Abstract
Let \({\varphi}\) be a Musielak–Orlicz function satisfying that, for any \({(x,\,t)\in{\mathbb R}^n \times [0, \infty)}\), \({\varphi(\cdot,\,t)}\) belongs to the Muckenhoupt weight class \({A_\infty({\mathbb R}^n)}\) with the critical weight exponent \({q(\varphi) \in [1,\,\infty)}\) and \({\varphi(x,\,\cdot)}\) is an Orlicz function with uniformly lower type \({p^{-}_{\varphi}}\) and uniformly upper type \({p^+_\varphi}\) satisfying \({q(\varphi) < p^{-}_{\varphi}\le p^{+}_{\varphi} < \infty}\). In this paper, the author obtains a sharp weighted bound involving \({A_\infty}\) constant for the Hardy–Littlewood maximal operator on the Musielak–Orlicz space \({L^{\varphi}}\). This result recovers the known sharp weighted estimate established by Hytönen et al. in [J. Funct. Anal. 263:3883–3899, 2012].
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The author is supported by the National Natural Science Foundation of China (Grant Nos. 11301534 and 11471042) and Da Bei Nong Education Fund (Grant No. 1101-2413002).
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Lin, H. Sharp weighted bounds for the Hardy–Littlewood maximal operators on Musielak–Orlicz spaces. Arch. Math. 106, 275–284 (2016). https://doi.org/10.1007/s00013-016-0877-3
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DOI: https://doi.org/10.1007/s00013-016-0877-3