Hardy-type Inequalities Research Articles

Unified Generalizations of Hardy-Type Inequalities Through the Nabla Framework on Time Scales

This research investigates innovative extensions of Hardy-type inequalities through the use of nabla Hölder’s and nabla Jensen’s inequalities, combined with the nabla chain rule and the characteristics of convex and submultiplicative functions. We extend these inequalities within a cohesive framework that integrates elements of both continuous and discrete calculus. Furthermore, our study revisits specific integral inequalities from the existing literature, showcasing the wide-ranging relevance of our results.

Quantitative Hardy inequalities for magnetic Hamiltonians

In this paper we present a new method of proof of Hardy type inequalities for two-dimensional quantum Hamiltonians with a magnetic field of finite flux. Our approach gives a quantitative lower bound on the best constant in these inequalities both for Schrödinger and Pauli operators. Pauli operators with Aharonov-Bohm magnetic field are discussed as well.

Existence of positive solution for a class of quasilinear Schrödinger equations with potential vanishing at infinity on nonreflexive Orlicz-Sobolev spaces

In this paper we investigate the existence of positive solution for a class of quasilinear problem on an Orlicz-Sobolev space that can be nonreflexive $$ - \Delta_{\Phi} u +V(x)\phi(|u|)u= K(x)f(u)\quad\mbox{in } \mathbb{R}^{N}, $$ where $ N \geq 2 $, $ V, K $ are nonnegative continuous functions and $f$ is a continuous function with a quasicritical growth. Here we extend the Hardy-type inequalities presented in \cite{AlvesandMarco} to nonreflexive Orlicz spaces. Through inequalities together with a variational method for non-differentiable functionals we will obtain a ground state solution. We analyze also the problem with $V=0$.

Fractional Hardy-type inequalities on homogeneous Lie groups in the case Q<sp

Fractional Hardy-type inequalities on homogeneous Lie groups in the case Q<sp

Optimal L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^{2}$$\\end{document}-growth of the generalized Rosenau equation

We report that the quantity measured in the L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document} norm of the solution itself of the generalized Rosenau equation, which was completely unknown in this equation, grows in the proper order at time infinity. It is also immediately apparent that this growth aspect does not occur in three or more spatial dimensions, so we will apply the results obtained in this study to provide another proof that Hardy-type inequalities do not hold in the case of one or two spatial dimensions.

Some Hardy and Rellich type inequalities for affine connections

Some Hardy and Rellich type inequalities for affine connections

Hardy-type inequalities for a class of iterated operators and their application to Morrey-type spaces

Abstract In this study, we discuss new Hardy-type inequalities for operators involving iteration and provide explicit characterizations of these inequalities. As an application of our results, we consider the problem of the boundedness of the multidimensional Hardy operator from a Lebesgue space to a Morrey-type space.

Some Generalizations of Dynamic Hardy-Knopp-Type Inequalities on Time Scales

In the present paper, some new generalizations of dynamic inequalities of Hardy-type in two variables on time scales are established. The integral and discrete Hardy-type inequalities that are given as special cases of main results are original. The main results are proved by using the dynamic Jensen inequality and the Fubini theorem on time scales.

Correction to: On weighted iterated Hardy-type inequalities

Correction to: On weighted iterated Hardy-type inequalities

ANISOTROPIC PICONE IDENTITIES FOR HALF LINEAR CONFORMABLE ELLIPTIC EQUATIONS

This study is devoted to investigating the anisotropic picone identities for half-linear Conformable elliptic equations and the Hardy-type inequality. Further, we provide some results for the nonlinear analogue to Picone identity.

Hardy type inequalities with mixed weights in cones

Hardy type inequalities with mixed weights in cones

A generic functional inequality and Riccati pairs: an alternative approach to Hardy-type inequalities

We present a generic functional inequality on Riemannian manifolds, both in additive and multiplicative forms, that produces well known and genuinely new Hardy-type inequalities. For the additive version, we introduce Riccati pairs that extend Bessel pairs developed by Ghoussoub and Moradifam (Proc. Natl. Acad. Sci. USA, 2008 & Math. Ann., 2011). This concept enables us to give very short/elegant proofs of a number of celebrated functional inequalities on Riemannian manifolds with sectional curvature bounded from above by simply solving a Riccati-type ODE. Among others, we provide alternative proofs for Caccioppoli inequalities, Hardy-type inequalities and their improvements, spectral gap estimates, interpolation inequalities, and Ghoussoub-Moradifam-type weighted inequalities. Concerning the multiplicative form, we prove sharp uncertainty principles on Cartan-Hadamard manifolds, i.e., Heisenberg-Pauli-Weyl uncertainty principles, Hydrogen uncertainty principles and Caffarelli-Kohn-Nirenberg inequalities. Some sharpness and rigidity phenomena are also discussed.

ON HARDY-TYPE INEQUALITIES AS AN INTELLECTUAL ADVENTURE FOR 100 YEARS

We describe some chosen ideas and results for more than 100 years prehistory and history of the remarkable development concerning Hardy-type inequalities. In particular, we present a newer convexity approach, which we believe could partly have changed this development if Hardy had discovered it. In order to emphasize the current very active interest in this subject, we finalize by presenting some examples of the recent results, which we believe have potential not only to be of interest for a broad audience from a historical perspective, but also to be useful in various applications.

Hardy type identities and inequalities with divergence type operators on smooth metric measure spaces

<abstract><p>We gave the Hardy type identities and inequalities for the divergence type operator $ L_{f, V} $ on smooth metric measure spaces. Additionally, we improved a Rellich type inequality by using the improved Hardy type inequality. Our results improved and included many previously known results as special cases.</p></abstract>

Oscillatory and spectral properties of a class of fourth-order differential operators via a new Hardy-type inequality

Oscillatory and spectral properties of a class of fourth-order differential operators via a new Hardy-type inequality

Advanced Hardy-type inequalities with negative parameters involving monotone functions in delta calculus on time scales

<p>In this study, we introduced several novel Hardy-type inequalities with negative parameters for monotone functions within the framework of delta calculus on time scales $ \mathbb{T} $. As an application, when $ \mathbb{T = N}_{0}, $ we derived discrete inequalities with negative parameters for monotone sequences, offering fundamentally new results. When $ \mathbb{T = R}, $ we established continuous analogues of inequalities that have appeared in previous literature. Additionally, we presented inequalities for other time scales, such as $ \mathbb{T} = q^{\mathbb{N}_{0}} $ for $ q > 1, $ which, to the best of the authors' knowledge, represented largely novel contributions.</p>

Some weighted dynamic inequalities of Hardy type with kernels on time scales nabla calculus

Some weighted dynamic inequalities of Hardy type with kernels on time scales nabla calculus

Weighted Hardy type inequalities with Robin boundary conditions

In this paper, we establish a general weighted Hardy type inequality for the $ p- $Laplace operator with Robin boundary condition. We provide various concrete examples to illustrate our results for different weights. Furthermore, we present some Heisenberg-Pauli-Weyl type inequalities with boundary terms on balls centred at the origin with radius $ R $ in $ \mathbb{R} ^n $.

Generalizations of Hardy-type inequalities by the Hermite interpolating polynomial

Generalizations of Hardy-type inequalities by the Hermite interpolating polynomial

On fractional Hardy-type inequalities in general open sets

We show that, when sp > N, the sharp Hardy constant hs,p of the punctured space ℝN \ {0} in the Sobolev–Slobodeckiĭ space provides an optimal lower bound for the Hardy constant hs,p(Ω) of an open set Ω ⊂ ℝN. The proof exploits the characterization of Hardy’s inequality in the fractional setting in terms of positive local weak supersolutions of the relevant Euler–Lagrange equation and relies on the construction of suitable supersolutions by means of the distance function from the boundary of Ω. Moreover, we compute the limit of hs,p as s ↗ 1, as well as the limit when p ↗ ∞. Finally, we apply our results to establish a lower bound for the non-local eigenvalue λs,p(Ω) in terms of hs,p when sp > N, which, in turn, gives an improved Cheeger inequality whose constant does not vanish as p ↗ ∞.