scholarly journals Orlicz spaces associated to a quasi-Banach function space: applications to vector measures and interpolation

2020 ◽  
Author(s):  
Ricardo del Campo ◽  
Antonio Fernández ◽  
Fernando Mayoral ◽  
Francisco Naranjo
Keyword(s):  
Function Space ◽  
Banach Function Space ◽  
Orlicz Spaces ◽  
Space Applications ◽  
Vector Measures

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.

A Stability Property of a Class of Banach Spaces Not Containing c0

1992 ◽  
Vol 35 (1) ◽  
pp. 56-60 ◽  
Author(s):  
Patrick N. Dowling
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Function Space ◽  
Stability Property ◽  
Banach Function Space ◽  
Ideal Space

AbstractLet E be a Banach ideal space and X be a Banach space. The Banach function space E(X) does not contain a copy of C0 if and only if neither E nor X contains a copy of c0. Some extensions of this result are also noted.

scholarly journals Banach Lattice Structures and Concavifications in Banach Spaces

Mathematics ◽  
2020 ◽  
Vol 8 (1) ◽  
pp. 127
Author(s):  
Lucia Agud ◽  
Jose Manuel Calabuig ◽  
Maria Aranzazu Juan ◽  
Enrique A. Sánchez Pérez
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Function Space ◽  
Lattice Structure ◽  
Banach Function Space ◽  
Finite Measure ◽  
Multiplication Operators ◽  
Spaces Of Operators ◽  
Finite Measure Space ◽  
Local Properties

Let ( Ω , Σ , μ ) be a finite measure space and consider a Banach function space Y ( μ ) . We say that a Banach space E is representable by Y ( μ ) if there is a continuous bijection I : Y ( μ ) → E . In this case, it is possible to define an order and, consequently, a lattice structure for E in such a way that we can identify it as a Banach function space, at least regarding some local properties. General and concrete applications are shown, including the study of the notion of the pth power of a Banach space, the characterization of spaces of operators that are isomorphic to Banach lattices of multiplication operators, and the representation of certain spaces of homogeneous polynomials on Banach spaces as operators acting in function spaces.

scholarly journals Sobolev type inequalities in ultrasymmetric spaces with applications to Orlicz-Sobolev embeddings

2005 ◽  
Vol 3 (2) ◽  
pp. 183-208 ◽  
Author(s):  
Evgeniy Pustylnik
Keyword(s):  
Function Space ◽  
Sobolev Spaces ◽  
Banach Function Space ◽  
Sobolev Type ◽  
Small Space ◽  
Partial Derivatives ◽  
Sobolev Embeddings ◽  
Derivatives Of

LetDkfmean the vector composed by all partial derivatives of orderkof a functionf(x),x∈Ω⊂ℝn. Given a Banach function spaceA, we look for a possibly small spaceBsuch that‖f‖B≤c‖|Dkf|‖Afor allf∈C0k(Ω). The estimates obtained are applied to ultrasymmetric spacesA=Lφ,E,B=Lψ,E, giving some optimal (or rather sharp) relations between parameter-functionsφ(t)andψ(t)and new results for embeddings of Orlicz-Sobolev spaces.

scholarly journals A NECESSARY AND SUFFICIENT CONDITION FOR CERTAIN MARTINGALE INEQUALITIES IN BANACH FUNCTION SPACES

2007 ◽  
Vol 49 (3) ◽  
pp. 431-447 ◽  
Author(s):  
MASATO KIKUCHI
Keyword(s):  
Function Space ◽  
Probability Space ◽  
Banach Function Space ◽  
Sufficient Conditions ◽  
Function Spaces ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Banach Function Spaces ◽  
Martingale Inequalities ◽  
Necessary And Sufficient

AbstractLet X be a Banach function space over a nonatomic probability space. We investigate certain martingale inequalities in X that generalize those studied by A. M. Garsia. We give necessary and sufficient conditions on X for the inequalities to be valid.

scholarly journals BMO Functions Generated by A X ℝ n Weights on Ball Banach Function Spaces

2021 ◽  
Vol 2021 ◽  
pp. 1-5
Author(s):  
Ruimin Wu ◽  
Songbai Wang
Keyword(s):  
Linear Operator ◽  
Function Space ◽  
Maximal Function ◽  
Lebesgue Space ◽  
Variable Exponent ◽  
Banach Function Space ◽  
Function Spaces ◽  
Banach Function Spaces ◽  
Bmo Functions ◽  
Littlewood Maximal Function

Let X be a ball Banach function space on ℝ n . We introduce the class of weights A X ℝ n . Assuming that the Hardy-Littlewood maximal function M is bounded on X and X ′ , we obtain that BMO ℝ n = α ln ω : α ≥ 0 , ω ∈ A X ℝ n . As a consequence, we have BMO ℝ n = α ln ω : α ≥ 0 , ω ∈ A L p · ℝ n ℝ n , where L p · ℝ n is the variable exponent Lebesgue space. As an application, if a linear operator T is bounded on the weighted ball Banach function space X ω for any ω ∈ A X ℝ n , then the commutator b , T is bounded on X with b ∈ BMO ℝ n .

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