewood (1936) [5] in their study of Lambert summability, and since then it has attracted attention of many researchers. In particular, this function has made a surprising appearance in the recent disproof by Alzer, Berg and Koumandos (2005) [3] of a conjecture by Clark and Ismail (2003) [14]. More precisely, Alzer et al. have shown that the Clark and Ismail conjecture is true if and only if Q(x)≥−π/2$Q\left(x\right)\ge -\pi /2$ for all x>0$x>0$. It is known that Q(x)$Q\left(x\right)$ is unbounded in the domain x∈(0,∞)$x\in (0,\infty )$ from above and below, which disproves the Clark and Ismail conjecture, and at the same time raises a natural question of whether we can exhibit at least one point x$x$ for which Q(x)<−π/2$Q\left(x\right)<-\pi /2$. This turns out to be a surprisingly hard problem, which leads to an interesting and non-trivial question of how to approximate Q(x)$Q\left(x\right)$ for very large values of x$x$. In this paper we continue the work started by Gautschi (2005) in [4] and develop several approximations to Q(x)$Q\left(x\right)$ for large values of x$x$. We use these approximations to find an explicit value of x$x$ for which Q(x)<−π/2$Q\left(x\right)<-\pi /2$.     

Hardy’s Inequalities for the Heisenberg Group

Potential Analysis - This article presents two types of Hardy’s inequalities for the Heisenberg group. The proofs are mainly based on estimates of the Fourier transform for atomic functions...     

On the quasilinear elliptic problem with a critical Hardy–Sobolev exponent and a Hardy term

In the present paper, a quasilinear elliptic problem with a critical Sobolev exponent and a Hardy-type term is considered. By means of a variational method, the existence of nontrivial solutions for the problem is obtained. The result depends crucially on the parameters p,t,s,λ$p,t,s,\lambda$ and μ$\mu$.     

Poletsky–Stessin–Hardy Spaces in the Plane

In this paper we give two characterizations of the Poletsky–Stessin–Hardy spaces in the complex plane: First we completely describe functions in these spaces by having a harmonic majorant with a certain growth condition and we prove some basic results about these spaces. Second we describe these functions in terms of their boundary values as a weighted subclass of the usual $L^p$ class with respect to the arclength measure on the boundary, when the boundary is $C^2$ . In particular, we extend the classical result of Beurling which describes the invariant subspaces of the shift operator. Additionally we provide non-trivial examples.     

Hardy’s Inequality for the n-Dimensional Hankel Transform

Analysis Mathematica - The purpose of this paper is to prove a Hardy type inequality associated with the n-dimensional Hankel transform (n ≥ 1) for the exponent σ0 = n(2 ? p) +...     

A representing system generated by the Szegö kernel for the Hardy space

In this paper we give an explicit construction of a representing system generated by the Szegö kernel for the Hardy space. Thus we answer an open question posed by Fricain, Khoi and Lefèvre. We use frame theory to prove the main result.     

Hardy–Littlewood–Sobolev inequality on the parabolic biangle

The Ramanujan Journal - We define the heat semigroup associated with a system of bivariate Jacobi polynomials which are orthogonal with respect to a probability measure on the parabolic biangle...     

Hoermander＇s Multipliers for the Weighted Herz-type Hardy Spaces

In this article, we apply the molecular characterization of the weighted Hardy space developed by the first two authors to show the boundedness of Hormander multiplier on the weighted Herz-type Hardy spaces HK^α,p 2（｜x｜^t; ｜x｜^t） and HK^α,P 2（｜x｜^t; ｜x｜^t）.     

Boundedness of the Cesàro Operator in Hardy Spaces

The Ces\aro operator $\mathcal{C}_{\alpha}$ is defined by \begin{equation*} (\mathcal{C}_{\alpha}f)(x) = \int_{0}^{1}t^{-1}f\left( t^{-1}x \right)\alpha (1-t)^{\alpha -1}\,dt~, \end{equation*} where $f$ denotes a function on $\mathbb{R}$. We prove that $\mathcal{C}_{\alpha}$, $\alpha >0$, is a bounded operator in the Hardy space $H^{p}$ for every $0 < p \leqq 1$.     

Composition Operators on Hardy–Orlicz Spaces on the Ball

We give embedding theorems for Hardy–Orlicz spaces on the ball and then apply our results to the study of the boundedness and compactness of composition operators in this context. As one of the motivations of this work, we show that there exist some Hardy–Orlicz spaces, different from H ^∞, on which every composition operator is bounded.     

Decompositions of the Hardy space Hz^1（Ω）

We give a decomposition of the Hardy space Hz^1（Ω） into ＂div-curl＂ quantities for Lipschitz domains in R^n. We also prove a decomposition of Hz^1（Ω） into Jacobians det Du, u ∈ W0^1,2 （Ω,R^2） for Ω in R^2. This partially answers a well-known open problem.     

The effect of curvature on the best constant in the Hardy–Sobolev inequalities

We address the question of attainability of the best constant in the following Hardy–Sobolev inequality on a smooth domain Ω of :

Sequences of Weighted Composition Operators on the Hardy–Hilbert Space

Let \(\varphi \) be an analytic self map of the open unit disc \(\mathbb {D}\). Assume that \(\psi \) is an analytic map of \(\mathbb {D}\). Suppose that f is in the Hardy–Hilbert space of the open unit disc \(H^2\). The operator that takes f into \(\psi \cdot f \circ \varphi \) is a weighted composition operator, and is denoted by \(C_{\psi ,\varphi }\). In this paper we relate the convergence of the sequence \(\{ C_{\psi _n,\varphi _n}\}\) in different operator topologies to the convergence of the two sequences of maps \(\{\varphi _n \}\) and \(\{ \psi _n\}\).     

General Hardy’s inequalities for functions nonzero on the boundary

Consider Hardy’s inequalities with general weight ϕ for functions nonzero on the boundary. By an integral identity in C ¹( ), define Hilbert spaces H _k ¹ (Ω, ϕ) called Sobolev-Hardy spaces with weight ϕ. As a corollary of this identity, Hardy’s inequalities with weight ϕ in C ¹ ( ) follow. At last, by Hardy’s inequalities with weight ϕ = 1, discuss the eigenvalue problem of the Laplace-Hardy operator with critical parameter (N − 2)²/4 in H ¹ ₁ (Ω).