A version of Calderón-Mityagin theorem for the class of rearrangement invariant groups

Arazy-Cwikel and Calderón-Mityagin type properties of the couples $(\ell^{p},\ell^{q})$, $0 \le p<q\le\infty$

We establish Arazy-Cwikel type properties for the family of couples $(\ell^{p},\ell^{q})$, $0\le p<q\le\infty$, and show that $(\ell^{p},\ell^{q}) $ is a Calderón-Mityagin couple if and only if $q\ge1$. Moreover, we identify interpolation orbits of elements with respect to this couple for all $p$ and $q$ such that $0\le p<q\le\infty$ and obtain a simple positive solution of a Levitina-Sukochev-Zanin problem, clarifying its connections with whether $(\ell^{p},\ell^{q})$ has the Calderón-Mityagin property or not.

Calderón-Mityagin couples of Banach spaces related to decreasing functions

A number of Calderón-Mityagin couples and relative Calderón-Mityagin pairs are identified among Banach function spaces defined in terms of the least decreasing majorant construction on the half line. The interpolation structure of such spaces is shown to closely parallel that of the rearrangement invariant spaces, and it is proved that a couple of these spaces is a Calderón-Mityagin couple if and only if the corresponding couple of rearrangement invariant spaces is a Calderón-Mityagin couple. Consequently, the class of all interpolation spaces for any couple of spaces of this type admits a com…

Optimal Gagliardo-Nirenberg interpolation inequality for rearrangement invariant spaces

We prove optimality of the Gagliardo-Nirenberg inequality $$ \|\nabla u\|_{X}\lesssim\|\nabla^2 u\|_Y^{1/2}\|u\|_Z^{1/2}, $$ where $Y, Z$ are rearrangement invariant Banach function spaces and $X=Y^{1/2}Z^{1/2}$ is the Calderón--Lozanovskii space. By optimality, we mean that for a certain pair of spaces on the right-hand side, one cannot reduce the space on the left-hand, remaining in the class of rearrangement invariant spaces. The optimality for the Lorentz and Orlicz spaces is given as a consequence, exceeding previous results. We also discuss pointwise inequalities, their importance and co…

The Rearrangement-Invariant space $Γ_{p,ϕ}$

Fix $b\in (0,\infty)$ and $p\in (1,\infty)$. Let $ϕ$ be a positive measurable function on $I_b:=(0,b)$. Define the Lorentz Gamma norm, $\r_{p,ϕ}$, at the measurable function $f:\R+\to\R+$ by $\rph(f):=[\int_0^bf^{**}(t)^pϕ(t)dt]^{\frac1p}$, in which $f^{**}(t):=t^{-1}\int_0^tf^{*}(s)ds$, where $f^*(t):=μ_f^{-1}(t)$, with $μ_f(s):=|\{x\in I_b: |f(x)|>s\}|$. Our aim in this paper is to study the rearrangement-invariant space determined by $\rph$. In particular, we determine its Köthe dual and its Boyd indices. Using the latter a sufficient condition is given for a Caldéron-Zygmund operator to ma…

通过Calderón型定理的潜在迹不等式

Potential trace inequalities via a Calderón-type theorem

本文发展了一种理论工具，用于建立难以处理的算子（如势算子）在特定重排不变函数空间上的有界性，通过更易处理的算子（如分数极大算子）的性质。主要贡献是推广Korobkov和Kristensen的结果，得到关于Riesz势的不等式。