Multidimensional rearrangement and Lorentz spaces

Wirtinger-type inequalities for some rearrangement invariant spaces

In this short paper we generalize the classical inequality between the norms in Lebesgue spaces of the functions and its derivatives, which in the multidimensional case are called Sobolev's inequalities, on the many popular classes pairs of rearrangement invariant (r.i.) spaces, namely, on the so-called moment rearrangement invariant spaces.

Mixed norm and multidimensional Lorentz spaces

In the last decade, the problem of characterizing the normability of the weighted Lorentz spaces has been completely solved (\cite{Sa}, \cite{CaSoA}). However, the question for multidimensional Lorentz spaces is still open. In this paper, we consider weights of product type, and give necessary and sufficient conditions for the Lorentz spaces, defined with respect to the two-dimensional decreasing rearrangement, to be normable. To this end, it is also useful to study the mixed norm Lorentz spaces. Finally, we prove embeddings between all the classical, multidimensional, and mixed norm Lorentz s…

The Fourier transform on Rearrangement-Invariant Spaces

We study inequalities of the form \begin{equation*} ρ( \lvert \hat{f} \rvert) \leq C σ(f) < \infty, \end{equation*} with $f \in L_{1}(\mathbb{R}^n)$, the Lebesgue-integrable functions on $\mathbb{R}^n$ and \begin{equation*} \hat{f}(ξ) := \int_{\mathbb{R}^n} f(x) \, e^{- 2 πi ξ\cdot x} dx, \ \ \ ξ\in \mathbb{R}^n. \end{equation*} The functionals $ρ$ and $σ$ are so-called rearrangement-invariant (r.i.) norms on $M_{+}(\mathbb{R}^n)$, the nonnegative measurable functions on $\mathbb{R}^n$. Results first proved in the general context of r.i. spaces are then both specialized and expanded on in the …

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…

Gradient Estimates via Rearrangements for Solutions of Some Schrödinger Equations

In this article, by applying the well known method for dealing with $p$-Laplace type elliptic boundary value problems, the authors establish a sharp estimate for the decreasing rearrangement of the gradient of solutions to the Dirichlet and the Neumann boundary value problems of a class of Schrödinger equations, under the weak regularity assumption on the boundary of domains. As applications, gradient estimates of these solutions in Lebesgue spaces and Lorentz spaces are obtained.

重排不变Banach型空间上粗糙奇异积分算子的复合

Composition of rough singular integral operators on rearrangement invariant Banach type spaces

本文研究满足零次齐次与球面消失条件的粗糙奇异积分算子T_Ω的复合算子，在重排不变Banach及拟Banach函数空间上的定量加权界与加权有界性，涵盖Lorentz、Orlicz等经典空间特例。