The Boundedness Of Singular Integral Operators In The Herz-Hardy Space Of Variable Index

This paper mainly studies the boundedness of singular integral operators and commu-tators on the Herz-type Hardy spaces with variable exponents.Firstly,it introduces many properties and conclusions of the variable exponent function spaces as the preliminary knowledge.Secondly,based on the boundedness for commuta-tors of Marcinkiewicz integrals on variable Lebesgue spaces and the atomic decomposition of Herz-Hardy spaces with variable exponents,by using the estimate of the classical in-equalities and the properties of the spaces with variable exponents,the boundedness for commutators of Marcinkiewicz integrals on the homogeneous Herz-type Hardy spaces with variable exponents is obtained.In the last part,by the molecular decomposition of Herz-Hardy spaces with variable exponents,it is proved that the boundedness of fractional inte-gral operator from the homogeneous Herz-type Hardy spaces with variable exponents to the homogeneous Herz-type Hardy spaces with variable exponents.And by the atomic decom-position of Herz-Hardy spaces with variable exponents,it is proved that the boundedness of fractional integral operator from the homogeneous Herz-type Hardy spaces with variable exponents to the homogeneous Herz spaces with variable exponents.