scholarly journals K-ENVELOPES FOR REAL INTERPOLATION METHODS

2012 ◽  
Vol 55 (2) ◽  
pp. 293-307
Author(s):  
MING FAN
Keyword(s):  
Function Space ◽  
Upper Bounds ◽  
Function Spaces ◽  
Real Interpolation ◽  
Rearrangement Invariant Space ◽  
Interpolation Spaces ◽  
Invariant Space ◽  
Bounded Operators ◽  
Fundamental Function ◽  
Interpolation Methods

AbstractIn this paper, we study the K-envelopes of the real interpolation methods with function space parameters in the sense of Brudnyi and Kruglyak [Y. A. Brudnyi and N. Ja. Kruglyak, Interpolation functors and interpolation spaces (North-Holland, Amsterdam, Netherlands, 1991)]. We estimate the upper bounds of the K-envelopes and the interpolation norms of bounded operators for the KΦ-methods in terms of the fundamental function of the rearrangement invariant space related to the function space parameter Φ. The results concerning the quasi-power parameters and the growth/continuity envelopes in function spaces are obtained.

scholarly journals A note on function spaces generated by Rademacher series

1997 ◽  
Vol 40 (1) ◽  
pp. 119-126 ◽  
Author(s):  
Guillermo P. Curbera
Keyword(s):  
Function Space ◽  
Invariant Function ◽  
Function Spaces ◽  
Lorentz Spaces ◽  
Rearrangement Invariant Space ◽  
Invariant Space ◽  
Fundamental Function ◽  
Rademacher Functions ◽  
Measurable Functions ◽  
Rademacher Series

Let X be a rearrangement invariant function space on [0,1] in which the Rademacher functions (rn) generate a subspace isomorphic to ℓ2. We consider the space Λ(R, X) of measurable functions f such that fg∈X for every function g=∑bnrn where (bn)∈ℓ2. We show that if X satisfies certain conditions on the fundamental function and on certain interpolation indices then the space Λ(R, X) is not order isomorphic to a rearrangement invariant space. The result includes the spaces Lp, q and certain classes of Orlicz and Lorentz spaces. We also study the cases X = Lexp and X = Lψ2 for ψ2) = exp(t2) – 1.

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.

scholarly journals Commutators in real interpolation with quasi-power parameters

2002 ◽  
Vol 7 (5) ◽  
pp. 239-257 ◽  
Author(s):  
Ming Fan
Keyword(s):  
Function Spaces ◽  
Higher Order ◽  
Real Interpolation ◽  
Hardy Inequalities ◽  
Commutator Estimates ◽  
The Real ◽  
Interpolation Methods ◽  
Commutator Theorem

The basic higher order commutator theorem is formulated for the real interpolation methods associated with the quasi-power parameters, that is, the function spaces on which Hardy inequalities are valid. This theorem unifies and extends various results given by Cwikel, Jawerth, Milman, Rochberg, and others, and incorporates some results of Kalton to the context of commutator estimates for the real interpolation methods.

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.

scholarly journals The Concepts of Well-Posedness and Stability in Different Function Spaces for the 1D Linearized Euler Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Stefan Balint ◽  
Agneta M. Balint
Keyword(s):  
Euler Equation ◽  
Function Space ◽  
Euler Equations ◽  
Small Perturbation ◽  
Function Spaces ◽  
Well Posedness ◽  
Small Solution ◽  
Null Solution ◽  
Linearized Euler Equations ◽  
The Stability

This paper considers the stability of constant solutions to the 1D Euler equation. The idea is to investigate the effect of different function spaces on the well-posedness and stability of the null solution of the 1D linearized Euler equations. It is shown that the mathematical tools and results depend on the meaning of the concepts “perturbation,” “small perturbation,” “solution of the propagation problem,” and “small solution, that is, solution close to zero,” which are specific for each function space.

Weighted Hardy inequalities, real interpolation methods and vector measures

2014 ◽  
Vol 109 (2) ◽  
pp. 337-352
Author(s):  
Ricardo del Campo ◽  
Antonio Fernández ◽  
Antonio Manzano ◽  
Fernando Mayoral ◽  
Francisco Naranjo
Keyword(s):  
Real Interpolation ◽  
Hardy Inequalities ◽  
Vector Measures ◽  
Interpolation Methods

scholarly journals Duals of vector-valued Köthe function spaces

1992 ◽  
Vol 112 (1) ◽  
pp. 165-174 ◽  
Author(s):  
Miguel Florencio ◽  
Pedro J. Paúl ◽  
Carmen Sáez
Keyword(s):  
Function Space ◽  
Normed Space ◽  
Function Spaces ◽  
Integrable Functions ◽  
Nikodym Property ◽  
Köthe Dual ◽  
Vector Valued

AbstractLet Λ be a perfect Köthe function space in the sense of Dieudonné, and Λ× its Köthe-dual. Let E be a normed space. Then the topological dual of the space Λ(E) of Λ-Bochner integrable functions equals the corresponding Λ×(E′) if and only if E′ has the Radon–Nikodým property. We also give some results concerning barrelledness for spaces of this kind.

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.

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