The Maximal Operator in Variable Spaces 𝐿 𝑝(·)(Ω, ρ) with Oscillating Weights

2006 ◽  
Vol 13 (1) ◽  
pp. 109-125 ◽  
Author(s):  
Vakhtang Kokilashvili ◽  
Natasha Samko ◽  
Stefan Samko
Keyword(s):  
Lipschitz Condition ◽  
Maximal Operator ◽  
Orlicz Spaces ◽  
Weight Functions ◽  
Power Functions ◽  
Index Numbers ◽  
Boyd Indices ◽  
Open Set ◽  
Class Weight ◽  
Young Functions

Abstract We study the boundedness of the maximal operator in the spaces 𝐿 𝑝(·)(Ω, ρ) over a bounded open set Ω in 𝑅𝑛 with the weight , where 𝑤𝑘 has the property that belongs to a certain Zygmund-type class. Weight functions 𝑤𝑘 may oscillate between two power functions with different exponents. It is assumed that the exponent 𝑝(𝑥) satisfies the Dini–Lipschitz condition. The final statement on the boundedness is given in terms of index numbers of functions 𝑤𝑘 (similar in a certain sense to the Boyd indices for the Young functions defining Orlicz spaces).

scholarly journals The maximal operator in weighted variable spacesLp(⋅)

2007 ◽  
Vol 5 (3) ◽  
pp. 299-317 ◽  
Author(s):  
Vakhtang Kokilashvili ◽  
Natasha Samko ◽  
Stefan Samko
Keyword(s):  
Maximal Operator ◽  
Type Condition ◽  
Index Numbers ◽  
Boyd Indices ◽  
Open Set ◽  
Muckenhoupt Class ◽  
Muckenhoupt Condition ◽  
Carleson Curve ◽  
Young Functions ◽  
Weighted Variable

We study the boundedness of the maximal operator in the weighted spacesLp(⋅)(ρ)over a bounded open setΩin the Euclidean spaceℝnor a Carleson curveΓin a complex plane. The weight function may belong to a certain version of a general Muckenhoupt-type condition, which is narrower than the expected Muckenhoupt condition for variable exponent, but coincides with the usual Muckenhoupt classApin the case of constantp. In the case of Carleson curves there is also considered another class of weights of radial type of the formρ(t)=∏k=1mwk(|t-tk|),tk∈Γ, wherewkhas the property thatr1p(tk)wk(r)∈Φ10, whereΦ10is a certain Zygmund-Bari-Stechkin-type class. It is assumed that the exponentp(t)satisfies the Dini–Lipschitz condition. For such radial type weights the final statement on the boundedness is given in terms of the index numbers of the functionswk(similar in a sense to the Boyd indices for the Young functions defining Orlich spaces).

Two-weight weak and extra-weak type inequalities for the one-sided maximal operator

1993 ◽  
Vol 123 (6) ◽  
pp. 1109-1118
Author(s):  
Pedro Ortega Salvador ◽  
Luboš Pick
Keyword(s):  
Maximal Operator ◽  
Weak Type ◽  
Orlicz Spaces ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Weight Functions ◽  
The One ◽  
Image Position ◽  
Necessary And Sufficient

SynopsisLet be the one-sided maximal operator and let Ф be a convex non-decreasing function on (0, ∞), Ф(0) = 0. We present necessary and sufficient conditions on a couple of weight functions (σ, ϱ) such that the integral inqualities of weak typeand of extra-weak typehold. Our proofs do not refer to the theory of Orlicz spaces.

Commutators of Fractional Maximal Operator on Orlicz Spaces

2018 ◽  
Vol 104 (3-4) ◽  
pp. 498-507
Author(s):  
V. S. Guliyev ◽  
F. Deringoz ◽  
S. G. Hasanov
Keyword(s):  
Maximal Operator ◽  
Orlicz Spaces ◽  
Fractional 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.

scholarly journals The maximal operator on generalized Orlicz spaces

2015 ◽  
Vol 269 (12) ◽  
pp. 4038-4048 ◽  
Author(s):  
Peter A. Hästö
Keyword(s):  
Maximal Operator ◽  
Orlicz Spaces

Sharpness of embeddings in logarithmic Bessel-potential spaces

1996 ◽  
Vol 126 (5) ◽  
pp. 995-1009 ◽  
Author(s):  
David E. Edmunds ◽  
Petr Gurka ◽  
Bohumír Opic
Keyword(s):  
Exponential Type ◽  
Orlicz Spaces ◽  
Bessel Potential ◽  
Double Exponential ◽  
Zygmund Spaces ◽  
Young Functions ◽  
Bessel Potential Spaces

This paper is a continuation of [4], where embeddings of certain logarithmic Bessel-potential spaces (modelled upon generalised Lorentz-Zygmund spaces) in appropriate Orlicz spaces (with Young functions of single and double exponential type) were derived. The aim of this paper is to show that these embedding results are sharp in the sense of [8].

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