Related papers: Hardy-Stein identity for pure-jump Dirichlet forms

We prove a Hardy-Stein type identity for the semigroups of symmetric, pure-jump L\'evy processes. Combined with the Burkholder-Gundy inequalities, it gives the $L^p$ two-way boundedness, for $1<p<\infty$, of the corresponding…

We prove the Hardy--Stein identity for vector functions in $L^p(\mathbb R^d;\mathbb R^n)$ with $1<p<\infty$ and for the canonical paring of two real functions in $L^p(\mathbb R^d)$ with $2\le p<\infty$. To this end we propose a notion of…

For an arbitrary regular Dirichlet form $\mathscr{E}$ and the associated symmetric Markovian semigroup $T_t$, we consider the corresponding Sobolev-Bregman form $\mathscr{E}_p(u) = -\tfrac{1}{p} \frac{d}{d t}\bigr\vert_{t = 0} \|T_t…

We show identities of Hardy-Stein type for harmonic functions relative to integro-differential operators corresponding to general symmetric regular Dirichlet forms satisfying the absolute continuity condition. The novelty is that we…

Let $1<p<\infty$. In this article we establish an $L^p$-Hodge decomposition theorem on sub-Riemannian compact contact manifolds without boundary, related to the Rumin complex of differential forms. Given an $L^p$- Rumin's form, we adopt an…

We obtain the boundedness in $L^p$ spaces for all $1<p<\infty$ of the so-called vertical Littlewood--Paley functions for non-local Dirichlet forms in the metric measure space under some mild assumptions. For $1<p\le 2$, the pseudo-gradient…

Given a bounded subanalytic submanifold of $\mathbb{R}^n$, possibly admitting singularities within its closure, we study the cohomology of $L^p$ differential forms having an $L^p$ exterior differential (in the sense of currents) and…

This paper is concerned with existence results for the singular $p$-biharmonic problem involving the Hardy potential and the critical Hardy-Sobolev exponent. More precisely, by using variational methods combined with the Mountain pass…

We obtain sharp fractional Hardy inequalities for the half-space and for convex domains. We extend the results of Bogdan and Dyda and of Loss and Sloane to the setting of Sobolev-Bregman forms.

We give sharp remainder terms of $L^{p}$ and weighted Hardy and Rellich inequalities on one of most general subclasses of nilpotent Lie groups, namely the class of homogeneous groups. As consequences, we obtain analogues of the generalised…

We prove a version of the $L^p$ hodge decomposition for differential forms in Euclidean space and a generalization to the class of Lizorkin currents. We also compute the $L_{qp}-$cohomology of $\mathbb{R}^n$.

In this paper we discuss cylindrical extensions of improved Hardy, Sobolev type and Caffarelli-Kohn-Nirenberg type inequalities with sharp constants and identities in the spirit of Badiale-Tarantello [2]. All identities are obtained in the…

Firstly, this paper establishes useful forms of the remainder term of Hardy-type inequalities on general domains where the weights are functions of the distance to the boundary. For weakly mean convex domains we use the resulting identities…

In this paper we develop a Fefferman-Stein theorem, a Hardy-Littlewood theorem and sharp function estimations in weighted Sobolev spaces. We also provide uniqueness and existence results for second-order elliptic and parabolic partial…

This paper is concerned with the Poisson transform of differential forms on the hyperbolic space $H^n(\mathbb R)$. Consider an integer $p$ such that $1\leqslant p\leqslant n$ and let $q$ be either $p-1$ or $p$. For $1<r<\infty$, we prove…

We present a unified and concise method for establishing L^p Hardy and Rellich inequalities for a broad class of subelliptic operators of divergence type. The approach, based on a fundamental algebraic identity, provides explicit control on…

In this paper, we provide a sharp remainder term for the general weighted discrete $p$-Hardy inequality. By simply choosing weights and specifying $1<p<\infty$, we are able to recover the identity by Krej{\v{c}}i{\v{r}}{\'\i}k-\v{S}tampach…

In the last few years the authors proved Poincar\'e and Sobolev type inequalities in Heisenberg groups $\mathbb{H}^n$ for differential forms in the Rumin's complex. The need to substitute the usual de Rham complex of differential forms for…

Let $M$ be a complete connected Riemannian manifold. Assuming that the Riemannian measure is doubling, we define Hardy spaces $H^p$ of differential forms on $M$ and give various characterizations of them, including an atomic decomposition.…

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…