Sobolev's inequality for Riesz potentials with variable exponent satisfying a log-Hölder condition at infinity
The authors study maximal functions in a variable \(L^{p(\cdot)}\)-space. They prove (weaker) boundedness of the maximal function under weaker assumptions on the exponent function \(p(\cdot)\) than in \textit{D. Cruz-Uribe, A. Firorenza} and \textit{C. J. Neugebauer} [Ann. Acad. Sci. Fenn., Math. 28, No. 1, 223--238 (2003; Zbl 1037.42023)]. As an application of the boundedness of the maximal function they show Sobolev's inequality for Riesz potentials with variable exponent.
Sobolev's imbedding theorem of variable exponents
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