scholarly journals Generalized Lebesgue Points for Hajłasz Functions

2018 ◽  
Vol 2018 ◽  
pp. 1-12
Author(s):  
Toni Heikkinen
Keyword(s):  
Function Space ◽  
Measure Space ◽  
Variable Exponent ◽  
Banach Function Space ◽  
Lebesgue Point ◽  
Metric Measure Space ◽  
Absolutely Continuous ◽  
Lebesgue Points ◽  
Boyd Index ◽  
Generalized Lebesgue Point

Let X be a quasi-Banach function space over a doubling metric measure space P. Denote by αX the generalized upper Boyd index of X. We show that if αX<∞ and X has absolutely continuous quasinorm, then quasievery point is a generalized Lebesgue point of a quasicontinuous Hajłasz function u∈M˙s,X. Moreover, if αX<(Q+s)/Q, then quasievery point is a Lebesgue point of u. As an application we obtain Lebesgue type theorems for Lorentz–Hajłasz, Orlicz–Hajłasz, and variable exponent Hajłasz functions.

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 Optimal maps in essentially non-branching spaces

2017 ◽  
Vol 19 (06) ◽  
pp. 1750007 ◽  
Author(s):  
Fabio Cavalletti ◽  
Andrea Mondino
Keyword(s):  
Measure Space ◽  
Metric Measure Space ◽  
Absolutely Continuous ◽  
Contraction Property ◽  
Transport Plan ◽  
Measure Contraction Property

In this paper we prove that in a metric measure space [Formula: see text] verifying the measure contraction property with parameters [Formula: see text] and [Formula: see text], any optimal transference plan between two marginal measures is induced by an optimal map, provided the first marginal is absolutely continuous with respect to [Formula: see text] and the space itself is essentially non-branching. In particular this shows that there exists a unique transport plan and it is induced by a map.

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.

The one-sided Ap conditions and local maximal operator

2011 ◽  
Vol 55 (1) ◽  
pp. 79-104 ◽  
Author(s):  
Ana L. Bernardis ◽  
Amiran Gogatishvili ◽  
Francisco Javier Martín-Reyes ◽  
Pedro Ortega Salvador ◽  
Luboš Pick
Keyword(s):  
Function Space ◽  
Maximal Operator ◽  
Variable Exponent ◽  
Banach Function Space ◽  
Lebesgue Spaces ◽  
Weighted Lebesgue Spaces ◽  
Variable Exponent Lebesgue Spaces ◽  
The One

AbstractWe introduce the one-sided local maximal operator and study its connection to the one-sided Ap conditions. We get a new characterization of the boundedness of the one-sided maximal operator on a quasi-Banach function space. We obtain applications to weighted Lebesgue spaces and variable-exponent Lebesgue spaces.

scholarly journals Lebesgue points of $$\ell _1$$-Cesàro summability of d-dimensional Fourier series

2021 ◽  
Vol 6 (3) ◽  
Author(s):  
Ferenc Weisz
Keyword(s):  
Fourier Series ◽  
Lebesgue Point ◽  
Cesàro Means ◽  
Cesàro Summability ◽  
Cesaro Means ◽  
Cesaro Summability ◽  
Lebesgue Points ◽  
Cesáro Means

AbstractWe generalize the classical Lebesgue’s theorem and prove that the $$\ell _1$$ ℓ 1 -Cesàro means of the Fourier series of the multi-dimensional function $$f\in L_1({{\mathbb {T}}}^d)$$ f ∈ L 1 ( T d ) converge to f at each strong $$\omega $$ ω -Lebesgue point.

scholarly journals Diffusion, Optimal Transport and Ricci Curvature for Metric Measure Space

2017 ◽  
Vol 2017-3 (103) ◽  
pp. 19-28
Author(s):  
Luigi Ambrosio ◽  
Nicola Gigli ◽  
Giuseppe Savaré
Keyword(s):  
Ricci Curvature ◽  
Measure Space ◽  
Optimal Transport ◽  
Metric Measure Space

scholarly journals Characterizations of Orlicz-Sobolev Spaces by Means of Generalized Orlicz-Poincaré Inequalities

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Toni Heikkinen
Keyword(s):  
Sobolev Space ◽  
Sobolev Spaces ◽  
Measure Space ◽  
Poincaré Inequality ◽  
Metric Measure Space ◽  
Poincare Inequality ◽  
Poincaré Inequalities ◽  
Space Setting ◽  
Poincare Inequalities

Let Φ be anN-function. We show that a functionu∈LΦ(ℝn)belongs to the Orlicz-Sobolev spaceW1,Φ(ℝn)if and only if it satisfies the (generalized) Φ-Poincaré inequality. Under more restrictive assumptions on Φ, an analog of the result holds in a general metric measure space setting.

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