Hardy type inequalities on multidimensional balls

We study the Hardy identities and inequalities on Cartan-Hadamard manifolds using the notion of a Bessel pair. These Hardy identities offer significantly more information on the existence/nonexistence of the extremal functions of the Hardy inequalities. These Hardy inequalities are in the spirit of Brezis-Vázquez in the Euclidean spaces. As direct consequences, we establish several Hardy type inequalities that provide substantial improvements as well as simple understandings to many known Hardy inequalities and Hardy-Poincaré-Sobolev type inequalities on hyperbolic spaces in the literature.

We prove generalized Hardy's type inequalities with sharp constants and Caffarelli-Kohn-Nirenberg inequalities with sharp constants on Riemannian manifolds $M$. When the manifold is Euclidean space we recapture the sharp Caffarelli-Kohn-Nirenberg inequality. By using a double limiting argument, we obtain an inequality that implies a sharp Hardy's inequality, for functions with compact support on the manifold $M $ (that is, not necessarily on a punctured manifold $ M \backslash \{ x_0 \} $ where $x_0$ is a fixed point in $M$). Some topological and geometric applications are discussed.

In this paper we establish improved Hardy and Rellich type inequalities on a Riemannian manifold M. Furthermore, we also obtain sharp constants for improved Hardy and Rellich type inequalities on the hyperbolic space ℍ n .

In this paper we present new results on two-weight Hardy, Hardy–Poincare 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.

The operator −(d/dx)2+ν where ν is a complex function, |J(x)(1+|x|1+e|>∞, has no positive eigenvalues. This follows from the existence of a triangular Green function for the operator −(d/dx)z−λ,gl>0, and from Hardy type inequalities. We give the multidimensional analogues of these inequalities and we prove the absence of positive eigenvalues for the operator −Δ+ν.

Using the notion of a Bessel pair, we study the Hardy type inequalities in the setting of Dunkl operator. We also establish a general symmetrization principle for weighted Hardy type inequalities with Dunkl operator in the situation that the standard Schwarz symmetrization is not applicable.

We study the Hardy type inequalities in the framework of equalities. We present equalities which immediately imply Hardy type inequalities by dropping the remainder term. Simultaneously we give a characterization of the class of functions which makes the remainder term vanish. A point of our observation is to apply an orthogonality properties in general Hilbert space, and which gives a simple and direct understanding of the Hardy type inequalities as well as the nonexistence of nontrivial extremizers.

Hardy’s inequality originated in the early twentieth century when G.H. Hardy introduced this fundamental result in real analysis to bound integral operators. Its elegant formulation and optimal constants spurred widespread interest, leading to numerous refinements. These developments laid the groundwork for further exploration and multidimensional extensions, deeply influencing harmonic analysis, partial differential equations, and mathematical physics. This historical evolution continues to inspire modern advancements in research. We discuss multidimensional generalizations of some improved Hardy inequalities based on the divergence theorem. The obtained Hardy-type inequalities extend a recent version of the one-dimensional Hardy inequality with the best constant to multidimensional cases.

New Hardy and Sobolev type inequalities involving L^1 -norms of scalar and vector valued functions in ℝ^n are obtained. The work is related to some problems stated in the recent paper by Bourgain and Brezis [BB2].

Poincaré-type estimates for a logarithmically concave measure μ on a convex set Ω are obtained. For this purpose, Ω is endowed with a Riemannian metric g in which the Riemannian manifold with measure (Ω, g, μ) has nonnegative Bakry–Emery tensor and, as a corollary, satisfies the Brascamp–Lieb inequality. Several natural classes of metrics (such as Hessian and conformal metrics) are considered; each of these metrics gives new weighted Poincare, Hardy, or log-Sobolev type inequalities and other results.

In this article, we obtain some new dynamic inequalities of Hardy type on time scales. The main results are derived using Fubini’s theorem and the chain rule on time scales. We apply the main results to the continuous calculus, discrete calculus, and q-calculus as special cases.

This paper presents a constructive description of all possible weighting factors in Hardytype inequalities. An exact description of these factors turned out to be possible within the framework of measures that are the sum of the regular Lebesgue-Stieltjes measure and the Dirac measure. As an application, multidimensional weighted inequalities in n , n > 1, of the Friedrichs and Poincare type are obtained.

New Hardy type inequalities in sectorial area and as a limit in an exterior of a ball are proved. Sharpness of the inequalities is shown as well.

In this paper, using weight functions as well as employing various techniques from real analysis, we establish a few equivalent conditions of two kinds of Hardy-type integral inequalities with nonhomogeneous kernel. To prove our results, we also deduce a few equivalent conditions of two kinds of Hardy-type integral inequalities with a homogeneous kernel in the form of applications. We additionally consider operator expressions. Analytic inequalities of this nature and especially the techniques involved have far reaching applications in various areas in which symmetry plays a prominent role, including aspects of physics and engineering.

This survey is devoted to a number of achievements in the theory of extremal problems in geometric function theory. The approaches to the solution of problems under consideration and the methods used are based on conformal isomorphisms and on the theory of univalent functions developed since the beginning of the 20th century. Results on integral means of conformal mappings of a disc are presented and, in particular, Dolzenko's inequality for rational functions is extended to arbitrary domains with rectifiable boundaries. Investigations in the field of Bohr-type inequalities are described. An emphasis is made on integral inequalities of Hardy and Rellich type, in which the analytic properties of inequalities are intertwined with geometric characteristics of the boundaries of domains. Results related to the solution of the Vuorinen problem on the behaviour of conformal moduli under unlimited dilations of the plane are presented. Formulae for the variation of Robin capacity are obtained. One-parameter families of rational and elliptic functions whose critical values vary in accordance with a prescribed law are characterized. The last results on Smale's conjecture and Smale's dual conjecture are described. Bibliography: 149 titles.