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We establish various Hardy inequalities involving the distance function from submanifolds of Riemannian manifolds, where the natural weights are expressed in terms of bounds of the mean curvature of the submanifold and sectional/Ricci…

We consider sharp inequalities for Bergman spaces of the unit disc, establishing analogues of the inequality in Carleman's proof of the isoperimetric inequality and of Weissler's inequality for dilations. By contractivity and a standard…

In this note we prove a class of sharp inequalities for singular integral operators in weighted Lebesgue spaces with angular integrability.

We prove Hardy inequalities for the conformally invariant fractional powers of the sublaplacian on the Heisenberg group $\mathbb{H}^n$. We prove two versions of such inequalities depending on whether the weights involved are non-homogeneous…

We obtain a trace Hardy inequality for the Euclidean space with a bounded cut $\Sigma\subset\mathbb R^d$, $d \ge 2$. In this novel geometric setting, the Hardy-type inequality non-typically holds also for $d = 2$. The respective Hardy…

The paper is devoted to weighted $L^p$-Hardy inequalities with best constants on Finsler metric measure manifolds. There are two major ingredients. The first, which is the main part of this paper, is the Hardy inequalities concerned with…

We establish existence of weighted Hardy and Rellich inequalities on the spaces $L_p(\Omega)$ where $\Omega= \Ri^d\backslash K$ with $K$ a closed convex subset of $\Ri^d$. Let $\Gamma=\partial\Omega$ denote the boundary of $\Omega$ and…

We consider a Schr\"odinger operator with complex-valued potentials on the line. The operator has essential spectrum on the half-line plus eigenvalues (counted with algebraic multiplicity) in the complex plane without the positive…

Let $\Omega$ be a smooth bounded domain in $\mahbb R^N$ with $N\ge 3$ and let $\Sigma_k$ be a closed smooth submanifold of $\delta \Omega$ of dimension $1\le k\le N-2$. In this paper we study the weighted Hardy inequality with weight…

In the paper two-weighted norm estimates with general weights for Hardy-type transforms, maximal functions, potentials and Calder\'on-Zygmund singular integrals in variable exponent Lebesgue spaces defined on quasimetric measure spaces $(X,…

In this paper, we derive a Reilly formula for differential forms on weighted manifolds with nonempty boundary. As an application of this formula, we prove a Poincar\'e-type inequality in the same context and explore several of its…

For doubling weights, we obtain a necessary and sufficient condition such that the one weighted inequality of the integral operator induced by Hardy kernels on the unit disk holds. This confirms a conjecture by Guo and Wang in such…

We obtain Hardy inequality for non-local diffusion operator with singular drift, in the case when the strength of attraction to the origin by the drift takes the critical value.

We deal with weighted Hardy-Sobolev type inequalities for functions on $\mathbb{R}^d$, $d\geq 2$. The weights involved are anisotropic, given by products of powers of the distance to the origin and to a nontrivial subspace. We establish…

We establish improved versions of the Hardy and Caffarelli-Kohn-Nirenberg inequalities by replacing the standard Dirichlet energy with some nonlocal nonconvex functionals which have been involved in estimates for the topological degree of…

The aim of this work is to study comparability of nonlocal Dirichlet forms. We provide sufficient conditions on the kernel for local and global comparability. As an application we prove a-priori estimates in H\"{o}lder spaces for solutions…

The behavior of certain weighted Hardy-type operators on rearrangement-invariant function spaces is thoroughly studied with emphasis being put on the optimality of the obtained results. First, the optimal rearrangement-invariant function…

Let $\O$ be a smooth bounded domain in $\R^N$ with $N\ge 1$. In this paper we study the Hardy-Poincar\'e inequalities with weight function singular at the boundary of $\O$. In particular we give sufficient conditions so that the best…

An inequality of Hardy type is established for quadratic forms involving Dirac operator and a weight $r^{-b}$ for functions in $\R^n$. The exact Hardy constant $c_b=c_b(n)$ is found and generalized minimizers are given. The constant $c_b$…

Inspired by the work of Cossetti and D'Arca [CD25], we show that the general weighted $L^{p}$-Hardy type inequalities [CD25, Theorems 1.1 and 1.2] and the corresponding identities hold for all $1<p<\infty$, thus extending their results…