Hardy-Littlewood maximal operator in generalized grand Lebesgue spaces

2014 ◽  
Author(s):  
Salaudin M. Umarkhadzhiev
Keyword(s):  
Maximal Operator ◽  
Lebesgue Spaces ◽  
Grand Lebesgue Spaces ◽  
Littlewood Maximal Operator

scholarly journals Fréchet Envelopes of Nonlocally Convex Variable Exponent Hörmander Spaces

2016 ◽  
Vol 2016 ◽  
pp. 1-9
Author(s):  
Joaquín Motos ◽  
María Jesús Planells ◽  
César F. Talavera
Keyword(s):  
Analytic Functions ◽  
Maximal Operator ◽  
Variable Exponent ◽  
Previous Article ◽  
Lebesgue Spaces ◽  
Variable Lebesgue Spaces ◽  
Complemented Subspace ◽  
Open Set ◽  
Hörmander Spaces ◽  
Littlewood Maximal Operator

We show that the dual Bp·locΩ′ of the variable exponent Hörmander space Bp(·)loc(Ω) is isomorphic to the Hörmander space B∞c(Ω) (when the exponent p(·) satisfies the conditions 0<p-≤p+≤1, the Hardy-Littlewood maximal operator M is bounded on Lp(·)/p0 for some 0<p0<p- and Ω is an open set in Rn) and that the Fréchet envelope of Bp(·)loc(Ω) is the space B1loc(Ω). Our proofs rely heavily on the properties of the Banach envelopes of the p0-Banach local spaces of Bp(·)loc(Ω) and on the inequalities established in the extrapolation theorems in variable Lebesgue spaces of entire analytic functions obtained in a previous article. Other results for p(·)≡p, 0<p<1, are also given (e.g., all quasi-Banach subspace of Bploc(Ω) is isomorphic to a subspace of lp, or l∞ is not isomorphic to a complemented subspace of the Shapiro space hp-). Finally, some questions are proposed.

An Almost Greedy Uniformly Bounded Orthonormal Basis in 𝐿𝑝(·)([0, 1]) Spaces

2009 ◽  
Vol 16 (3) ◽  
pp. 465-474
Author(s):  
Ana Danelia ◽  
Ekaterine Kapanadze
Keyword(s):  
Maximal Operator ◽  
Variable Exponent ◽  
Orthonormal Basis ◽  
Lebesgue Spaces ◽  
Variable Exponent Lebesgue Spaces ◽  
Greedy Basis ◽  
Littlewood Maximal Operator ◽  
Uniformly Bounded

Abstract We construct a uniformly bounded orthonormal almost greedy basis for the variable exponent Lebesgue spaces 𝐿𝑝(·)([0, 1]), 1 < 𝑝– ≤ 𝑝+ ≤ 2 (or 2 ≤ 𝑝– ≤ 𝑝+ < ∞), when the diadic Hardy–Littlewood maximal operator is bounded on these spaces.

scholarly journals Local Muckenhoupt class for variable exponents

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Toru Nogayama ◽  
Yoshihiro Sawano
Keyword(s):  
Maximal Operator ◽  
New Method ◽  
Variable Exponents ◽  
Weighted Inequality ◽  
Lebesgue Spaces ◽  
Extension Method ◽  
Extrapolation Technique ◽  
Muckenhoupt Class ◽  
Vector Valued ◽  
Littlewood Maximal Operator

AbstractThis work extends the theory of Rychkov, who developed the theory of $A_{p}^{\mathrm{loc}}$ A p loc weights. It also extends the work by Cruz-Uribe SFO, Fiorenza, and Neugebauer. The class $A_{p(\cdot )}^{\mathrm{loc}}$ A p ( ⋅ ) loc is defined. The weighted inequality for the local Hardy–Littlewood maximal operator on Lebesgue spaces with variable exponents is proven. Cruz-Uribe SFO, Fiorenza, and Neugebauer considered the Muckenhoupt class for Lebesgue spaces with variable exponents. However, due to the setting of variable exponents, a new method for extending weights is needed. The proposed extension method differs from that by Rychkov. A passage to the vector-valued inequality is realized by means of the extrapolation technique. This technique is an adaptation of the work by Cruz-Uribe and Wang. Additionally, a theory of extrapolation adapted to our class of weights is also obtained.

Maximal Operator in Variable Exponent Lebesgue Spaces on Unbounded Quasimetric Measure Spaces

2015 ◽  
Vol 116 (1) ◽  
pp. 5 ◽  
Author(s):  
Tomasz Adamowicz ◽  
Petteri Harjulehto ◽  
Peter Hästö
Keyword(s):  
Maximal Operator ◽  
Measure Space ◽  
Variable Exponent ◽  
Hölder Condition ◽  
Lebesgue Spaces ◽  
General Measure ◽  
Radon Measures ◽  
Measure Spaces ◽  
Littlewood Maximal Operator ◽  
Special Case

We study the Hardy-Littlewood maximal operator $M$ on $L^{p({\cdot})}(X)$ when $X$ is an unbounded (quasi)metric measure space, and $p$ may be unbounded. We consider both the doubling and general measure case, and use two versions of the $\log$-Hölder condition. As a special case we obtain the criterion for a boundedness of $M$ on $L^{p({\cdot})}({\mathsf{R}^n},\mu)$ for arbitrary, possibly non-doubling, Radon measures.

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