scholarly journals Menchoff-Rademacher type theorems in vector-valued Banach function spaces

2007 ◽  
Vol 14 (3) ◽  
pp. 473-482
Author(s):  
Andreas Defant ◽  
Marius Junge
Keyword(s):  
Function Spaces ◽  
Banach Function Spaces ◽  
Vector Valued

scholarly journals Topological Dual Systems for Spaces of Vector Measurep-Integrable Functions

2016 ◽  
Vol 2016 ◽  
pp. 1-8
Author(s):  
P. Rueda ◽  
E. A. Sánchez Pérez
Keyword(s):  
Type Theorem ◽  
Classical Case ◽  
Function Spaces ◽  
Dual Systems ◽  
Banach Function Spaces ◽  
Infinite Dimensional ◽  
Local Compactness ◽  
Finite Dimensional ◽  
Integrable Functions ◽  
Vector Valued

We show a Dvoretzky-Rogers type theorem for the adapted version of theq-summing operators to the topology of the convergence of the vector valued integrals on Banach function spaces. In the pursuit of this objective we prove that the mere summability of the identity map does not guarantee that the space has to be finite dimensional, contrary to the classical case. Some local compactness assumptions on the unit balls are required. Our results open the door to new convergence theorems and tools regarding summability of series of integrable functions and approximation in function spaces, since we may find infinite dimensional spaces in which convergence of the integrals, our vector valued version of convergence in the weak topology, is equivalent to the convergence with respect to the norm. Examples and applications are also given.

scholarly journals Compactness in Vector-valued Banach Function Spaces

Positivity ◽  
2007 ◽  
Vol 11 (3) ◽  
pp. 461-467 ◽  
Author(s):  
Jan van Neerven
Keyword(s):  
Function Spaces ◽  
Banach Function Spaces ◽  
Vector Valued

scholarly journals Sparse domination implies vector-valued sparse domination

2022 ◽  
Author(s):  
Emiel Lorist ◽  
Zoe Nieraeth
Keyword(s):  
Function Space ◽  
Hilbert Transform ◽  
Maximal Operator ◽  
Banach Function Space ◽  
Orlicz Spaces ◽  
Function Spaces ◽  
Banach Function Spaces ◽  
Bilinear Hilbert Transform ◽  
Vector Valued ◽  
Littlewood Maximal Operator

AbstractWe prove that scalar-valued sparse domination of a multilinear operator implies vector-valued sparse domination for tuples of quasi-Banach function spaces, for which we introduce a multilinear analogue of the $${{\,\mathrm{UMD}\,}}$$ UMD condition. This condition is characterized by the boundedness of the multisublinear Hardy-Littlewood maximal operator and goes beyond examples in which a $${{\,\mathrm{UMD}\,}}$$ UMD condition is assumed on each individual space and includes e.g. iterated Lebesgue, Lorentz, and Orlicz spaces. Our method allows us to obtain sharp vector-valued weighted bounds directly from scalar-valued sparse domination, without the use of a Rubio de Francia type extrapolation result. We apply our result to obtain new vector-valued bounds for multilinear Calderón-Zygmund operators as well as recover the old ones with a new sharp weighted bound. Moreover, in the Banach function space setting we improve upon recent vector-valued bounds for the bilinear Hilbert transform.

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.

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