Abstract
In this paper, we first introduce the weak Lipschitz spaces WLipq,α, 1 < q < ∞, 0 < α < 1 which are the analog of weak Lebesgue spaces Lq,∞ in the setting of Lipschitz space. We obtain the equivalence between the norm \({\left\| {\, \cdot \,} \right\|_{{\rm{Li}}{{\rm{p}}_\alpha }}}\) and \({\left\| {\, \cdot \,} \right\|_{{\rm{WLi}}{{\rm{p}}_{q,\alpha }}}}\). As an application, we show that the commutator M bβ is bounded from Lp to Lq,∞ for some p ∈ (1, ∞) and \({1 \over p} - {1 \over q} = {{\alpha + \beta } \over n}\) if and only if b is in Lipα. We also introduce the weak central bounded mean oscillation space WCBMOq,α and give a characterization of WCBMOq,α via the boundedness of the commutators of Hardy type operators.
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Supported by the National Natural Science Foundation of China (Grant No. 11871452), the Natural Science Foundation of Henan Province (Grant No. 202300410338), and the Nanhu Scholar Program for Young Scholars of Xinyang Normal University
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Wei, M.Q., Yan, D.Y. Two New Lipschitz Type Spaces and Their Characterizations. Acta. Math. Sin.-English Ser. 38, 1523–1536 (2022). https://doi.org/10.1007/s10114-022-1090-x
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DOI: https://doi.org/10.1007/s10114-022-1090-x
Keywords
- Weak Lipschitz space
- commutator
- fractional maximal operator
- weak central bounded mean oscillation space
- Hardy operator