Weighted bilinear Hardy inequalities

We characterize the weights w, $w_1$, $w_2$ such that the weighted bilinear Hardy inequality$\left(\underset{a}{\overset{b}{\int }}\right(\underset{a}{\overset{x}{\int }}f{)}^q(\underset{a}{\overset{x}{\int }}g{)}^{q}w(x)dx{)}^{\frac{1}{q}}?C(\underset{a}{\overset{b}{\int }}{f}^{p_1}{w}_1{)}^{\frac{1}{p_1}}(\underset{a}{\overset{b}{\int }}{g}^{p_2}{w}_2{)}^{\frac{1}{p_2}}$ holds for all nonnegative functions f and g, with a positive constant C independent of f and g, for all possible values of q, $p_1$ and $p_2$ with $1<q,p_1,p_2<\infty$. We also characterize the good weights for the weighted bilinear n-dimensional Hardy inequality to hold.     

Weighted korn inequalities in paraboloidal domains

A weighted Korn inequality in a domain Ω ⊂ ℝ ⁿ with paraboloidal exit II to infinity is obtained. Asymptotic sharpness of the inequality is achieved by using different weight factors for the longitudinal (with respect to the axis of II) and transversal displacement vector components and by making the weight factors of the derivatives depend on the direction of differentiation. The solvability of the elasticity problem in the energy class (the closure of in the norm generated by the elastic energy functional) is studied; the dimensions of the kernel and the cokerned of the corresponding operator depend on the exponents∈(−∞, 1) in the “rate of expansion” of the paraboloid II. Translated fromMatematicheskie Zametki, Vol. 62, No. 5, pp. 751–765, November, 1997. Translated by V. N. Dubrovsky     

Weighted inequalities of Hardy type

Khabarovsk. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 30, No. 1, pp. 13–22, January–February, 1989.     

Weighted Hardy and Opial-type inequalities

An integral condition on weights u and v is given which is equivalent to the boundedness of the Hardy operator between the weighted Lebesgue spaces L_u^p and L_v^q with 0 < q < 1 < p < ∞. The Hardy inequalities are applied to give easily verified weight conditions which imply inequalities of Opial type.     

Some new inequalities involving the Hardy operator

In this paper we derive some new inequalities involving the Hardy operator, using some estimates of the Jensen functional, continuous form generalization of the Bellman inequality and a Banach space variant of it. Some results are generalized to the case of Banach lattices on $(0,b],0<b\le \infty$.     

Pointwise Hardy inequalities

If is an open set with the sufficiently regular boundary, then the Hardy inequality holds for and , where . The main result of the paper is a pointwise inequality , where on the right hand side there is a kind of maximal function. The pointwise inequality combined with the Hardy-Littlewood maximal theorem implies the Hardy inequality. This generalizes some recent results of Lewis and Wannebo.

     

Hardy inequalities with non-standard remainder terms

Improved Hardy inequalities, involving remainder terms, are established both in the classical and in the limiting case. The relevant remainders depend on a suitable distance from the families of the “virtual” extremals.     

A Generalization of the Weighted Hardy Inequality for One Class of Integral Operators

We consider the problem of finding necessary and sufficient conditions for validity of an estimate for a function by some differential operation containing a weight function rather than the ordinary derivative. This operation is referred to as the -weighted derivative.     

Weighted weak type inequalities for modified Hardy operators and geometric means operators in dimensions one and greater

We characterize the pairs of weights (u,v) such that the geometric mean operator G₁, defined for positive functions f on (0,∞) by

     

Weighted inequalities for iterated convolutions

Given a fixed exponent , , and suitable nonnegative weight functions , , an optimal associated weight function is constructed for which the iterated convolution product satisfies

for all complex valued measurable functions with . Here and for each , . Analogous results are given when is replaced by and also when the convolution on is taken instead to be . The extremal functions are also discussed.

     

Weighted variational inequalities in normed spaces

In this article, we consider weighted variational inequalities over a product of sets and a system of weighted variational inequalities in normed spaces. We extend most results established in Ansari, Q.H., Khan, Z. and Siddiqi, A.H., (Weighted variational inequalities, Journal of Optimization Theory and Applications, 127(2005), pp. 263–283), from Euclidean spaces ordered by their respective non-negative orthants to normed spaces ordered by their respective non-trivial closed convex cones with non-empty interiors.     

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.     

Multidimensional Hardy-type inequalities with general kernels

Some new multidimensional Hardy-type inequalities involving arithmetic mean operators with general positive kernels are derived. Our approach is mainly to use a convexity argument and the results obtained improve some known results in the literature and, in particular, some recent results in [S. Kaijser, L. Nikolova, L.-E. Persson, A. Wedestig, Hardy-type inequalities via convexity, Math. Inequal. Appl. 8 (3) (2005) 403-417] are generalized and complemented.     

Weighted Hardy inequalities for decreasing sequences and functions

We obtain a complete characterization of the weights for which Hardy's inequality holds on the cone of non-increasing sequences. Our proofs translate immediately to the analogous inequality for non-increasing functions, thereby also completing the investigation in that direction. As an application of our results we characterize the boundedness of the Hardy-Littlewood maximal operator on Lorentz sequence spaces.     

On weighted weak type inequalities for modified Hardy operators

We characterize the pairs of weights for which the modified Hardy operator applies into weak- where is a monotone function and .

     

A simple approach to Hardy inequalities

We describe a simple method of proving Hardy-type inequalities of second and higher order with weights for functions defined in ℝ ⁿ . It is shown that we can obtain such inequalities with sharp constants by applying the divergence theorem to specially chosen vector fields. Another approach to Hardy inequalities based on the application of identities of Rellich-Pokhozhaev type is also proposed. Translated fromMatematicheskie Zametki, Vol. 67, No. 4, pp. 563–572, April, 2000.     

Hardy and Rellich type inequalities on complete manifolds

We prove some Hardy and Rellich type inequalities on complete noncompact Riemannian manifolds supporting a weight function which is not very far from the distance function in the Euclidean space.