Abstract
This work extends the theory of Rychkov, who developed the theory of A_{p}^{mathrm{loc}} weights. It also extends the work by Cruz-Uribe SFO, Fiorenza, and Neugebauer. The class A_{p(cdot )}^{mathrm{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.
Highlights
1 Introduction This paper develops the theory of local Muckenhoupt weights in a variable exponent setting
The definition of Lp(·)(w) slightly differs from that in [3], where the authors considered the theory of Muckenhoupt weights for the Hardy–Littlewood maximal operator M for Lebesgue spaces with variable exponents
< ∞, which is used in the proof of [3, Theorem 1.5], fails for local Muckenhoupt class with variable exponents
Summary
This paper develops the theory of local Muckenhoupt weights in a variable exponent setting. We use the following notation of variable exponents: Let p(·) : Rn → [1, ∞) be a measurable function, and let w be a weight. The definition of Lp(·)(w) slightly differs from that in [3], where the authors considered the theory of Muckenhoupt weights for the Hardy–Littlewood maximal operator M for Lebesgue spaces with variable exponents. < ∞, which is used in the proof of [3, Theorem 1.5], fails for local Muckenhoupt class with variable exponents. The following theorem is the weighted vector-valued inequality for the local variable weight: Theorem 1.11 Suppose that p(·) satisfies conditions (1.1) and (1.2), as well as 1 < p– ≤ p+ < ∞, and let w ∈ Alpo(c·) and 1 < q ≤ ∞.
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
