Hardy-type Inequalities Research Articles

Some dynamic Hardy-type inequalities with negative parameters on time scales nabla calculus

<abstract><p>In this paper, we establish some new dynamic Hardy-type inequalities with negative parameters on time scales nabla calculus by applying the reverse H ölder's inequality, integration by parts, and chain rule on time scales nabla calculus. As special cases of our results (when $ \mathbb{ T = R} $), we get the continuous analouges of inequalities proven by Benaissa and Sarikaya, and when $ \mathbb{T = N}_{0} $, the results to the best of the authors' knowledge are essentially new.</p></abstract>

Iterated discrete Hardy-type inequalities with three weights for a class of matrix operators

Iterated Hardy-type inequalities are one of the main objects of current research on the theory of Hardy inequalities. These inequalities have become well-known after study boundedness properties of the multidimensional Hardy operator acting from the weighted Lebesgue space to the local Morrie-type space. In addition, the results of quasilinear inequalities can be applied to study bilinear Hardy inequalities. In the paper, we discussed weighted discrete Hardy-type inequalities containing some quasilinear operators with a matrix kernel where matrix entries satisfy discrete Oinarov condition. The research of weighted Hardy-type inequalities depends on the relations between parameters p, q and θ, so we considered the cases 1 < p ≤ q < θ < ∞ and p ≤ q < θ < ∞, 0 < p ≤1, criteria for the fulfillment of iterated discrete Hardy-type inequalities are obtained. Moreover, an alternative method of proof was shown in the work.

Generalised Hardy type and Rellich type inequalities on the Heisenberg group

Generalised Hardy type and Rellich type inequalities on the Heisenberg group

Sharpness of some Hardy-type inequalities

The current status concerning Hardy-type inequalities with sharp constants is presented and described in a unified convexity way. In particular, it is then natural to replace the Lebesgue measure dx with the Haar measure dx/x\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$dx/x$\\end{document}. There are also derived some new two-sided Hardy-type inequalities for monotone functions, where not only the two constants are sharp but also the involved function spaces are (more) optimal. As applications, a number of both well-known and new Hardy-type inequalities are pointed out. And, in turn, these results are used to derive some new sharp information concerning sharpness in the relation between different quasi-norms in Lorentz spaces.

Boundedness of Wolff-type potentials and applications to PDEs

We provide a short proof of a sharp rearrangement estimate for a generalized version of a potential of Wolff–Havin–Maz’ya type. As a consequence, we prove a reduction principle for that integral operators, that is, a characterization of those rearrangement invariant spaces between which the potentials are bounded via a one-dimensional inequality of Hardy-type.Since the special case of the mentioned potential is known to control precisely very weak solutions to a broad class of quasilinear elliptic PDEs of non-standard growth, we infer the local regularity properties of the solutions in rearrangement invariant spaces for prescribed classes of data.

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.

Exploring Generalized Hardy-Type Inequalities via Nabla Calculus on Time Scales

In this research, we aim to explore generalizations of Hardy-type inequalities using nabla Hölder’s inequality, nabla Jensen’s inequality, chain rule on nabla calculus and leveraging the properties of convex and submultiplicative functions. Nabla calculus on time scales provides a unified framework that unifies continuous and discrete calculus, making it a powerful tool for studying various mathematical problems on time scales. By utilizing this approach, we seek to extend Hardy-type inequalities beyond their classical continuous or discrete settings to a more general time scale domain. As specific instances of our discoveries, we have the integral inequalities previously established in the existing literature.

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.

Novel Hardy-Type Inequalities with Submultiplicative Functions on Time Scales Using Delta Calculus

In this study, we apply Hölder’s inequality, Jensen’s inequality, chain rule and the properties of convex functions and submultiplicative functions to develop an innovative category of dynamic Hardy-type inequalities on time scales delta calculus. A time scale, denoted by T, is any closed nonempty subset of R. In time scale calculus, results are unified and extended. As particular cases of our findings (when T=R), we have the continuous analogues of inequalities established in some the literature. Furthermore, we can find other inequalities in different time scales, such as T=N, which, to the best of the authors’ knowledge, is a largely novel conclusion.

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.

On the mixed regularity of N-body Coulombic wavefunctions

In this paper, we prove a new mixed regularity of Coulombic wavefunction taking into account the Pauli exclusion principle. We also study the hyperbolic cross space approximation of eigenfunctions associated with this new regularity, and deduce the corresponding error estimates in L2-norm and H1-semi-norm. The proofs are based on the study of extended Hardy-type inequalities for Coulomb-type potentials.

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.

Generalizations of Hardy-Type Inequalities by Montgomery Identity and New Green Functions

In this paper we extend general Hardy’s inequality by appropriately combining Montgomery’s identity and Green functions. Related Grüss and Ostrowski-type inequalities are also derived.

Improved Poincaré-Hardy inequalities on certain subspaces of the Sobolev space

We prove an improved version of Poincaré-Hardy inequality in suitable subspaces of the Sobolev space on the hyperbolic space via Bessel pairs. As a consequence, we obtain a new Hardy type inequality with an improved constant (than the usual Hardy constant). Furthermore, we derive a new kind of improved Caffarelli-Kohn-Nirenberg inequality on the hyperbolic space.

Quantitative dependence of some discrete limiting classes on the Muckenhoupt \calA 1 ( u ) class

UDC 517.5 We prove some relations between the discrete Gehring classes 𝒢 q and the discrete Muckenhoupt classes 𝒜 p . Specifically, by using some known Hardy-type and Carleman-type inequalities, we study the relationship between 𝒢 1 , 𝒜 ∞ and 𝒜 1 for nonincreasing and nondecreasing weights. Finally, we establish some general results by introducing the notions of 𝒢 φ classes defined for nonnegative convex function φ .

Quasilinear equation with critical exponential growth in the zero mass case

In this work we study the existence of solutions for the following class of quasilinear elliptic equations −divA(|x|)|∇u|N−2∇u=Q(|x|)f(u),inRN,where N≥2 and f has critical exponential growth. We establish conditions on the non-homogeneous weights A and Q to introduce a suitable function space where we are able to apply Variational Methods to obtain weak solutions. The main key is a Hardy type inequality for radial functions. Our approach is based on a new Trudinger–Moser type inequality, a version of the Symmetric Criticality Principle and Mountain Pass Theorem.

A half-discrete Hilbert-type inequality in the whole plane with the constant factor related to a cotangent function

In this work, by the introduction of some parameters, a new half-discrete kernel function in the whole plane is defined, which involves both the homogeneous and the nonhomogeneous cases. By employing some techniques of real analysis, especially the method of a weight function, a new half-discrete Hilbert-type inequality with the new kernel function, as well as its equivalent Hardy-type inequalities are established. Moreover, it is proved that the constant factors of the newly obtained inequalities are the best possible. Finally, assigning special values to the parameters, some new half-discrete Hilbert-type inequalities with special kernels are presented at the end of the paper.

Horizontal magnetic fields and improved Hardy inequalities in the Heisenberg group

In this article, we introduce a notion of magnetic field in the Heisenberg group and we study its influence on spectral properties of the corresponding magnetic (sub-elliptic) Laplacian. We show that uniform magnetic fields uplift the bottom of the spectrum. For magnetic fields vanishing at infinity, including Aharonov–Bohm potentials, we derive magnetic improvements to a variety of Hardy-type inequalities for the Heisenberg sub-Laplacian. In particular, we establish a sub-Riemannian analogue of Laptev and Weidl sub-criticality result for magnetic Laplacians in the plane. Instrumental for our argument is the validity of a Hardy-type inequality for the Folland–Stein operator, that we prove in this article and has an interest on its own.

Extension of Hardy-Type Inequalities on Time Scales

The concept of superquadratic functions of several variables with general kernels in the setting of time scales play a fundamental role in mathematical analysis. In this article, we derive some new results related to Hardy-type inequalities through Jensen inequality for multivariate superquadratic functions on time scales.

Improved Hardy Inequalities with a Class of Weights

In the framework of Hardy type inequalities and their applications to evolution problems, the paper deals with local and nonlocal weighted improved Hardy inequalities related to the study of Kolmogorov operators perturbed by singular potentials. The class of weights is wide enough. We focus our attention on weighted Hardy inequalities with potentials obtained by inverse square potentials adding a nonnegative correction term. The method used to get the results is based on the introduction of a suitable vector-valued function and on a generalized vector field method. The local estimates show some examples of this type of potentials and extend some known results to the weighted case.