A Hardy Type Inequality and Inde Nite Eigenvalue Problems on the Homogeneous Group

In this paper we present a Hardy type inequality and a Picone type identity for the real sub-Laplacian on the homogeneous group. The existence of the indefinite eigenvalue problem and the simplicity of the principal eigenvalue are proved.     

两个变量的Hardy型积分不等式

建立了两个独立变量函数的新的Hardy型积分不等式.     

Weighted Modular Inequalities for Hardy Type Operators

Given weight functions , w, and v, the weighted modular inequality is characterized. Here Qis a strictly increasing function with Q(0) = 0, Q() = and2Q(x) Q(C x), P is a Young's function, and T is the Hardy operatoror a Hardy type operator. In particular, a characterizing conditionfor the Hardy type operator to map L^p(w) to L^q(v) when 0 <q < 1 p < is deduced. In addition, a new proof for theMaz'ja-Sinnamon theorem is given, and weighted Lorentz norminequalities for Hardy type operators are established. 1991Mathematics Subject Classification: primary 26D15, 42B25; secondary26A33, 46E30.     

Heisenberg型群上一类精确Hardy型不等式和Hardy-Sobolev型不等式及应用(英文)

本文在Heisenberg型群上建立了一类精确的Hardy型不等式。采用的技巧是逼近及正则化的方法。进一步利用这个结果,本文建立了一类精确的Hardy-Sobolev型不等式。这两个结果包括了已有的相关结果。作为应用,讨论了一类具有Hardy位势的非线性算子的正定性与下无界性。     

海森堡群上半空间内最佳Hardy不等式

证明了一类海森堡群上半空间内与次拉普拉斯算子相关的最佳Hardy不等式.作为应用,我们得到了相应的最佳Rellich型不等式.     

关于一类向量场的Picone恒等式和Hardy不等式1

本文给出了构成Baouendi-Grushin算子 的向量场的Picone恒 式,由此导出了Hardy不等式.已有文献中得到的不等式成为本文结果的特殊情形。     

Some Hardy type integral inequalities

We present new kinds of Hardy integral inequalities involving some generalization and improvement.     

Hardy and Rellich-Type Inequalities for Metrics Defined by Vector Fields

Let {X _i }, i=1,...,m be a system of locally Lipschitz vector fields on DR ⁿ , such that the corresponding intrinsic metric is well-defined and continuous w.r.t. the Euclidean topology. Suppose that the Lebesgue measure is doubling w.r.t. the intrinsic balls, that a scaled L¹ Poincaré inequality holds for the vector fields at hand (thus including the case of Hörmander vector fields) and that the local homogeneous dimension near a point x ₀ is sufficiently large. Then weighted Sobolev–Poincaré inequalities with weights given by power of (,x ₀) hold; as particular cases, they yield non-local analogues of both Hardy and Sobolev–Okikiolu inequalities. A general argument which shows how to deduce Rellich-type inequalities from Hardy inequalities is then given: this yields new Rellich inequalities on manifolds and even in the uniformly elliptic case. Finally, applications of Sobolev–Okikiolu inequalities to heat kernel estimates for degenerate subelliptic operators and to criteria for the absence of bound states for Schrödinger operators H=–L+V are given.     

与广义Baouendi-Grushin向量场相联系的Hardy不等式及其应用

本文建立一类与广义Baouendi—Grushin向量场联系的Hardy不等式．采用的技巧是延伸欧氏空间上的散度定理推出的基本积分不等式和选定适当的向量场．Hardy不等式相应的最佳常数也得到证明．本文结果包括了已有广义Baouendi-Grushin向量场的Hardy不等式．作为应用，讨论了由Baouendi-GrusMn向量场构成一退化次椭圆算子的一些性质和刻画了这类向量场构成的非线性算子的一个正解．     

Sobolev Hardy不等式与临界双重调和问题

该文讨论一类带有奇异系数的双重调和方程〖JB({〗△2u-μ[SX(]u[]|x|s[SX)]=f(x,u),\=u=[SX(]u[]ν[SX)]=0,〖JB)〗\ \ 〖JB(〗x∈Ω,x∈Ω,[JB)] 这里ΩRN是包含0的有界光滑区域，u∈H20(Ω)，μ∈R是参数，0≤s≤2，△2=△△表示双重拉普拉斯算子.当f(x,u)=up,p=[SX(]2N[]N-4[SX)]时，上述问题就是一个临界双重调和问题. 该文运用Sobolev Hardy不等式和变分方法，得到它的解的存在性的一些结果.     

Singular quasilinear elliptic problems with indefinite weights and critical potential

The Dirichlet problem for a quasilinear sub-critical inhomogeneous elliptic equation with critical potential and singular coefficients, which has indefinite weights in RN , is studied in this paper. We discuss the corresponding eigenvalue problems by the variational techniques and Picone’s identity, and obtain the existence of non-trivial solutions for the inhomogeneous Dirichlet problem by using Hardy inequality, Mountain Pass Lemma in conjunction with the property of eigenvalues.     

Bounds for Hardy type differences

Let A _k be an integral operator defined by

广义Greiner算子的几类Hardy型不等式

对构成广义Greiner算子的向量场$X_j = \frac{\partial }{\partial x_j} + 2ky_j \vert z\vert ^{2k - 2}\frac{\partial }{\partialt}$, $Y_j = \frac{\partial }{\partial y_j } - 2kx_j \vert z\vert^{2k - 2}\frac{\partial }{\partial t}$, j = 1,... ,n, x,y∈ Rⁿ, $z = x + \sqrt { - 1} \,y$, t ∈ R, k ≥1, 得到了拟球域内和拟球域外的Hardy型不等式;建立了广义Picone型恒等式,并由此导出比文献[3]更一般的全空间上的Hardy型不等式;并在$p = 2$时建立了具最佳常数的Hardy型不等式.     

The Martingale Hardy Type Inequalities for Dyadic Derivative and Integral

Since the Leibniz-Newton formula for derivatives cannot be used in local fields, it is important to investigate the new concept of derivatives in Walsh-analysis, or harmonic analysis on local fields. On the basis of idea of derivatives introduced by Butzer, Schipp and Wade, Weisz has proved that the maximal operators of the one-dimensional dyadic derivative and integral are bounded from the dyadic Hardy space Hp,q to Lp,q, of weak type （L1,L1）, and the corresponding maximal operators of the two-dimensional case are of weak type （Hi, L1）. In this paper, we show that these maximal operators are bounded both on the dyadic Hardy spaces Hp and the hybrid Hardy spaces H^#p 0〈p≤1.     

Finsler Hardy inequalities

In this paper we present a unified simple approach to anisotropic Hardy inequalities in various settings. We consider Hardy inequalities which involve a Finsler distance from a point or from the boundary of a domain. The sharpness and the non-attainability of the constants in the inequalities are also proved.     

Hardy inequalities related to Grushin type operators

We prove some Hardy type inequalities related to the Grushin type operator .

     

$p=-1$的Hardy型不等式的最佳常数

For p＞1,many improved or generalized results of the well-known Hardy's inequality have been established.In this paper,by means of the weight coefficient method,we establish the following Hardy type inequality for P=-1:n∑i=1(1/ii∑j=1aj)-1＜2n∑i=1(1-π2-9/3i)ai-1,Cn such that the inequality ∑ni=1(1/i∑ij=1 aj)-1≤Cn∑ni=1ai-1 holds.Moreover,by means of the Mathematica software,we give some examples.     

华罗庚-王中烈不等式的一些推广

在数学上和美学上都有意义的一些不等式链,是已知的一类华罗庚-王中烈型不等式的推广.在建立不等式的方法中有两种是近些年出现的:其一是基于最近Sándor的恰当想法;其二是利用非线性正泛函的特性.本文展示了具有新形式的不等式,还展示了建立它们的若干方法。