scholarly journals New classes of rearrangement-invariant spaces appearing in extreme cases of weak interpolation

2006 ◽  
Vol 4 (3) ◽  
pp. 275-304 ◽  
Author(s):  
Evgeniy Pustylnik ◽  
Teresa Signes
Keyword(s):  
Weak Type ◽  
Interpolation Theorem ◽  
Optimal Interpolation ◽  
Rearrangement Invariant Space ◽  
Invariant Space ◽  
Type Interpolation ◽  
Rearrangement Invariant Spaces ◽  
Invariant Spaces ◽  
Quasiconcave Function

We study weak type interpolation for ultrasymmetric spacesL?,Ei.e., having the norm??(t)f*(t)?E˜, where?(t)is any quasiconcave function andE˜is arbitrary rearrangement-invariant space with respect to the measuredt/t. When spacesL?,Eare not “too close” to the endpoint spaces of interpolation (in the sense of Boyd), the optimal interpolation theorem was stated in [13]. The case of “too close” spaces was studied in [15] with results which are optimal, but only among ultrasymmetric spaces. In this paper we find better interpolation results, involving new types of rearrangement-invariant spaces,A?,b,EandB?,b,E, which are described and investigated in detail.

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 $W^{1}X(\Omega,\mathcal{R}^2)$.

scholarly journals The strong Fatou property of risk measures

2018 ◽  
Vol 6 (1) ◽  
pp. 183-196 ◽  
Author(s):  
Shengzhong Chen ◽  
Niushan Gao ◽  
Foivos Xanthos
Keyword(s):  
Risk Measures ◽  
Lower Semicontinuous ◽  
Rearrangement Invariant Space ◽  
Invariant Space ◽  
Fatou Property ◽  
Rearrangement Invariant Spaces ◽  
Dual Representations ◽  
Wide Range ◽  
Invariant Functional ◽  
Invariant Spaces

AbstractIn this paper, we explore several Fatou-type properties of risk measures. The paper continues to reveal that the strong Fatou property,whichwas introduced in [19], seems to be most suitable to ensure nice dual representations of risk measures. Our main result asserts that every quasiconvex law-invariant functional on a rearrangement invariant space X with the strong Fatou property is (X, L1) lower semicontinuous and that the converse is true on a wide range of rearrangement invariant spaces. We also study inf-convolutions of law-invariant or surplus-invariant risk measures that preserve the (strong) Fatou property.

Hermite Conjugate Functions and Rearrangement Invariant Spaces

1973 ◽  
Vol 16 (3) ◽  
pp. 377-380 ◽  
Author(s):  
Kenneth F. Andersen
Keyword(s):  
Weak Type ◽  
Conjugate Function ◽  
Poisson Integral ◽  
Conjugate Functions ◽  
Rearrangement Invariant Spaces ◽  
Image Position ◽  
Invariant Spaces

The Hermite conjugate Poisson integral of a given f ∊ L1(μ), dμ(y)= exp(—y2) dy, was defined by Muckenhoupt [5, p. 247] aswhereIf the Hermite conjugate function operator T is defined by (Tf) a.e., then one of the main results of [5] is that T is of weak-type (1, 1) and strongtype (p,p) for all p>l.

scholarly journals Atomic decompositions of Lorentz martingale spaces and applications

2009 ◽  
Vol 7 (2) ◽  
pp. 153-166 ◽  
Author(s):  
Jiao Yong ◽  
Peng Lihua ◽  
Liu Peide
Keyword(s):  
Atomic Decomposition ◽  
Weak Type ◽  
Interpolation Theorem ◽  
Sublinear Operator ◽  
Sufficient Condition ◽  
Atomic Decompositions ◽  
Type Interpolation ◽  
Marcinkiewicz Interpolation Theorem ◽  
Decomposition Theorems

In the paper we present three atomic decomposition theorems of Lorentz martingale spaces. With the help of atomic decomposition we obtain a sufficient condition for sublinear operator defined on Lorentz martingale spaces to be bounded. Using this sufficient condition, we investigate some inequalities on Lorentz martingale spaces. Finally we discuss the restricted weak-type interpolation, and prove the classical Marcinkiewicz interpolation theorem in the martingale setting.

scholarly journals Optimal Couples of Rearrangement Invariant Spaces for Generalized Maximal Operators

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
Irshaad Ahmed ◽  
Waqas Nazeer
Keyword(s):  
Maximal Operator ◽  
Maximal Operators ◽  
Rearrangement Invariant Spaces ◽  
Quasiconcave Functions ◽  
Invariant Spaces ◽  
Quasiconcave Function

The optimal couples of rearrangement invariant spaces for boundedness of a generalized maximal operator, associated with a quasiconcave function, have been characterized in terms of certain indices connected with rearrangement invariant spaces and quasiconcave functions.

scholarly journals Weak-Type Boundedness of the Fourier Transform on Rearrangement Invariant Function Spaces

2018 ◽  
Vol 61 (3) ◽  
pp. 879-890
Author(s):  
Santiago Boza ◽  
Javier Soria
Keyword(s):  
Fourier Transform ◽  
Weak Type ◽  
Bounded Operator ◽  
Invariant Function ◽  
Rearrangement Invariant Space ◽  
Rearrangement Invariant Function Spaces ◽  
Weighted Lorentz Spaces ◽  
The Fourier Transform ◽  
Invariant Spaces ◽  
Maximal Space

AbstractWe study several questions about the weak-type boundedness of the Fourier transform ℱ on rearrangement invariant spaces. In particular, we characterize the action of ℱ as a bounded operator from the minimal Lorentz space Λ(X) into the Marcinkiewicz maximal space M(X), both associated with a rearrangement invariant space X. Finally, we also prove some results establishing that the weak-type boundedness of ℱ, in certain weighted Lorentz spaces, is equivalent to the corresponding strong-type estimates.

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.

Indices of Function Spaces and their Relationship to Interpolation

1969 ◽  
Vol 21 ◽  
pp. 1245-1254 ◽  
Author(s):  
David W. Boyd
Keyword(s):  
Linear Operator ◽  
Function Space ◽  
Weak Type ◽  
General Theorem ◽  
Function Spaces ◽  
Rearrangement Invariant Space ◽  
Invariant Space ◽  
Special Case

A special case of the theorem of Marcinkiewicz states that if T is a linear operator which satisfies the weak-type conditions (p, p) and (q,q), then T maps Lr continuously into itself for any r with p < r < q. In a recent paper (5), as part of a more general theorem, Calderόn has characterized the spaces X which can replace Lr in the conclusion of this theorem, independent of the operator T. The conditions which X must satisfy are phrased in terms of an operator S(σ) which acts on the rearrangements of the functions in X.One of Calderόn's results implies that if X is a function space in the sense of Luxemburg (9), then X must be a rearrangement-invariant space.

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