The sharpness of various Hardy-type inequalities is well-understood in the reversible Finsler setting; while infinite reversibility implies the failure of these functional inequalities, cf. Kristály et al. [Trans. Am. Math. Soc., 2020]. However, in the remaining case of irreversible manifolds with finite reversibility, there is no evidence on the sharpness of Hardy-type inequalities. In fact, we are not aware of any particular examples where the sharpness persists. In this paper, we present two such examples involving two celebrated inequalities: the classical/weighted Hardy inequality (assuming non-positive flag curvature) and the McKean-type spectral gap estimate (assuming strong negative flag curvature). In both cases, we provide a family of Finsler metric measure manifolds on which these inequalities are sharp. We also establish some sufficient conditions, which guarantee the sharpness of more involved Hardy-type inequalities on these spaces. Our relevant technical tool is a Finslerian extension of the method of Riccati pairs (for proving Hardy inequalities), which also inspires the main ideas of our constructions.

In this paper, we prove the existence of minimizers for the sharp stability constant of Caffarelli–Kohn–Nirenberg inequality near the new curve $b^*_{\mathrm{FS}}(a)$ (which lies above the well-known Felli–Schneider curve $b_{\mathrm{FS}}(a)$), extending the work of Wei and Wu [Math. Z., 2024] to a slightly larger region. Moreover, we provide an upper bound for the Caffarelli–Kohn–Nirenberg inequality with an explicit sharp constant, which may have its own interest.

Recently, the discrete reversed Hardy–Littlewood–Sobolev inequality with infinite terms was proved. In this article, we study the attainability of its best constant. For this purpose, we introduce a discrete reversed Hardy–Littlewood–Sobolev inequality with finite terms. The constraint of parameters of this inequality is more relaxed than that of parameters of inequality with infinite terms. We here show the limit relations between their best constants and between their extremal sequences. Based on these results, we obtain the attainability of the best constant of the inequality with infinite terms in the noncritical case.

In this paper, we establish a new version of one-dimensional discrete improved Hardy’s inequality with shifts by introducing a shifting discrete Dirichlet’s Laplacian. We prove that the general discrete Hardy’s inequality as well as its variants in some special cases admit improvements. Further, it is proved that two-variable discrete $p$-Hardy inequality can also be improved via improved discrete $p$-Hardy inequality in one dimension. The result is also extended to the multivariable cases.

This article aims to establish fractional Sobolev trace inequalities, logarithmic Sobolev trace inequalities, and Hardy trace inequalities associated with time-space fractional heat equations. The key steps involve establishing dedicated estimates for the fractional heat kernel, regularity estimates for the solution of the time-space fractional equations, and characterizing the norm of $\dot {W}^{\nu /2}_p(\mathbb {R}^n)$ in terms of the solution $u(x,t)$. Additionally, fractional logarithmic Gagliardo–Nirenberg inequalities are proven, leading to $L^p-$logarithmic Sobolev inequalities for $\dot {W}^{\nu /2}_{p}(\mathbb R^{n})$. As a byproduct, Sobolev affine trace-type inequalities for $\dot {H}^{-\nu /2}(\mathbb {R}^n)$ and local Sobolev-type trace inequalities for $Q_{\nu /2}(\mathbb {R}^n)$ are established.

We establish a new improvement of the classical Lp-Hardy inequality on the multidimensional Euclidean space in the supercritical case. Recently, in [14], there has been a new kind of development of the one-dimensional Hardy inequality. Using some radialisation techniques of functions and then exploiting symmetric decreasing rearrangement arguments on the real line, the new multidimensional version of the Hardy inequality is given. Some consequences are also discussed.

In this paper, we prove some weighted sharp inequalities of Trudinger–Moser type. The weights considered here have a logarithmic growth. These inequalities are completely new and are established in some new Sobolev spaces where the norm is a mixture of the norm of the gradient in two different Lebesgue spaces. This fact allowed us to prove a very interesting result of sharpness for the case of doubly exponential growth at infinity. Some improvements of these inequalities for the weakly convergent sequences are also proved using a version of the Concentration-Compactness principle of P.L. Lions. Taking profit of these inequalities, we treat in the last part of this work some elliptic quasilinear equation involving the weighted $(N,q)-$Laplacian operator where $1 < q < N$ and a nonlinearities enjoying a new type of exponential growth condition at infinity.

We investigate the continuity and differentiability of the Hardy constant with respect to perturbations of the domain in the case where the problem involves the distance from a boundary submanifold. We also investigate the case where only the submanifold is deformed.

We prove a reversed Hardy–Littlewood–Pólya inequality with finite terms. We also give the limit of the best constant.

We introduce a notion of barycenter of a probability measure related to the symmetric mean of a collection of non-negative real numbers. Our definition is inspired by the work of Halász and Székely, who in 1976 proved a law of large numbers for symmetric means. We study the analytic properties of this Halász–Székely barycenter. We establish fundamental inequalities that relate the symmetric mean of a list of non-negative real numbers with the barycenter of the measure uniformly supported on these points. As consequence, we go on to establish an ergodic theorem stating that the symmetric means of a sequence of dynamical observations converge to the Halász–Székely barycenter of the corresponding distribution.

In this paper, by the introduction of several parameters, we construct a new kernel function which is defined in the whole plane and includes some classical kernel functions. Estimating the weight functions with the techniques of real analysis, we establish a new Hilbert-type inequality in the whole plane, and the constant factor of the newly obtained inequality is proved to be the best possible. Additionally, by means of the partial fraction expansion of the tangent function, some special and interesting inequalities are presented at the end of the paper.

In this paper, we establish an infinite series expansion of Leray–Trudinger inequality, which is closely related with Hardy inequality and Moser Trudinger inequality. Our result extends early results obtained by Mallick and Tintarev [A. Mallick and C. Tintarev. An improved Leray-Trudinger inequality. Commun. Contemp. Math. 20 (2018), 17501034. OP 21] to the case with many logs. It should be pointed out that our result is about series expansion of Hardy inequality under the case $p=n$, which case is not considered by Gkikas and Psaradakis in [K. T. Gkikas and G. Psaradakis. Optimal non-homogeneous improvements for the series expansion of Hardy's inequality. Commun. Contemp. Math. doi:10.1142/S0219199721500310]. However, we can't obtain the optimal form by our method.

Based on the Gale–Ryser theorem [2, 6], for the existence of suitable $(0,1)$-matrices for different partitions of a natural number, we revisit the classical result of Lorentz [4] regarding the characterization of a plane measurable set, in terms of its cross-sections, and extend it to general measure spaces.

In this paper we obtain some improved $L^p$-Hardy and $L^p$-Rellich inequalities on bounded domains of Riemannian manifolds. For Cartan–Hadamard manifolds we prove the inequalities with sharp constants and with weights being hyperbolic functions of the Riemannian distance.

Weight criteria for embedding of the weighted Sobolev–Lorentz spaces to the weighted Besov–Lorentz spaces built upon certain mixed norms and iterated rearrangement are investigated. This gives an improvement of some known Sobolev embedding. We achieve the result based on different norm inequalities for the weighted Besov–Lorentz spaces defined in some mixed norms.

We show that the sequence of moments of order less than 1 of averages of i.i.d. positive random variables is log-concave. For moments of order at least 1, we conjecture that the sequence is log-convex and show that this holds eventually for integer moments (after neglecting the first $p^2$ terms of the sequence).

The notion of the capacity of a polynomial was introduced by Gurvits around 2005, originally to give drastically simplified proofs of the van der Waerden lower bound for permanents of doubly stochastic matrices and Schrijver’s inequality for perfect matchings of regular bipartite graphs. Since this seminal work, the notion of capacity has been utilised to bound various combinatorial quantities and to give polynomial-time algorithms to approximate such quantities (e.g. the number of bases of a matroid). These types of results are often proven by giving bounds on how much a particular differential operator can change the capacity of a given polynomial. In this paper, we unify the theory surrounding such capacity-preserving operators by giving tight capacity preservation bounds for all nondegenerate real stability preservers. We then use this theory to give a new proof of a recent result of Csikvári, which settled Friedland’s lower matching conjecture.

We determine the asymptotic behavior of the eigenvalues of finite sections of the multiplicative Hilbert matrices.