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 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.

scholarly journals Uniform boundedness of Kantorovich operators in variable exponent Lebesgue spaces

Filomat ◽  
2019 ◽  
Vol 33 (18) ◽  
pp. 5755-5765 ◽  
Author(s):  
Rabil Ayazoglu ◽  
Sezgin Akbulut ◽  
Ebubekir Akkoyunlu
Keyword(s):  
Variable Exponent ◽  
Closed Interval ◽  
Uniform Boundedness ◽  
Lebesgue Spaces ◽  
Variable Exponent Lebesgue Spaces ◽  
The Difference ◽  
Upper Estimate ◽  
Kantorovich Operators ◽  
Uniformly Bounded

In this paper, the Kantorovich operators Kn, n ? N are shown to be uniformly bounded in variable exponent Lebesgue spaces on the closed interval [0; 1]. Also an upper estimate is obtained for the difference Kn(f)-f for functions f of regularity of order 1 and 2 measured in variable exponent Lebesgue spaces, which is of interest on its own and can be applied to other problems related to the Kantorovich operators.

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.

scholarly journals Uniform boundedness of Szász-Mirakjan-Kantorovich operators in Morrey spaces with variable exponents

Filomat ◽  
2020 ◽  
Vol 34 (7) ◽  
pp. 2109-2121
Author(s):  
Yoshihiro Sawano ◽  
Xinxin Tian ◽  
Jingshi Xu
Keyword(s):  
Maximal Operator ◽  
Uniform Boundedness ◽  
Morrey Spaces ◽  
Variable Exponents ◽  
Lebesgue Spaces ◽  
Littlewood Maximal Operator ◽  
Kantorovich Operators ◽  
Uniformly Bounded

The Sz?sz-Mirakjan-Kantorovich operators and the Baskakov-Kantorovich operators are shown to be controlled by the Hardy-Littlewood maximal operator. The Sz?sz-Mirakjan-Kantorovich operators and the Baskakov-Kantorovich operators turn out to be uniformly bounded in Lebesgue spaces and Morrey spaces with variable exponents when the integral exponent is global log-H?lder continuous.

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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