Lipschitz estimates for commutator of fractional integral operators on non-homogeneous metric measure spaces
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Lipschitz estimates for commutator of fractional integral operators on non-homogeneous metric measure spaces WANG Ding-huai∗
ZHOU Jiang
MA Bo-lin
Abstract. In this paper, the authors establish the (Lp (µ), Lq (µ))-type estimate for fractional commutator generated by fractional integral operators Tα with Lipschitz functions (b ∈ Lipβ (µ)), where 1 < p < 1/(α + β) and 1/q = 1/p − (α + β), and obtain their weak (L1 (µ), L1/(1−α−β) (µ))type. Moreover, the authors also consider the boundedness in the case that 1/(α+β) < p < 1/α, 1/α ≤ p ≤ ∞ and the endpoint cases, namely, p = 1/(α + β).
§1
Introduction and Notation
It is well known that the doubling condition is a key assumption in the analysis on spaces of homogeneous type. However, some theories have been proved still valid with non-doubling measure (see[6-8]). In 2010, Hyt¨onen [4] introdeced a new class of metric measure spaces satisfying both the so-called geometrically doubling and the upper doubling conditions (see the definition below), which are called non-homogeneous spaces. Recently, many classical results have been proved still valid if the underlying spaces are replaced by the non-homogeneous spaces, for example, the theory of Carlder´on-Zygmund operators(see [1,3,6]). In 2014, J. Zhou and D. Wang [10] established the definition of fractional operator and the definition of Lipschitz space on non-homogeneous metric measure spaces and they also establish some equivalent characterizations for the Lipschitz spaces. Motivated by [10], we consider the endpoint estimates for commutator generated by fractional integral operators with Lipschitz functions. In this paper, we will prove the (Lp (µ), Lq (µ)) boundedness of commutator generated by fractional integral operator Tα (see the definition below) with Lipschitz functions b ∈ Lipβ (µ), where 1 < p < 1/(α + β) and 1/q = 1/p − (α + β), and their weak (L1 (µ), L1/(1−α−β) (µ)). We also consider the boundedness in the case that 1/(α + β) < p < 1/α, 1/α ≤ p ≤ ∞ and the endpoint case of p = 1/(α + β). Received: 2014-09-03. Revised: 2019-12-30. MR Subject Classification: 47B47, 42B25. Keywords: Non-homogeneous space, Fractional integral, Lipschitz function, Commutator, Endpoint estimate. Digital Object Identifier(DOI): https://doi.org/10.1007/s11766-020-3319-8. Supported by the National Natural Science Foundation of China (Grant No.11661075). ∗ Corresponding author.
254
Appl. Math. J. Chinese Univ.
Vol. 35, No. 3
To state the main results of this paper, we first recall some necessary notions and remarks. Firstly, we make some conventions on notation. Throughout the whole paper, C stands for a positive constant, which is independent of the main parameters, but it may vary from line to line. Definition 1.1. [4] A metric space (X , d, µ) is said to be geometrically doubling if there exists some N0 ∈ N such that, for any ball B(x, r) ⊂ X , there exist a finite ball covering{B(xi , r/2)}i of B(x, r) such that the cardinality of this covering is at most N0 . Definition 1.2. [4] A metric measure space (X , d, µ) is said to
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