Recent Developments of Function Spaces With Variable Exponents

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.