Abstract
The main target of this article is the boundedness of the fractional maximal operator , on Musielak–Orlicz spaces \(L^{\Phi }(X)\) over unbounded quasi-metric measure spaces as an extension of recent results by Cruz-Uribe and Shukla (Studia Math 242(2):109–139, 2018) and the authors (2019), where \(\eta \) is the order of the fractional maximal operator and \(\lambda \) is its modification rate. Our results are new even for the Hardy–Littlewood maximal operator \(M_{\lambda }\) or for the Orlicz spaces \(L^{p(\cdot )}(\log L)^{q(\cdot )}(X)\). Usually, for the proof of the boundedness, the three-line theorem is employed. This new technique of using the three-line theorem enables us to extend the function spaces with ease. An example explains why we can not remove the modification parameter \(\lambda \).
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Yoshihiro Sawano was supported by Grant-in-Aid for Scientific Research (C) (19K03546), the Japan Society for the Promotion of Science and People’s Friendship University of Russia. Tetsu Shimomura is partially supported by Grant-in-Aid for Scientific Research (C), No. 18K03332, Japan Society for the Promotion of Science.
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Appendix–An Estimate of Welland Type
Appendix–An Estimate of Welland Type
The following equivalence is well known. But for the sake of self-containedness, we supply its proof. The symbol \(g\sim h\) means that \(C^{-1}h\le g\le Ch\) for some constant \(C>0\).
Lemma 6.1
Let \(\varepsilon ,A,B>0\). Then
This estimate is standard and somewhat well known. Here for the sake of convenience, we supply the whole proof.
Proof
Let \(j_0\equiv \left[ \frac{1}{2 \varepsilon } \log _2 \frac{A}{B}\right] \). Then \(2^{-j_0\varepsilon }A \sim 2^{j_0\varepsilon }B\). Thus,
\(\square \)
The following lemma extends [9, Proposition 5.1]. We employ the idea of Welland [43].
Lemma 6.2
Let \(0<\eta <1\), \(0<\varepsilon <\min (1-\eta ,\eta )\) and \(\lambda \ge 1\). Then
for any non-negative \(\mu \)-measurable function f.
Noteworthy is the fact that we do not have to postulate any condition on \(\mu \) such as the reverse doubling condition. In [9, Proposition 5.1], they used the assumption that \(\mu \) is a reverse doubling measure, that is,
for every \(x \in X\), \(r>0\) such that \(B(x,r) \subset X\) and some constant \(0<\gamma <1\).
Proof
Let us set
Then we have \(r_j \le r_{j+1}\) for all \(j \in {{\mathbb {Z}}}\). We decompose
Since \(\mu (B(x,2\lambda r_{j-1}))>2^{j-1}\) thanks to the definition of \(r_{j-1}\), we obtain
Since \(\mu (B(x,\lambda r_j)) \le 2^j\), we obtain
thanks to Lemma 6.1, as required. \(\square \)
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Sawano, Y., Shimomura, T. Fractional Maximal Operator on Musielak–Orlicz Spaces Over Unbounded Quasi-Metric Measure Spaces. Results Math 76, 188 (2021). https://doi.org/10.1007/s00025-021-01490-7
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DOI: https://doi.org/10.1007/s00025-021-01490-7