Maximal function on generalized Lebesgue spaces $L^{p(\cdot)}$

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VARIABLE BESOV AND TRIEBEL-LIZORKIN SPACES (2008, background): “omit the details here, one can see [3], [4], [ 5 ], [7], [14], [15].”
On Lipschitz truncations of Sobolev functions (with variable exponent) and their selected applications (2008, result): “following proposition is proved in [ Die04a , CUFN03].”
An atomic decomposition of variable Besov and Triebel-Lizorkin spaces (2009, background): “Moreover, these spaces have been applied to partial differential equations and the calculus of variation, see [1, 3, 4, 5, 6, 7, 8, 10, 11, 12, 16, 17, 18, 19, 21, 22, 29] and references therein.”
Anisotropic parabolic equations with variable nonlinearity (2009, background): “Spaces Lp(x)() and W 1,p(x) 0 () . The definitions of the function spaces used throughout the paper and a brief description of their properties follow [ 24 , 25, 33, 37].”
ATOMIC DECOMPOSITION OF HARDY-MORREY SPACES WITH VARIABLE EXPONENTS (2015, background): “One of the important results in the variable exponent analysis is the identification of the class of exponent functions for which the Hardy–Littlewood maximal function is bounded on the corresponding Lebesgue spaces with variable exponent [10, 11, 13, 15, 16, 36, 38, 39, 47, 48, 49].”