Extrapolation of Lp Data from a Modular Inequality

2002 ◽  
Vol 45 (1) ◽  
pp. 25-35
Author(s):  
Steven Bloom ◽  
Ron Kerman
Keyword(s):  
Orlicz Spaces ◽  
Rearrangement Invariant Space ◽  
Invariant Space ◽  
Partial Converse

AbstractIf an operator T satisfies a modular inequality on a rearrangement invariant space Lρ(Ω, μ), and if p is strictly between the indices of the space, then the Lebesgue inequality holds. This extrapolation result is a partial converse to the usual interpolation results. A modular inequality for Orlicz spaces takes the form , and here, one can extrapolate to the (finite) indices i(Φ) and I(Φ) aswell.

scholarly journals Embeddings of homogeneous Sobolev spaces on the entire space

2020 ◽  
pp. 1-33 ◽  
Author(s):  
Zdeněk Mihula
Keyword(s):  
Sobolev Spaces ◽  
Orlicz Spaces ◽  
Function Spaces ◽  
The Other ◽  
Rearrangement Invariant Space ◽  
Entire Space ◽  
Invariant Space ◽  
One Dimensional ◽  
Homogeneous Sobolev Spaces ◽  
Invariant Spaces

Abstract We completely characterize the validity of the inequality $\| u \|_{Y(\mathbb R)} \leq C \| \nabla^{m} u \|_{X(\mathbb R)}$ , where X and Y are rearrangement-invariant spaces, by reducing it to a considerably simpler one-dimensional inequality. Furthermore, we fully describe the optimal rearrangement-invariant space on either side of the inequality when the space on the other side is fixed. We also solve the same problem within the environment in which the competing spaces are Orlicz spaces. A variety of examples involving customary function spaces suitable for applications is also provided.

Variable exponents and grand Lebesgue spaces: Some optimal results

2015 ◽  
Vol 17 (06) ◽  
pp. 1550023 ◽  
Author(s):  
Alberto Fiorenza ◽  
Jean Michel Rakotoson ◽  
Carlo Sbordone
Keyword(s):  
Variable Exponent ◽  
Bounded Function ◽  
Rearrangement Invariant Space ◽  
Invariant Space ◽  
Lebesgue Spaces ◽  
Bounded Set ◽  
Decreasing Rearrangement ◽  
Grand Lebesgue Spaces ◽  
Measurable Bounded Function ◽  
Grand Lebesgue Space

Consider p : Ω → [1, +∞[, a measurable bounded function on a bounded set Ø with decreasing rearrangement p* : [0, |Ω|] → [1, +∞[. We construct a rearrangement invariant space with variable exponent p* denoted by [Formula: see text]. According to the growth of p*, we compare this space to the Lebesgue spaces or grand Lebesgue spaces. In particular, if p*(⋅) satisfies the log-Hölder continuity at zero, then it is contained in the grand Lebesgue space Lp*(0))(Ω). This inclusion fails to be true if we impose a slower growth as [Formula: see text] at zero. Some other results are discussed.

Convolution in Weighted Lorentz Spaces of Type $\Gamma$

2016 ◽  
Vol 119 (1) ◽  
pp. 113 ◽  
Author(s):  
Martin Křepela
Keyword(s):  
Convolution Operator ◽  
Lorentz Spaces ◽  
Rearrangement Invariant Space ◽  
Invariant Space ◽  
Weighted Lorentz Spaces

We characterize boundedness of the convolution operator between weighted Lorentz spaces $\Gamma^p(v)$ and $\Gamma^q(w)$ for the range of parameters $p,q\in[1,\infty]$, or $p\in(0,1)$ and $q\in\{1,\infty\}$, or $p=\infty$ and $q\in(0,1)$. We provide Young-type convolution inequalities of the form \[ \|f\ast g\|_{\Gamma^q(w)} \le C \|f\|_{\Gamma^p(v)}\|g\|_Y, \quad f\in\Gamma^p(v), g\in Y, \] characterizing the optimal rearrangement-invariant space $Y$ for which the inequality is satisfied.

The rearrangement-invariant space $$\Gamma _{p,\phi }$$ Γ p , ϕ

Positivity ◽  
2013 ◽  
Vol 18 (2) ◽  
pp. 319-345 ◽  
Author(s):  
Amiran Gogatishvili ◽  
Ron Kerman
Keyword(s):  
Rearrangement Invariant Space ◽  
Invariant Space

scholarly journals Indices, convexity and concavity of Calderón-Lozanovskii spaces

2003 ◽  
Vol 92 (1) ◽  
pp. 141 ◽  
Author(s):  
A. Kamińska ◽  
L. Maligranda ◽  
L. E. Persson
Keyword(s):  
Banach Space ◽  
Orlicz Function ◽  
Sufficient Conditions ◽  
Lorentz Spaces ◽  
Rearrangement Invariant Space ◽  
Invariant Space ◽  
Boyd Indices ◽  
Type And Cotype ◽  
Convexity And Concavity ◽  
Necessary And Sufficient

In this article we discuss lattice convexity and concavity of Calderón-Lozanovskii space $E_\varphi$, generated by a quasi-Banach space $E$ and an increasing Orlicz function $\varphi$. We give estimations of convexity and concavity indices of $E_\varphi$ in terms of Matuszewska-Orlicz indices of $\varphi$ as well as convexity and concavity indices of $E$. In the case when $E_\varphi$ is a rearrangement invariant space we also provide some estimations of its Boyd indices. As corollaries we obtain some necessary and sufficient conditions for normability of $E_\varphi$, and conditions on its nontrivial type and cotype in the case when $E_\varphi$ is a Banach space. We apply these results to Orlicz-Lorentz spaces receiving estimations, and in some cases the exact values of their convexity, concavity and Boyd indices.

scholarly journals Diffeomorphic approximation of Planar Sobolev Homeomorphisms in rearrangement invariant spaces

2021 ◽  
Author(s):  
Daniel Campbell ◽  
Luigi Greco ◽  
Roberta Schiattarella ◽  
Filip Soudsky
Keyword(s):  
Rearrangement Invariant Space ◽  
Invariant Space ◽  
Rearrangement Invariant Spaces ◽  
Invariant Spaces

Let $\Omega\subseteq\mathcal{R}^2$ be a domain, let $X$ be a rearrangement invariant space and let $f\in W^{1}X(\Omega,\mathcal{R}^2)$ be a homeomorphism between $\Omega$ and $f(\Omega)$. Then there exists a sequence of diffeomorphisms $f_k$ converging to $f$ in the space $W^{1}X(\Omega,\mathcal{R}^2)$.

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