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.

scholarly journals New Herz Type Besov and Triebel-Lizorkin Spaces with Variable Exponents

2012 ◽  
Vol 2012 ◽  
pp. 1-27 ◽  
Author(s):  
Baohua Dong ◽  
Jingshi Xu
Keyword(s):  
Maximal Operator ◽  
Maximal Functions ◽  
Variable Exponents ◽  
Herz Spaces ◽  
Vector Valued ◽  
Littlewood Maximal Operator

The authors establish the boundedness of vector-valued Hardy-Littlewood maximal operator in Herz spaces with variable exponents. Then new Herz type Besov and Triebel-Lizorkin spaces with variable exponents are introduced. Finally, characterizations of these new spaces by maximal functions are given.

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.

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.

Characterization of a Two-Weighted Vector-Valued Inequality for Fractional Maximal Operators

1998 ◽  
Vol 5 (6) ◽  
pp. 583-600
Author(s):  
Y. Rakotondratsimba
Keyword(s):  
Maximal Operator ◽  
Maximal Operators ◽  
Lebesgue Spaces ◽  
Fractional Maximal Operator ◽  
Weighted Lebesgue Spaces ◽  
Vector Valued

Abstract We give a characterization of the weights 𝑢(·) and 𝑣(·) for which the fractional maximal operator 𝑀𝑠 is bounded from the weighted Lebesgue spaces 𝐿𝑝(𝑙𝑟, 𝑣𝑑𝑥) into 𝐿𝑞(𝑙𝑟, 𝑢𝑑𝑥) whenever 0 ≤ 𝑠 < 𝑛, 1 < 𝑝, 𝑟 < ∞, and 1 ≤ 𝑞 < ∞.

Endpoint Regularity of Multisublinear Fractional Maximal Functions

2017 ◽  
Vol 60 (3) ◽  
pp. 586-603 ◽  
Author(s):  
Feng Liu ◽  
Huoxiong Wu
Keyword(s):  
Maximal Operator ◽  
Maximal Functions ◽  
Maximal Operators ◽  
One Dimensional ◽  
Regularity Properties ◽  
The One ◽  
Vector Valued Function ◽  
Vector Valued ◽  
Littlewood Maximal Operator

AbstractIn this paper we investigate the endpoint regularity properties of the multisublinear fractional maximal operators, which include the multisublinear Hardy–Littlewood maximal operator. We obtain some new bounds for the derivative of the one-dimensional multisublinear fractional maximal operators acting on the vector-valued function with all ƒ j being BV-functions.

Sign in / Sign up

Export Citation Format

Share Document