Local Muckenhoupt class for variable exponents

2019/12/03 by Nogayama, Toru, Sawano, Yoshihiro
#42B25 #42B35 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1912.01295

Abstract

We define A_p(⋅)^{\rm loc} and show that the weighted inequality for local Hardy--Littlewood maximal operator on the Lebesgue spaces with variable exponent. This work will extend the theory of Rychkov, who developed the theory of A_p^{\rm loc} weights. It will also extend the work by Cruz-Uribe. SFO, Fiorenza and Neugebaucer, who considered the Muckenhoupt class for Lebesgue spaces with variable exponents. Due to the setting of variable exponents, a new method of extension of weights will be needed; the extension method is different from the one by Rychkov. A passage to the vector-valued inequality is also done by means of the extrapolation technique. This technique is an adaptation of the work by Cruz-Uribe and Wang. We develop the theory of extrapolation adapted to our class of weights.

Related

• Local Hardy-Littlewood maximal opeator in variable Lebesgue spaces
• Maximal operator on variable exponent spaces
• Factorizations for variable exponent Muckenhoupt weights
• Variable Muckenhoupt weights with applications in approximation
• A boundedness criterion for the maximal operator on variable Lebesgue spaces