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 .

scholarly journals Singular integrals and potentials in some Banach function spaces with variable exponent

2003 ◽  
Vol 1 (1) ◽  
pp. 45-59 ◽  
Author(s):  
Vakhtang Kokilashvili ◽  
Stefan Samko
Keyword(s):  
Function Space ◽  
Singular Integral ◽  
Variable Exponent ◽  
Singular Integrals ◽  
Banach Function Space ◽  
Function Spaces ◽  
Power Weight ◽  
Banach Function Spaces ◽  
Dini Condition ◽  
Potential Type Operators

We introduce a new Banach function space - a Lorentz type space with variable exponent. In this space the boundedness of singular integral and potential type operators is established, including the weighted case. The variable exponentp(t)is assumed to satisfy the logarithmic Dini condition and the exponentβof the power weightω(t)=|t|βis related only to the valuep(0). The mapping properties of Cauchy singular integrals defined on Lyapunov curves and on curves of bounded rotation are also investigated within the framework of the introduced 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 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 Banach algebra of the Fourier multipliers on weighted Banach function spaces

2015 ◽  
Vol 2 (1) ◽  
Author(s):  
Alexei Karlovich
Keyword(s):  
Integral Operator ◽  
Banach Algebra ◽  
Function Space ◽  
Singular Integral ◽  
Banach Function Space ◽  
Fourier Multipliers ◽  
Banach Function Spaces ◽  
Continuous Embedding ◽  
Cauchy Singular Integral ◽  
Cauchy Singular Integral Operator

AbstractLet MX,w(ℝ) denote the algebra of the Fourier multipliers on a separable weighted Banach function space X(ℝ,w).We prove that if the Cauchy singular integral operator S is bounded on X(ℝ, w), thenMX,w(ℝ) is continuously embedded into L∞(ℝ). An important consequence of the continuous embedding MX,w(ℝ) ⊂ L∞(ℝ) is that MX,w(ℝ) is a Banach algebra.

Compactness Properties of Carleman and Hille-Tamarkin Operators

1985 ◽  
Vol 37 (5) ◽  
pp. 921-933 ◽  
Author(s):  
Anton R. Schep
Keyword(s):  
Linear Operator ◽  
Integral Operator ◽  
Function Space ◽  
Measurable Function ◽  
Banach Function Space ◽  
Integral Operators ◽  
Carleman Integral Operator ◽  
Measurable Functions ◽  
Associate Space ◽  
Image Position

In this paper we study integral operators with domain a Banach function space Lρ1 and range another Banach function space Lρ2 or the space L0 of all measurable functions. Recall that a linear operator T from Lρ1 into L0 is called an integral operator if there exists a μ × v-measurable function T(x, y) on X × Y such thatSuch an integral operator is called a Carleman integral operator if for almost every x ∊ X the functionis an element of the associate space L′ρ1, i.e.,

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 A Decomposition of the Dual Space of Some Banach Function Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Claudia Capone ◽  
Maria Rosaria Formica
Keyword(s):  
Lebesgue Space ◽  
Dual Space ◽  
Function Spaces ◽  
Zygmund Space ◽  
Marcinkiewicz Space ◽  
Banach Function Spaces ◽  
Integrable Functions ◽  
Grand Lebesgue Space

We give a decomposition for the dual space of some Banach Function Spaces as the Zygmund space of the exponential integrable functions, the Marcinkiewicz space , and the Grand Lebesgue Space .

SEPARATION PROPERTIES AND OPERATING FUNCTIONS ON A SPACE OF CONTINUOUS FUNCTIONS

1993 ◽  
Vol 04 (04) ◽  
pp. 551-600 ◽  
Author(s):  
OSAMU HATORI
Keyword(s):  
Function Space ◽  
Real Function ◽  
Banach Function Space ◽  
Function Algebra ◽  
Continuous Functions ◽  
Positive Answer ◽  
Banach Function Spaces ◽  
The Real ◽  
Affine Functions ◽  
Real Function Algebra

Characterizations of the space CR (X) of all real-valued continuous functions on a compact Hausdorff space X among its subspaces are investigated under the circumstances of operating functions. One of the main purpose in this paper is to disprove the following conjecture: if a non-affine function operates on an ultraseparating real Banach function space E on X, then E = CR (X). A positive answer is given in the case that E satisfies a stronger separation axiom than ultraseparation one, which the real part of an ultraseparating Banach function algebra satisfies. For the original conjecture a counterexample is given; there is an ultraseparating real Banach function space on a compact metric space Y on which the function |·| operates, but it does not coincide with CR (Y). A characterization is given for non-affine functions which operate only on CR (X) among ultraseparating real Banach function spaces on X. By using these results the symbolic calculus on real Banach function spaces is investigated without extra hypothesis of ultraseparation. Several non-local-Lipschitz functions are shown not to operate on a real Banach function space E on X unless E = CR (X). In particular, the function tp defined on [0,1) for a p with 0 < p < 1 or the standard Cantor function on [0, 1] never operates on a real Banach function space E on X unless E = CR (X). Functions which operate on the real part of a real function algebra are also investigated. A positive answer is given for the conjecture that only affine functions operate on the real part of a non-trivial real function algebra.

FOURIER-TYPE TRANSFORMS ON REARRANGEMENT-INVARIANT QUASI-BANACH FUNCTION SPACES

2018 ◽  
Vol 61 (1) ◽  
pp. 231-248 ◽  
Author(s):  
KWOK-PUN HO
Keyword(s):  
Banach Function Space ◽  
Integral Operators ◽  
Oscillatory Integral ◽  
Function Spaces ◽  
Lebesgue Spaces ◽  
Banach Function Spaces ◽  
Oscillatory Integral Operators ◽  
Interpolation Functor ◽  
The Laplace Transform ◽  
Fourier Type

AbstractWe establish the mapping properties of Fourier-type transforms on rearrangement-invariant quasi-Banach function spaces. In particular, we have the mapping properties of the Laplace transform, the Hankel transforms, the Kontorovich-Lebedev transform and some oscillatory integral operators. We achieve these mapping properties by using an interpolation functor that can explicitly generate a given rearrangement-invariant quasi-Banach function space via Lebesgue spaces.

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