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Consider one-sided Hardy-Littlewood maximal operator on the general Lebesgue space with variable exponent. It is known a local sufficient condition to the function p(.) for the boundedness of the one-sided maximal operator on Lp(.)(R) provided p(.) is a constant function in a neighborhood of infinity. Our main aim is to find a weaker condition to p(.) at infinity to guarantee the boundedness of the one-sided maximal operator on Lp(.)(R) . We will show two different sufficient conditions to the behavior of p(.) at infinity under which the one-sided maximal operator is bounded on Lp(.)(R) . Mathematics subject classification (2010): 46E30, 26D15.

A note on one-sided maximal operator in L^p(.) (ℝ)