Hardy Inequality Research Articles

Anisotropic Sobolev Spaces with Weights

We study anisotropic weighted Sobolev spaces defined on a half-space. These spaces are important in the study of singular elliptic operators which degenerate at the boundary. We prove trace properties and density of smooth functions. Hardy inequalities are studied in detail.

One‐dimensional sharp discrete Hardy–Rellich inequalities

AbstractIn this paper, we establish discrete Hardy–Rellich inequalities on with and optimal constants, for any . As far as we are aware, these sharp inequalities are new for . Our approach is to use weighted equalities to get some sharp Hardy inequalities using shifting weights, then to settle the higher order cases by iteration. We provide also a new Hardy–Leray–type inequality on with the same constant as the continuous setting. Furthermore, the main ideas work also for general graphs or the setting.

Rellich inequalities via Riccati pairs on model space forms

We present a simple method for proving Rellich inequalities on Riemannian manifolds with constant, non-positive sectional curvature. The method is built upon simple convexity arguments, integration by parts, and the so-called Riccati pairs, which are based on the solvability of a Riccati-type ordinary differential inequality. These results can be viewed as the higher order counterparts of the recent work by Kajántó, Kristály, Peter, and Zhao, discussing Hardy inequalities using Riccati pairs.

Generalizations of integral inequalities similar to Hardy inequality on time scales

Generalizations of integral inequalities similar to Hardy inequality on time scales

The Hardy inequality and large time behaviour of the heat equation on [formula omitted

In this paper we study the large time asymptotic behaviour of the heat equation with Hardy inverse-square potential on corner spaces RN−k×(0,∞)k, k≥0. We first show a new improved Hardy-Poincaré inequality for the quantum harmonic oscillator with Hardy potential. In view of that, we construct the appropriate functional setting in order to pose the Cauchy problem. Then we obtain optimal polynomial large time decay rates and subsequently the first term in the asymptotic expansion of the solutions in L2(RN−k×(0,∞)k). Particularly, we extend and improve the results obtained by Vázquez and Zuazua (J. Funct. Anal. 2000), which correspond to the case k=0, to any k≥0. We emphasize that the higher the value of k the better time decay rates are. We employ a different and simplified approach than Vázquez and Zuazua, managing to remove the usage of spherical harmonics decomposition in our analysis.

On Hardy and Hermite-Hadamard inequalities for $N$-tuple diamond-alpha integral

In this paper, we aim to construct $n$ dimensional Jensen, Hardy and Hermite-Hadamard type inequalities for multiple diamond-alpha integral on time scales. The cases of Hardy type inequality with a weighted function and Hermite-Hadamard type inequality with three variables are also considered minutely.

A generalization of some integral inequalities similar to Hardy inequality on time scales

In this paper, we prove some new dynamic inequalities similar to Hardy's inequality on time scales T. The results as special cases when T=R contain continuous inequalities similar to Hardy's inequality, and when T=Z, the results contain discrete inequalities that are essentially new to the best of the authors' knowledge. R and Z denote the set of all real numbers and the set of all integers, respectively. Our main results are proved by using the Hölder inequality, Chain rule, and some properties of multiple integrals on time scales. Furthermore, some applications and examples are given to illustrate the investigated results.

Stein-Weiss-Adams inequality on Morrey spaces

We establish Adams type Stein-Weiss inequality on global Morrey spaces on general homogeneous groups. Special properties of homogeneous norms and some boundedness results on global Morrey spaces play key roles in our proofs. As consequence, we obtain fractional Hardy, Hardy-Sobolev, Rellich and Gagliardo-Nirenberg inequalities on Morrey spaces on stratified groups. While the results are obtained in the setting of general homogeneous groups, they are new already for the Euclidean space RN.

Integral inequalities with an extended Poisson kernel and the existence of the extremals

Abstract In this article, we first apply the method of combining the interpolation theorem and weak-type estimate developed in Chen et al. to derive the Hardy-Littlewood-Sobolev inequality with an extended Poisson kernel. By using this inequality and weighted Hardy inequality, we further obtain the Stein-Weiss inequality with an extended Poisson kernel. For the extremal problem of the corresponding Stein-Weiss inequality, the presence of double-weighted exponents not being necessarily nonnegative makes it impossible to obtain the desired existence result through the usual technique of symmetrization and rearrangement. We then adopt the concentration compactness principle of double-weighted integral operator, which was first used by the authors in Chen et al. to overcome this difficulty and obtain the existence of the extremals. Finally, the regularity of the positive solution for integral system related with the extended kernel is also considered in this article. Our regularity result also avoids the nonnegativity condition of double-weighted exponents, which is a common assumption in dealing with the regularity of positive solutions of the double-weighted integral systems in the literatures.

Lp-Mapping Properties of a Class of Spherical Integral Operators

In this paper, we study a class of spherical integral operators IΩf. We prove an inequality that relates this class of operators with some well-known Marcinkiewicz integral operators by using the classical Hardy inequality. We also attain the boundedness of the operator IΩf for some 1<p<2 whenever Ω belongs to a certain class of Lebesgue spaces. In addition, we introduce a new proof of the optimality condition on Ω in order to obtain the L2-boundedness of IΩ. Generally, the purpose of this work is to set up new proofs and extend several known results connected with a class of spherical integral operators.

Improved Hardy inequalities on Riemannian manifolds

We study the following version of Hardy-type inequality on a domain Ω in a Riemannian manifold : We provide sufficient conditions on and V for which the above inequality holds. This generalizes earlier well-known works on Hardy inequalities on Riemannian manifolds. The functional setup covers a wide variety of particular cases, which are discussed briefly: for example, with p<N, with , , etc.

Hardy and Sobolev inequalities on antisymmetric functions

We obtain sharp Hardy inequalities on antisymmetric functions, where antisymmetry is understood for multi-dimensional particles. Partially it is an extension of the paper [Th. Hoffmann-Ostenhof and A. Laptev, Hardy inequalities with homogeneous weights, J. Funct. Anal. 268 (2015) 3278–3289], where Hardy’s inequalities were considered for the antisymmetric functions in the case of the 1D particles. As a byproduct we obtain some Sobolev and Gagliardo–Nirenberg type inequalities that are applied to the study of spectral properties of Schrödinger operators.

Best constants in bipolar Lp Hardy-type inequalities

In this work we prove sharp Lp versions of the multipolar Hardy inequalities in [8,10], in the case of a bipolar potential and p≥2. Our results are sharp and minimizers do exist in the energy space. New features appear when p>2 compared to the linear case p=2 at the level of criticality of the p-Laplacian −Δp perturbed by a singular Hardy bipolar potential.

Weighted discrete Hardy's inequalities

UDC 517.5 We give a short proof of a weighted version of the discrete Hardy inequality. This includes the known case of classical monomial weights with optimal constant. The proof is based on the ideas of the short direct proof given recently in [P. Lefèvre, Arch. Math. (Basel), &lt;strong&gt;114&lt;/strong&gt;, № 2, 195–198 (2020)].

Sharp Hardy inequalities for Sobolev–Bregman forms

AbstractWe obtain sharp fractional Hardy inequalities for the half‐space and for convex domains. We extend the results of Bogdan and Dyda and of Loss and Sloane to the setting of Sobolev–Bregman forms.

New proofs of Fejer's and discrete Hermite-Hadamard inequalities with applications

New proofs of the classical Fejer inequality and discrete Hermite-Hadamard inequality (HH) are presented and several applications are given, including (HH)-type inequalities for the functions, whose derivatives have inflection points. Morever, some estimates from below and above for the first moments of functions $f:[a,b]\rightarrow \mathbb{R}$ about the midpoint $c=(a+b)/2$ are obtained and the reverse Hardy inequality for convex functions $f:(0,\infty )\rightarrow (0,\infty )$ is established.

Some characterizations of dynamic weighted Hardy-type inequalities with applications

The pairs of non-negative weights (u,v) for which dynamic inequalities of Hardy's type on a time scale T hold are (p,q) characterized in two different time-scale spaces Lvp(T) and Luq(T). We consider two distinct scenarios of values of the exponents p and q. These results complement the classical strong (p,p) results and their generalizations considered by some authors. For applications, we provide the corresponding continuous and discrete results when T=R, T=N, and T is the quantum space ℓN0 for ℓ>1, respectively. As special cases, we capture some consequences for the results obtained by Anderson, Heinig, Kufner, Flett, and Saker.

A non-local quasi-linear ground state representation and criticality theory

We study energy functionals associated with quasi-linear Schrödinger operators on infinite weighted graphs, and develop a ground state representation. Using the representation, we develop a criticality theory, and show characterisations for a Hardy inequality to hold true. As an application, we show a Liouville comparison principle.

Quantum Lp-spaces and inequalities of Hardy's type

Quantum Lp-spaces and inequalities of Hardy's type

Characterization of Fractional Sobolev–Poincaré and (Localized) Hardy Inequalities

In this paper, we prove capacitary versions of the fractional Sobolev–Poincaré inequalities. We characterize localized variant of the boundary fractional Sobolev–Poincaré inequalities through uniform fatness condition of the domain in $$\mathbb {R}^n$$ . Existence type results on the fractional Hardy inequality in the supercritical case $$sp>n$$ for $$s\in (0,1)$$ , $$p>1$$ are established.