Abstract
We shall verify that the Kakeya (Nikodym) maximal operator K N , \({N \gg 1}\), is bounded on the variable Lebesgue space \({L^{p(\cdot)}(\mathbb{R}^2)}\) when the exponent function \({p(\cdot)}\) is N-modified locally log-Hölder continuous and log-Hölder continuous at infinity.
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The second author is supported by the FMSP program at Graduate School of Mathematical Sciences, the University of Tokyo, and Grant-in-Aid for Scientific Research (C) (No. 23540187), the Japan Society for the Promotion of Science.
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Saito, H., Tanaka, H. The Kakeya maximal operator on the variable Lebesgue spaces. Arch. Math. 103, 481–491 (2014). https://doi.org/10.1007/s00013-014-0709-2
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DOI: https://doi.org/10.1007/s00013-014-0709-2