scholarly journals Octahedrality in Lipschitz-free Banach spaces

2017 ◽  
Vol 148 (3) ◽  
pp. 447-460 ◽  
Author(s):  
Julio Becerra Guerrero ◽  
Ginés López-Pérez ◽  
Abraham Rueda Zoca
Keyword(s):  
Banach Spaces ◽  
Metric Space ◽  
Lipschitz Function ◽  
Function Spaces ◽  
Weak Star ◽  
Vector Valued

The aim of this note is to study octahedrality in vector-valued Lipschitz-free Banach spaces on a metric space, under topological hypotheses on it, by analysing the weak-star strong diameter 2 property in Lipschitz function spaces. Also, we show an example that proves that our results are optimal and that octahedrality in vector-valued Lipschitz-free Banach spaces actually relies on the underlying metric space as well as on the Banach one.

scholarly journals 2-Local standard isometries on vector-valued Lipschitz function spaces

2018 ◽  
Vol 461 (2) ◽  
pp. 1287-1298 ◽  
Author(s):  
Antonio Jiménez-Vargas ◽  
Lei Li ◽  
Antonio M. Peralta ◽  
Liguang Wang ◽  
Ya-Shu Wang
Keyword(s):  
Lipschitz Function ◽  
Function Spaces ◽  
Local Standard ◽  
Vector Valued

On the Basis Problem for Vector Valued Function Spaces

1956 ◽  
Vol 8 ◽  
pp. 417-422 ◽  
Author(s):  
H. W. Ellis
Keyword(s):  
Banach Spaces ◽  
Function Spaces ◽  
Length Function ◽  
General Measure ◽  
Basis Problem ◽  
Measure Spaces ◽  
Vector Valued Function ◽  
Vector Valued

1. Introduction. In a recent paper (2) Halperin and the author considered separable Banach spaces Lλ of real valued functions on general measure spaces and proved the existence of 1-regular (§2) Haar or σ-Haar bases when λ was the classical p-norm or any levelling length function (3) and, more generally, of K-regular Haar or σ-Haar bases when λ was a continuous length function satisfying certain additional conditions (2, Theorem 3.2).

On l1 subspaces of Orlicz vector-valued function spaces

1987 ◽  
Vol 101 (1) ◽  
pp. 107-112 ◽  
Author(s):  
Fernando Bombal
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
Function Spaces ◽  
Complete Characterization ◽  
Complemented Subspace ◽  
Vector Valued Function ◽  
Vector Valued ◽  
General Banach Space

The purpose of this paper is to characterize the Orlicz vector-valued function spaces containing a copy or a complemented copy of l1. Pisier proved in [13] that if a Banach space E contains no copy of l1, then the space Lp(S, Σ, μ, E) does not contain it either, for 1 < p < ∞. We extend this result to the case of Orlicz vector valued function spaces, by reducing the problem to the situation considered by Pisier. Next, we pass to study the problem of embedding l1 as a complemented subspace of LΦ(E). We obtain a complete characterization when E is a Banach lattice and only partial results in case of a general Banach space. We use here in a crucial way a result of E. Saab and P. Saab concerning the embedding of l1 as a complemented subspace of C(K, E), the Banach space of all the E-valued continuous functions on the compact Hausdorff space K (see [14]). Finally, we use these results to characterize several classes of Banach spaces for which LΦ(E) has some Banach space properties, namely the reciprocal Dunford-Pettis property and Pelczyński's V property.

On Vector-Valued Lipschitz Function Spaces

1987 ◽  
Vol 30 (1) ◽  
pp. 43-48 ◽  
Author(s):  
Pablo Galindo
Keyword(s):  
Lipschitz Condition ◽  
Open Subset ◽  
Sequence Space ◽  
Lipschitz Function ◽  
Function Spaces ◽  
Vector Valued ◽  
Compact Subsets

AbstractThis paper is devoted to obtaining sequence space representations of spaces of vector-valued Ck-functions defined on an open subset, Ω, of ℝn, whose kth derivatives satisfy a Lipschitz condition on compact subsets of Ω.

scholarly journals A note on singular integrals with angular integrability

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Feng Liu
Keyword(s):  
Singular Integral ◽  
Singular Integrals ◽  
Function Spaces ◽  
Riesz Transforms ◽  
Hilbert Transforms ◽  
Rotation Method ◽  
Vector Valued Function ◽  
Vector Valued ◽  
L Log L

Abstract In this note we study the rough singular integral $$ T_{\varOmega }f(x)=\mathrm{p.v.} \int _{\mathbb{R}^{n}}f(x-y)\frac{\varOmega (y/ \vert y \vert )}{ \vert y \vert ^{n}}\,dy, $$ T Ω f ( x ) = p . v . ∫ R n f ( x − y ) Ω ( y / | y | ) | y | n d y , where $n\geq 2$ n ≥ 2 and Ω is a function in $L\log L(\mathrm{S} ^{n-1})$ L log L ( S n − 1 ) with vanishing integral. We prove that $T_{\varOmega }$ T Ω is bounded on the mixed radial-angular spaces $L_{|x|}^{p}L_{\theta }^{\tilde{p}}( \mathbb{R}^{n})$ L | x | p L θ p ˜ ( R n ) , on the vector-valued mixed radial-angular spaces $L_{|x|}^{p}L_{\theta }^{\tilde{p}}(\mathbb{R}^{n},\ell ^{\tilde{p}})$ L | x | p L θ p ˜ ( R n , ℓ p ˜ ) and on the vector-valued function spaces $L^{p}(\mathbb{R}^{n}, \ell ^{\tilde{p}})$ L p ( R n , ℓ p ˜ ) if $1<\tilde{p}\leq p<\tilde{p}n/(n-1)$ 1 < p ˜ ≤ p < p ˜ n / ( n − 1 ) or $\tilde{p}n/(\tilde{p}+n-1)< p\leq \tilde{p}<\infty $ p ˜ n / ( p ˜ + n − 1 ) < p ≤ p ˜ < ∞ . The same conclusions hold for the well-known Riesz transforms and directional Hilbert transforms. It should be pointed out that our proof is based on the Calderón–Zygmund’s rotation method.

Vector-valued variational principle in fuzzy metric space and its applications

2001 ◽  
Vol 119 (2) ◽  
pp. 343-354 ◽  
Author(s):  
Jiang Zhu ◽  
Cheng-Kui Zhong ◽  
Ge-Ping Wang
Keyword(s):  
Variational Principle ◽  
Metric Space ◽  
Fuzzy Metric Space ◽  
Fuzzy Metric ◽  
Vector Valued
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