Abstract
In this paper,we give a summary of the boundedness of(local) Hardy-Littlewood maximal operator and(variable) fractional integrals on variable Lebesgue spaces.Then we present recent developments of the theory of function spaces with variable exponents,which include Besov and Triebel-Lizorkin spaces,Morrey type Besov and Triebel-Lizorkin spaces,Herz type Besov and Triebel-Lizorkin spaces,Hardy spaces and Herz type Hardy spaces with variable exponents.Along the way several open problems are also given.
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