Hardy-type Inequalities Research Articles

Quantum strips in higher dimensions

We consider the Dirichlet Laplacian in unbounded strips on ruled surfaces in any space dimension. We locate the essential spectrum under the condition that the strip is asymptotically flat. If the Gauss curvature of the strip equals zero, we establish the existence of discrete spectrum under the condition that the curve along which the strip is built is not a geodesic. On the other hand, if it is a geodesic and the Gauss curvature is not identically equal to zero, we prove the existence of Hardy-type inequalities. We also derive an effective operator for thin strips, which enables one to replace the spectral problem for the Laplace-Beltrami operator on the two-dimensional surface by a one-dimensional Schroedinger operator whose potential is expressed in terms of curvatures. In the appendix, we establish a purely geometric fact about the existence of relatively parallel adapted frames for any curve under minimal regularity hypotheses.

Multivariate Hardy and Littlewood inequalities on time scales

In the paper we extend some Hardy and Littlewood type inequalities on time scales for the function of n variables. Special cases of obtained results include generalized Wirtinger, Hardy and Littlewood type inequalities.

Refinement multidimensional dynamic inequalities with general kernels and measures

Using the properties of superquadratic and subquadratic functions, we establish some new refinement multidimensional dynamic inequalities of Hardy’s type on time scales. Our results contain some of the recent results related to classical multidimensional Hardy’s and Pólya–Knopp’s inequalities on time scales. To show motivation of the paper, we apply our results to obtain some particular multidimensional cases and provide refinements of some Hardy-type inequalities known in the literature.

Ground state solutions for Choquard equations with Hardy-Littlewood-Sobolev upper critical growth and potential vanishing at infinity

In this paper, we investigate the following non-autonomous Choquard equation−Δu+V(x)u=(Iα⁎(|u|2α⁎+K(x)F(u)))(2α⁎|u|2α⁎−2u+K(x)f(u))inRN, where N⩾3, Iα is the Riesz potential of order α∈(0,N), 2α⁎=N+αN−2 is the upper critical exponent due to the Hardy-Littlewood-Sobolev inequality, the functions V,K∈C(RN,R+) may vanish at infinity, and the function f∈C(R,R) is noncritical and F(t)=∫0tf(s)ds. Based on variational methods and the Hardy-type inequalities, using the mountain pass theorem and the Nehari manifold approach, we prove the existence of ground state solution.

Fractional Korn and Hardy-type inequalities for vector fields in half space

We prove a fractional Hardy-type inequality for vector fields over the half space based on a modified fractional semi-norm. A priori, the modified semi-norm is not known to be equivalent to the standard fractional semi-norm and in fact gives a smaller norm, in general. As such, the inequality we prove improves the classical fractional Hardy inequality for vector fields. We will use the inequality to establish the equivalence of a space of functions (of interest) defined over the half space with the classical fractional Sobolev spaces, which amounts to prove a fractional version of the classical Korn’s inequality.

Factorizations and Hardy’s type identities and inequalities on upper half spaces

Motivated and inspired by the improved Hardy inequalities studied in their well-known works by Brezis and Vazquez (Rev Mat Univ Complut Madrid 10:443–469, 1997) and Brezis and Marcus (Ann Scuola Norm Sup Pisa Cl Sci 25(1–2):217–237, 1997), we establish in this paper several identities that imply many sharpened forms of the Hardy type inequalities on upper half spaces $$\left\{ x_{N}>0\right\} $$. We set up these results for the distance to the origin, the distance to the boundary of any strip $$ \mathbb {R} ^{N-1}\times \left( 0,R\right) $$ and the distance to the hyperplane $$\left\{ x_{N}=0\right\} $$, using both the usual full gradient and radial derivative (in the case of distance to the origin) or only the partial derivative $$\frac{\partial u}{\partial x_{N}}$$ (in the case of distance to the boundary of the strip or hyperplane). One of the applications of our main results is that when $$\Omega $$ is the strip $$\mathbb {R}^{N-1}\times \left( 0,2R\right) $$, the bound $$\lambda \left( \Omega \right) $$ given by Brezis and Marcus in Brezis and Marcus (1997) can be improved to $$\frac{z_{0}^{2}}{R^{2}}$$, where $$z_{0} =2.4048 \ldots $$ is the first zero of the Bessel function $$J_{0}\left( z\right) $$. Our approach makes use of the notion of Bessel pairs introduced by Ghoussoub and Moradifam (Math Ann 349(1):1–57, 2011) and (Functional inequalities: new perspectives and new applications. Mathematical Surveys and Monographs, American Mathematical Society, Providence, 2013) and the method of factorizations of differential operators. In particular, our identities and inequalities offer sharpened and more precise estimates of the second remainder term in the existing Hardy type inequalities on upper half spaces in the literature, including the Hardy-Sobolev-Maz’ya type inequalities.

Multidimensional Hardy Type Inequalities with Remainders

Hardy type inequalities with an additional nonnegative-term are established for compactly supported smooth functions on arbitrary open subsets and on convex domains of the Euclidean space. We prove Hardy-type inequalities in spatial domains with finite inner radius. Weight functions depend on the distance function to the boundary of the domain. We obtain one-dimensional L1-inequalities. In particular cases we obtained sharp constants. Also new Hardy type inequality with remainders for the Riemann-Liouville fractional integrals is proved.

Hardy Type Inequalities on Domains with Convex Complement and Uncertainty Principle of Heisenberg

We prove new integral inequalities for real-valued test functions defined on subdomains of the Euclidean space. We assume that the complement of the subdomain is a non-empty convex set. We prove an extension of the Hadwiger theorems about approximations of convex compact sets by polytopes and obtain some generalizations and improvements of several Hardy type multidimensional inequalities. In particular, in the last section we present an improvement of a two-dimensional inequality, connected with the uncertainty principle of Heisenberg.

New sharp Hardy and Rellich type inequalities on Cartan–Hadamard manifolds and their improvements

In this paper, we prove several new Hardy type inequalities (such as the weighted Hardy inequality, weighted Rellich inequality, critical Hardy inequality and critical Rellich inequality) related to the radial derivation (i.e., the derivation along the geodesic curves) on the Cartan–Hadamard manifolds. By Gauss lemma, our new Hardy inequalities are stronger than the classical ones. We also establish the improvements of these inequalities in terms of sectional curvature of the underlying manifolds which illustrate the effect of curvature to these inequalities. Furthermore, we obtain some improvements of Hardy and Rellich inequalities on the hyperbolic space ℍn. Especially, we show that our new Rellich inequalities are indeed stronger than the classical ones on the hyperbolic space ℍn.

Conformally Invariant Inequalities

We study conformally invariant, integral inequalities of Hardy and Rellich type in the case where the weight functions are powers of coefficients of the Poincare metric.

Properties of eigenvalues and some regularities on fractional [formula omitted]-Laplacian with singular weights

We provide fundamental properties of the first eigenpair for fractional p-Laplacian eigenvalue problems under singular weights, which is related to Hardy type inequality, and also show that the second eigenvalue is well-defined. We obtain a-priori bounds and the continuity of solutions to problems with such singular weights with some additional assumptions. Moreover, applying the above results, we show a global bifurcation emanating from the first eigenvalue, the Fredholm alternative for non-resonant problems, and obtain the existence of infinitely many solutions for some nonlinear problems involving singular weights. These are new results, even for (fractional) Laplacian.

New Characterizations of Weights in Hardy and Opial Type Inequalities via Solvability of Dynamic Equations

In this paper, we prove that the solvability of dynamic equations of second order is sufficient for the validity of some Hardy and Opial type inequalities with two different weights on time scales. In particular, the results give new characterizations of two different weights in inequalities containing Hardy and Opial operators. The main contribution in this paper is the characterizations of weights in discrete inequalities that will be formulated from our results as special cases.

On the isoperimetric constant, covariance inequalities and $L_{p}$-Poincaré inequalities in dimension one

First, we derive in dimension one a new covariance inequality of $L_{1}-L_{\infty}$ type that characterizes the isoperimetric constant as the best constant achieving the inequality. Second, we generalize our result to $L_{p}-L_{q}$ bounds for the covariance. Consequently, we recover Cheeger’s inequality without using the co-area formula. We also prove a generalized weighted Hardy type inequality that is needed to derive our covariance inequalities and that is of independent interest. Finally, we explore some consequences of our covariance inequalities for $L_{p}$-Poincaré inequalities and moment bounds. In particular, we obtain optimal constants in general $L_{p}$-Poincaré inequalities for measures with finite isoperimetric constant, thus generalizing in dimension one Cheeger’s inequality, which is a $L_{p}$-Poincaré inequality for $p=2$, to any real $p\geq1$.

Sobolev embeddings, rearrangement-invariant spaces and Frostman measures

Sobolev embeddings, of arbitrary order, are considered into function spaces on domains of Rn endowed with measures whose decay on balls is dominated by a power d of their radius. Norms in arbitrary rearrangement-invariant spaces are contemplated. A comprehensive approach is proposed based on the reduction of the relevant n-dimensional embeddings to one-dimensional Hardy-type inequalities. Interestingly, the latter inequalities depend on the involved measure only through the power d. Our results allow for the detection of the optimal target space in Sobolev embeddings, for broad families of norms, in situations where customary techniques do not apply. In particular, new embeddings, with augmented target spaces, are deduced even for standard Sobolev spaces.

Generalized Steffensen\u2019s inequality by Montgomery identity

By using generalized Montgomery identity and Green functions we proved several identities which assist in developing connections with Steffensen’s inequality. Under the assumptions of n-convexity and n-concavity many inequalities, which generalize Steffensen’s inequality, inequalities from (Fahad et al. in J. Math. Inequal. 9:481–487, 2015; Pečarić in Southeast Asian Bull. Math. 13:89–91, 1989; Rabier in Proc. Am. Math. Soc. 140:665–675, 2012), and their reverse, have been proved. Generalization of some inequalities (and their reverse) which are related to Hardy-type inequality (Fahad et al. in J. Math. Inequal. 9:481–487, 2015) have also been proved. New bounds of Ostrowski and Grüss type inequalities have been developed. Moreover, we formulate generalized Steffensen-type linear functionals and prove their monotonicity for the generalized class of (n+1)-convex functions at a point. At the end, we present some applications of our study to the theory of exponentially convex functions.

A Hardy-type inequality and some spectral characterizations for the Dirac–Coulomb operator

We prove a sharp Hardy-type inequality for the Dirac operator. We exploit this inequality to obtain spectral properties of the Dirac operator perturbed with Hermitian matrix-valued potentials \({\mathbf {V}}\) of Coulomb type: we characterise its eigenvalues in terms of the Birman–Schwinger principle and we bound its discrete spectrum from below, showing that the ground-state energy is reached if and only if \({\mathbf {V}}\) verifies some rigidity conditions. In the particular case of an electrostatic potential, these imply that \({\mathbf {V}}\) is the Coulomb potential.

Many-Particle Hardy Type Inequalities

We study a variant of the many-particle Hardy type inequality with the number of particles N ≥ 3 and discuss the sharpness of the constant in the inequality.

Caccioppoli-type estimates and Hardy-type inequalities derived from weighted p-harmonic problems

We obtain Caccioppoli-type estimates for nontrivial and nonnegative solutions to anticoercive partial differential inequalities of elliptic type involving weighted p-Laplacian: -Delta _{p,a} u:= -{mathrm {div}}(a(x)|{nabla }u|^{p-2}{nabla }u)ge b(x)Phi (u)chi _{{u>0}}, where u is defined in a domain Omega . Using Caccioppoli-type estimates, we obtain several variants of Hardy-type inequalities in weighted Sobolev spaces.

Generalized Steffensen’s Inequality by Fink’s Identity

By using Fink’s Identity, Green functions, and Montgomery identities we prove some identities related to Steffensen’s inequality. Under the assumptions of n-convexity and n-concavity, we give new generalizations of Steffensen’s inequality and its reverse. Generalizations of some inequalities (and their reverse), which are related to Hardy-type inequality. New bounds of Gr u ¨ ss and Ostrowski-type inequalities have been proved. Moreover, we formulate generalized Steffensen’s-type linear functionals and prove their monotonicity for the generalized class of ( n + 1 ) -convex functions at a point. At the end, we present some applications of our study to the theory of exponentially convex functions. .

Weighted anisotropic Hardy and Rellich type inequalities for general vector fields

In this paper, we establish the weighted anisotropic Hardy and Rellich type inequalities with boundary terms for general (real-valued) vector fields. As consequences, we derive new as well as many of the fundamental Hardy and Rellich type inequalities which are known in different settings.