On the boundedness of singular integral operators on grand variable Herz-Hardy spaces

This study investigates the boundedness of various commutators on grand variable Herz-Hardy spaces, including the Marcinkiewicz integral operator, the Calderón-Zygmund singular integral operator and the fractional integral operator. Firstly we define the Lebesgue spaces with variable exponent and some basic lemmas including Hölder's inequality for Lebesgue spaces. We use the definition of variable Herz spaces to give the definition of grand variable Herz spaces. Then we apply the atomic decomposition of grand variable Herz-Hardy spaces to obtain the boundedness of Marcinkiewicz integral operator, Calderón-Zygmund singular integral operator and fractional integral operator on grand variable Herz-Hardy spaces.