Hardy and Rellich type inequalities with remainders

Czechoslovak Mathematical Journal - Hardy and Rellich type inequalities with an additional term are proved for compactly supported smooth functions on open subsets of the Euclidean space. We obtain...     

On a class of Rellich inequalities

We prove the Rellich and the improved Rellich inequalities that involve the distance function from a hypersurface of codimension k, under a certain geometric assumption. In case, the distance is taken from the boundary, that assumption is the convexity of the domain. We also discuss the best constant of these inequalities.     

Interpolation inequalities with weights

We use elementary methods to prove a sufficient and necessary condition for a Sobolev interpolation inequalities with weight [ILLM0001] where [ILLM0001] are real numbers, and [ILLM0001]     

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.     

Sharp Nikolskii inequalities with exponential weights

w _a(x)=exp(–x^a), xR, a0. , N _n (a,p,q) — (2), _n P _nw_ap, CN_n(a,p, q)P_nw_aq. , — , {P _n}, .

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This material is based upon research supported by the National Science Foundation under Grant No. DMS-84-19525, by the United States Information Agency under Senior Research Fulbright Grant No. 85-41612, and by the Hungarian Ministry of Education (first author). The work was started while the second author visited The Ohio State University between 1983 and 1985, and it was completed during the first author's visit to Hungary in 1985.     

Weighted Hardy and Rellich type inequalities on Riemannian manifolds

In this paper we present new results on two‐weight Hardy, Hardy–Poincaré and Rellich type inequalities with remainder terms on a complete noncompact Riemannian Manifold M. The method we use is flexible enough to obtain more weighted Hardy type inequalities. Our results improve and include many previously known results as special cases.     

Conjugate Hardy's inequalities with decreasing weights

We prove that for a decreasing weight on , the conjugate Hardy transform is bounded on () if and only if it is bounded on the cone of all decreasing functions of . This property does not depend on .

     

Improved Poincaré inequalities with weights

In this paper we prove that if Ω∈Rn is a bounded John domain, the following weighted Poincaré-type inequality holds:

     

Rellich type inequalities in domains of the Euclidean space

For smooth functions supported in a domain of the Euclidean space we investigate two Rellich type inequalities with weights which are powers of the distance function. We prove that for an arbitrary plane domain there exist positive Rellich constants in these inequalities if and only if the boundary of the domain is a uniformly perfect set. Moreover, we obtain explicit estimates of constants in function of geometric domain characteristics. Also, we find sharp constants in these Rellich type inequalities for all non-convex domains of dimension d ≥ 2 provided that the domains satisfy the exterior sphere condition with certain restriction on the radius of spheres.     

Integral inequalities with weights for maxiamal operators

Let μ be a measure on the upper half-space R ₊ ⁿ⁺¹ , and v a weight onR ⁿ, we give a characterization for the pair (v, μ) such that ∥M(fv)∥_{L Θ (μ}) ⩽ c ∥f∥_{L Θ (μ}), where Φ is an N-function satisfying Δ₂ condition andMf(x,t), is the maximal function onR ₊ ⁿ⁺¹ , which was introduced by Ruiz, F. and Torrea, J.. Supported by NSFC.     

Hardy type and Rellich type inequalities on the Heisenberg group

This paper contains some interesting Hardy type inequalities and Rellich type inequalities for the left invariant vector fields on the Heisenberg group.

     

Stechkin-Marchaud-Type inequalities with Jacobi weights for Bernstein Operators

Using the modulus of smoothnessω _? ^{λ 2}, Stechkin-Marchaud-Type inequalities with Jacobi weights of Bernstein Operators is established.     

Rellich type inequalities related to Grushin type operator and Greiner type operator

We prove some sharp Hardy inequality associated with the gradient ? _?? = (? _x ,|x| ^?? ? _y ) by a direct and simple approach. Moreover, similar method is applied to obtain some weighted sharp Rellich inequality related to the Grushin operator in the setting of L ^p . We also get some weighted Hardy and Rellich type inequalities related to a class of Greiner type operators.     

Sharp inequalities for dyadic A 1 weights

We show how the Bellman function method can be used to obtain sharp inequalities for the maximal operator of a dyadic A ₁ weight on ${\mathbb{R}^n}$ R n . Using this approach, we determine the optimal constants in the corresponding weak-type estimates. Furthermore, we provide an alternative, simpler proof of the related maximal L ^p -inequalities, originally shown by Melas.     

On first order interpolation inequalities with weights on the Heisenberg group

In this paper, sufficient and necessary conditions for the first order interpolation inequalities with weights on the Heisenberg group are given. The necessity is discussed by polar coordinates changes of the Heisenberg group. Establishing a class of Hardy type inequalities via a new representation formula for functions and Hardy-Sobolev type inequalities by interpolation, we derive the sufficiency. Finally, sharp constants for Hardy type inequalities are determined.     

On first order interpolation inequalities with weights on the H-type group

In this paper, sufficient and necessary conditions for a class of first order interpolation inequalities with weights on the H-type group are given. By polar coordinate changes of the H-type group, the necessity is verified. A class of Hardy type inequalities is established via a representation formula for functions, Hardy-Sobolev type inequalities are obtained by interpolation and then the sufficiency is completed through discussion of parameter σ.     

Inequalities of Rellich Type

Functional Analysis and Its Applications - Hardy inequalities have been important topics of research for a century, and in the past twenty years or so, there has been a deluge of important papers...     

Hardy type inequalities with exponential weights for a class of convolution operators

In this paper we consider operators of the form H=λ(-i∇), with λ analytic in a strip and with some specific growth conditions at infinity, and prove Hardy type estimates in L ²(ℝ ⁿ ) with exponential weights. In fact we extend our previous results [19] from bounded analytic functions on a strip to analytic functions with polynomial growth in that strip.