On the Weighted Variable Exponent Spaces of Differential Forms

2017 ◽  
Vol 10 (1) ◽  
pp. 1-9
Author(s):  
Lifeng Guo ◽  
Jing Wang ◽  
Jinzi Liu
Keyword(s):  
Variable Exponent ◽  
Differential Forms ◽  
Variable Exponent Spaces ◽  
Weighted Variable

The Maximal Operator in Weighted Variable Exponent Spaces on Metric Spaces

2008 ◽  
Vol 15 (4) ◽  
pp. 683-712
Author(s):  
Vakhtang Kokilashvili ◽  
Stefan Samko
Keyword(s):  
Metric Space ◽  
Maximal Operator ◽  
Variable Exponent ◽  
Metric Spaces ◽  
Doubling Measure ◽  
Lower Dimension ◽  
Variable Exponent Spaces ◽  
Muckenhoupt Class ◽  
Muckenhoupt Condition ◽  
Weighted Variable

Abstract We study the boundedness of the maximal operator in the weighted variable exponent spaces 𝐿𝑝(·)(𝑋, ϱ) on a doubling measure metric space 𝑋. When 𝑋 is bounded, the weight belongs to a version of a Muckenhoupt-type class, which is narrower than the expected Muckenhoupt condition for a variable exponent, but coincides with the usual Muckenhoupt class 𝐴𝑝 in the case of a constant 𝑝. For the bounded 𝑋 we also consider the class of weights of the form , where the functions 𝑤𝑘(𝑟) have finite upper and lower indices 𝑚(𝑤) and 𝑀(𝑤) satisfying the condition , where 𝔡𝔦𝔪(𝑋) is a version of lower dimension of the space 𝑋. In the case of unbounded 𝑋 we admit weights of the form . Some of the results are new even in the case of a constant 𝑝. We also deal with some new notions of upper and lower local dimensions of measure metric spaces.

scholarly journals Variable Exponent Spaces of Analytic Functions

2020 ◽  
Author(s):  
Gerardo A. Chacón ◽  
Gerardo R. Chacón
Keyword(s):  
Hardy Spaces ◽  
Measurable Function ◽  
Analytic Functions ◽  
Variable Exponent ◽  
Bergman Spaces ◽  
Lebesgue Spaces ◽  
Basic Properties ◽  
Variable Exponent Spaces ◽  
Real Functions ◽  
Spaces Of Analytic Functions

Variable exponent spaces are a generalization of Lebesgue spaces in which the exponent is a measurable function. Most of the research done in this topic has been situated under the context of real functions. In this work, we present two examples of variable exponent spaces of analytic functions: variable exponent Hardy spaces and variable exponent Bergman spaces. We will introduce the spaces together with some basic properties and the main techniques used in the context. We will show that in both cases, the boundedness of the evaluation functionals plays a key role in the theory. We also present a section of possible directions of research in this topic.

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