Hardy Inequality Research Articles

Some Fractional Dynamic Inequalities of Hardy’s Type via Conformable Calculus

In this article, we prove some new fractional dynamic inequalities on time scales via conformable calculus. By using chain rule and Hölder’s inequality on timescales we establish the main results. When α = 1 we obtain some well-known time-scale inequalities due to Hardy, Copson, Bennett and Leindler inequalities.

The Hardy inequality and fractional Hardy inequality for the Dunkl Laplacian

We prove the Lp Hardy inequality and Lp fractional Hardy inequality for the Dunkl Laplacian on ℝN. Further, we prove the same kind of inequalities for a half-space and cone.

Inequalities for Convex and Concave Functions and a New Concept of Majorization Intended for Two Pairs of Vectors

In this paper, some inequalities for convex and concave functions are demonstrated. A new concept of majorization is applied for two pairs of vectors. Instead of the standard approach based on one doubly stochastic matrix, here two matrices are employed: one is both row stochastic and column substochastic, while the other is both column stochastic and row superstochastic. An inequality of Hardy–Littlewood–Polya–Karamata type is derived for a row stochastic–column substochastic matrix.

On Hardy's Inequality for Hermite Expansions

Sharp multi-dimensional Hardy's inequality for the Laguerre functions of Hermite type is proved for the type parameter $\al\in[-1/2,\infty)^d$. As a consequence we obtain the corresponding result for the generalized Hermite expansions. In particular, it validate that the known version of Hardy's inequality for the Hermite functions is sharp.

Hardy-Type Inequalities for the Carnot–Carathéodory Distance in the Heisenberg Group

In this paper we study various Hardy inequalities in the Heisenberg group $\mathbb H^n$, w.r.t. the Carnot-Carath\'eodory distance $\delta$ from the origin. We firstly show that the optimal constant for the Hardy inequality is strictly smaller than $n^2 = (Q-2)^2/4$, where $Q$ is the homogenous dimension. Then, we prove that, independently of $n$, the Heisenberg group does not support a radial Hardy inequality, i.e., a Hardy inequality where the gradient term is replaced by its projection along $\nabla_{\mathbb H}\delta$. This is in stark contrast with the Euclidean case, where the radial Hardy inequality is equivalent to the standard one, and has the same constant. Motivated by these results, we consider Hardy inequalities for non-radial directions, i.e., directions tangent to the Carnot-Carath\'eodory balls. In particular, we show that the associated constant is bounded on homogeneous cones $C_\Sigma$ with base $\Sigma\subset \mathbb S^{2n}$, even when $\Sigma$ degenerates to a point. This is a genuinely sub-Riemannian behavior, as such constant is well-known to explode for homogeneous cones in the Euclidean space.

Lyapunov-type inequalities for the fractional p-sub-Laplacian

In this paper we study the fractional Dirichlet p-sub-Laplacian in a Haar measurable set on homogeneous Lie groups. We show analogues of the fractional Sobolev and Hardy inequalities and we also present a Lyapunov-type inequality for the fractional p-sub-Laplacian. As a consequence of the Lyapunov-type inequality we obtain an estimate of the first eigenvalue in a quasi-ball for the Dirichlet fractional p-sub-Laplacian.

Hardy type inequalities with spherical derivatives

A Hardy type inequality is presented with spherical derivatives in $${\mathbb {R}}^{n}$$ with $$n\ge 2$$ in the framework of equalities. This clarifies the difference between contribution by radial and spherical derivatives in the improved Hardy inequality as well as nonexistence of nontrivial extremizers without compactness arguments.

A Generalization of the Hardy Inequality

A Generalization of the Hardy Inequality

A weighted anisotropic Sobolev type inequality and its applications to Hardy inequalities

In this paper we focus our attention on an embedding result for a weighted Sobolev space that involves as weight the distance function from the boundary taken with respect to a general smooth gauge function F. Starting from this type of inequalities we prove some refined Hardy-type inequalities.

Nonlinear dynamics of full-range CNNs with time-varying delays and variable coefficients

In the article, the dynamical behaviours of the full-range cellular neural networks (FRCNNs) with variable coefficients and time-varying delays are considered. Firstly, the improved model of the FRCNNs is proposed, and the existence and uniqueness of the solution are studied by means of differential inclusions and set-valued analysis. Secondly, by using the Hardy inequality, the matrix analysis, and the Lyapunov functional method, we get some criteria for achieving the globally exponential stability (GES). Finally, some examples are provided to verify the correctness of the theoretical results.

Multiple Lagrange stability and Lyapunov asymptotical stability of delayed fractional-order Cohen–Grossberg neural networks**Project supported by the Natural Science Foundation of Zhejiang Province of China (Grant Nos. LY18F030023, LY17F030016, LQ18F030015, and LY18F020028) and the National Natural Science Foundation of China (Grant Nos. 61503338, 61773348, and 61972354).

This paper addresses the coexistence and local stability of multiple equilibrium points for fractional-order Cohen–Grossberg neural networks (FOCGNNs) with time delays. Based on Brouwer’s fixed point theorem, sufficient conditions are established to ensure the existence of equilibrium points for FOCGNNs. Through the use of Hardy inequality, fractional Halanay inequality, and Lyapunov theory, some criteria are established to ensure the local Lagrange stability and the local Lyapunov asymptotical stability of equilibrium points for FOCGNNs. The obtained results encompass those of integer-order Hopfield neural networks with or without delay as special cases. The activation functions are nonlinear and nonmonotonic. There could be many corner points in this general class of activation functions. The structure of activation functions makes FOCGNNs could have a lot of stable equilibrium points. Coexistence of multiple stable equilibrium points is necessary when neural networks come to pattern recognition and associative memories. Finally, two numerical examples are provided to illustrate the effectiveness of the obtained results.

Some new integral inequalities via Steklov operator

Hardy and Copson type inequalities have been studied by a large number of authors during the twentieth century and has motivated some important lines of study which are currently active. A large number of papers have been appeared involving Copson and Hardy inequalities (see [2-16] for more details). In this paper some Hardy-Steklov and Copson-Steklov type integral inequalities were established.

The optimal constant in generalized Hardy's inequality

We obtain the sharp factor of the two-sides estimates of the optimal constant in generalized Hardy's inequality with two general Borel measures on $\mathbb{R}$, which generalizes and unifies the known continuous and discrete cases.

Weighted Hardy's inequalities and the variational problem with compact perturbations

Weighted Hardy's inequalities and the variational problem with compact perturbations

One dimensional weighted Hardy's inequalities and application

One dimensional weighted Hardy's inequalities and application

A simple proof of the Hardy inequality on Carnot groups and for some hypoelliptic families of vector fields

We give an elementary proof of the classical Hardy inequality on any Carnot group, using only integration by parts and a fine analysis of the commutator structure, which was not deemed possible until now. We also discuss the conditions under which this technique can be generalized to deal with hypoelliptic families of vector fields, which, in this case, leads to an open problem regarding the symbol properties of the gauge norm.

Remarks on a limiting case of Hardy type inequalities

The classical Hardy inequality holds in Sobolev spaces $W_0^{1,p}$ when $1\le p< N$. In the limiting case where $p=N$, it is known that by adding a logarithmic function to the Hardy potential, some inequality which is called the critical Hardy inequality holds in $W_0^{1,N}$. In this note, in order to give an explanation of appearance of the logarithmic function at the potential, we derive the logarithmic function from the classical Hardy inequality with the best constant via some limiting procedure as $p \nearrow N$. And we show that our limiting procedure is also available for the classical Rellich inequality in second order Sobolev spaces $W_0^{2,p}$ with $p \in (1, \frac{N}{2})$ and the Poincare inequality.

Hardy inequalities and Caffarelli-Kohn-Nirenberg inequalities with radial derivative

Hardy inequalities and Caffarelli-Kohn-Nirenberg inequalities with radial derivative

Hardy and Rellich inequalities for anisotropic p-sub-Laplacians

In this paper we establish the subelliptic Picone type identities. As consequences, we obtain Hardy and Rellich type inequalities for anisotropic p-sub-Laplacians which are operators of the form $$\begin{aligned} {\mathcal {L}}_{p}f:= \sum _{i=1}^{N} X_i\left( |X_i f|^{p_i-2} X_i f \right) ,\quad 1<p_i<\infty , \end{aligned}$$where \(X_i\), \(i=1,\ldots , N\), are the generators of the first stratum of a stratified (Lie) group. Moreover, analogues of Hardy type inequalities with multiple singularities and many-particle Hardy type inequalities are obtained on stratified groups.

Hardy inequalities for the fractional powers of the Grushin operator

We establish uncertainty principles and Hardy inequalities for the fractional Grushin operator, which are reduced to those inequalities for the fractional generalized sublaplacian. The key ingredients to obtain them are an explicit integral representation and a ground state representation for the fractional powers of generalized sublaplacian.