The maximal operator in Lebesgue spaces with variable exponent near 1

2007 ◽  
Vol 280 (1-2) ◽  
pp. 74-82 ◽  
Author(s):  
Peter A. Hästö
Keyword(s):  
Maximal Operator ◽  
Variable Exponent ◽  
Lebesgue Spaces

scholarly journals Estimates of Fractional Integral Operators on Variable Exponent Lebesgue Spaces

2016 ◽  
Vol 2016 ◽  
pp. 1-7
Author(s):  
Canqin Tang ◽  
Qing Wu ◽  
Jingshi Xu
Keyword(s):  
Integral Operator ◽  
Maximal Operator ◽  
Variable Exponent ◽  
Fractional Integral ◽  
Weak Type ◽  
Type Inequality ◽  
Integral Operators ◽  
Lebesgue Spaces ◽  
Fractional Integral Operators ◽  
Variable Exponent Lebesgue Spaces

By some estimates for the variable fractional maximal operator, the authors prove that the fractional integral operator is bounded and satisfies the weak-type inequality on variable exponent Lebesgue spaces.

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.

Bn-maximal operator and Bn-singular integral operators on variable exponent Lebesgue spaces

2020 ◽  
Vol 70 (4) ◽  
pp. 893-902
Author(s):  
Ismail Ekincioglu ◽  
Vagif S. Guliyev ◽  
Esra Kaya
Keyword(s):  
Differential Operator ◽  
Singular Integral ◽  
Maximal Operator ◽  
Variable Exponent ◽  
Integral Operators ◽  
Singular Integral Operators ◽  
Lebesgue Spaces ◽  
Variable Exponent Lebesgue Spaces ◽  
Bessel Differential Operator

AbstractIn this paper, we prove the boundedness of the Bn maximal operator and Bn singular integral operators associated with the Laplace-Bessel differential operator ΔBn on variable exponent Lebesgue spaces.

The maximal operator on weighted variable Lebesgue spaces

2011 ◽  
Vol 14 (3) ◽  
Author(s):  
David Cruz-Uribe ◽  
Lars Diening ◽  
Peter Hästö
Keyword(s):  
Maximal Operator ◽  
Variable Exponent ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Lebesgue Spaces ◽  
Hölder Continuous ◽  
Variable Lebesgue Spaces ◽  
Muckenhoupt Condition ◽  
Necessary And Sufficient ◽  
Weighted Variable

AbstractWe study the boundedness of the maximal operator on the weighted variable exponent Lebesgue spaces L ωp(·) (Ω). For a given log-Hölder continuous exponent p with 1 < inf p ⩽ supp < ∞ we present a necessary and sufficient condition on the weight ω for the boundedness of M. This condition is a generalization of the classical Muckenhoupt condition.

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.

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.

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 A note on maximal operators related to Laplace-Bessel differential operators on variable exponent Lebesgue spaces

2021 ◽  
Vol 19 (1) ◽  
pp. 306-315
Author(s):  
Esra Kaya
Keyword(s):  
Differential Operator ◽  
Maximal Function ◽  
Maximal Operator ◽  
Variable Exponent ◽  
Differential Operators ◽  
Necessary Condition ◽  
Lebesgue Spaces ◽  
Fractional Maximal Function ◽  
Variable Exponent Lebesgue Spaces ◽  
Bessel Differential Operator

Abstract In this paper, we consider the maximal operator related to the Laplace-Bessel differential operator ( B B -maximal operator) on L p ( ⋅ ) , γ ( R k , + n ) {L}_{p\left(\cdot ),\gamma }\left({{\mathbb{R}}}_{k,+}^{n}) variable exponent Lebesgue spaces. We will give a necessary condition for the boundedness of the B B -maximal operator on variable exponent Lebesgue spaces. Moreover, we will obtain that the B B -maximal operator is not bounded on L p ( ⋅ ) , γ ( R k , + n ) {L}_{p\left(\cdot ),\gamma }\left({{\mathbb{R}}}_{k,+}^{n}) variable exponent Lebesgue spaces in the case of p − = 1 {p}_{-}=1 . We will also prove the boundedness of the fractional maximal function associated with the Laplace-Bessel differential operator (fractional B B -maximal function) on L p ( ⋅ ) , γ ( R k , + n ) {L}_{p\left(\cdot ),\gamma }\left({{\mathbb{R}}}_{k,+}^{n}) variable exponent Lebesgue spaces.

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