A Nonseparable Invariant Extension of Lebesgue Measure -- A Generalized and Abstract Approach

某些不可分抽象插值空间的$2$-圆性

$2$-rotundity of some nonseparable abstract interpolation spaces

推广Argyros-Motakis构造，定义与具有无条件基的自反空间及Schreier空间相关的不可分抽象插值空间，在族复杂度假设下证明该空间具有含不可数$1$-无条件基的$2$-圆范数。

抽象测度大勒贝格空间及其应用

Abstract Measure Grand Lebesgue Spaces and Applications

介绍具有球基结构的抽象测度大勒贝格空间，研究其基本性质及\(\mathfrak{BO}\)算子行为，利用稀疏控制和范数不等式得出算子范数估计，并证明\(\mathfrak{BO}\)算子包含多种算子。

Generalized vectorial Lebesgue and Bochner integration theory

This paper contains a development of the Theory of Lebesgue and Bochner spaces of summable functions. It represents a synthesis of the results due to H. Lebesgue, S. Banach, S. Bochner, G. Fubini, S. Saks, F. Riesz, N. Dunford, P. Halmos, and other contributors to this theory. The construction of the theory is based on the notion of a measure on a prering of sets in any abstract space X. No topological structure of the space X is required for the development of the theory. Measures on prerings generalize the notion of abstract Lebesgue measures. These measures are readily available and it is n…

Use of abstract Hardy spaces, Real interpolation and Applications to bilinear operators

This paper can be considered as the sequel of [6], where the authors have proposed an abstract construction of Hardy spaces H^1. They shew an interpolation result for these Hardy spaces with the Lebesgue spaces. Here we describe a more precise result using the real interpolation theory and we clarify the use of Hardy spaces. Then with the help of the bilinear interpolation theory, we then give applications to study bilinear operators on Lebesgue spaces. These ideas permit us to study singular operators with singularities similar to those of bilinear Calderon-Zygmund operators in a far more abs…

Generalized almost periodic and generalized asymptotically almost periodic type functions in Lebesgue spaces with variable exponents $L^{p(x)}$

In the paper under review, we introduce the notions of various types of generalized (asymptotical) almost periodicity with variable exponents. We define and thoroughly analyze an important subclass of (asymptotically) Stepanov almost periodic functions which contains all (asymptotically) almost periodic functions. We provide a great number of relevant applications to abstract Volterra integro-differential equations in Banach spaces.

One-dimensional Schrödinger operators with $δ'$-interactions on a set of Lebesgue measure zero

We give an abstract definition of a one-dimensional Schrödinger operator with $δ'$-interaction on an arbitrary set~$Γ$ of Lebesgue measure zero. The number of negative eigenvalues of such an operator is at least as large as the number of those isolated points of the set~$Γ$ that have negative values of the intensity constants of the $δ'$-interaction. In the case where the set~$Γ$ is endowed with a Radon measure, we give constructive examples of such operators having an infinite number of negative eigenvalues.