Some remarks on orthogonally additive operators on Riesz spaces

A few remarks on bounded operators on topological vector spaces

We give a few observations on different types of bounded operators on a topological vector space X and their relations with compact operators on X. In particular, we investigate when these bounded operators coincide with compact operators. We also consider similar types of bounded bilinear mappings between topological vector spaces. Some properties of tensor product operators between locally convex spaces are established. In the last part of the paper we deal with operators on topological Riesz spaces.

On disjointness, bands and projections in partially ordered vector spaces

Disjointness, bands, and band projections are a classical and essential part of the structure theory of vector lattices. If $X$ is such a lattice, those notions seem - at first glance - intimately related to the lattice operations on $X$. The last fifteen years, though, have seen an extension of all those concepts to a much larger class of ordered vector spaces. In fact if $X$ is an Archimedean ordered vector space with generating cone, or a member of the slightly larger class of pre-Riesz spaces, then the notions of disjointness, bands and band projections can be given proper meaning and give…

Some remarks on $L^1$ embeddings in the subelliptic setting

In this paper we establish an optimal Lorentz estimate for the Riesz potential in the $L^1$ regime in the setting of a stratified group $G$: Let $Q\geq 2$ be the homogeneous dimension of $G$ and $\mathcal{I}_α$ denote the Riesz potential of order $α$ on $G$. Then, for every $α\in (0,Q)$, there exists a constant $C=C(α,Q)>0$ such that \begin{align} \| \mathcal{I}_αf \|_{L^{Q/(Q-α),1}(G)} \leq C\| X \mathcal{I}_1 f \|_{L^1(G)} \end{align} for distributions $f$ such that $X \mathcal{I}_1 f \in L^1(G)$, where $X$ denotes the horizontal gradient.

Some remarks on characterization of t-normed integrals on compacta

A characterization of t-normed integrals was obtained in \cite{CLM} for finite compacta and in \cite{Rad} for the general case. Such characterization establishes a correspondence between the space of capacities and homogeneous respect t-norm monotone normed functionals preserving the maximum operation of comonotone functions. In fact these theorems we can consider as non-additive and non-linear analogues of well-known Riesz Theorem about a correspondence between the set of $σ$-additive regular Borel measures and the set of linear positively defined functionals. We discuss optimality of such ch…

一个反向里斯估计与谱隙相结合蕴含庞加莱不等式

A reverse Riesz estimate combined with a spectral gap implies a Poincaré inequality

研究在巴拿赫空间中，由阿贝尔遍历扇形算子\(A\)及无界算子\(\partial\)构成的系统，通过反向里斯估计与谱隙条件，推出庞加莱不等式及伴随散度不等式，该方法统一且灵活，能恢复并扩展相关定理，适用于多种几何情形。