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We introduce new Besov and Triebel-Lizorkin spaces with variableintegrable exponent, which are different from those introduced by the second author early. Then we characterize these spaces by the boundedness of the local Hardy-Littlewoodmaximal operator on variable exponent Lebesgue space. Finally the completeness andthe lifting property of these spaces are also given.

Local Characterizations of Besov and Triebel-Lizorkin Spaces with Variable Exponent