On the boundedness of fractional integral operators

This thesis consists of two chapters. In the first chapter, basic concepts and fundamental theorems in the second section will be used was included.In the second part, the boundedness of the Riesz potential and its general form (fractional integral) were studied. So first, the (L^p,L^q )-boundedness of the Riesz potential and then the weighted boundedness of Riesz potential were examined. Then, weighted boundedness of fractional integral operator with homogeneous kernel which is defined by? T ? _(?,?) f(x)=?_(R^n)^ ? ? ?(x-y)/|x-y|^(n-?) f(y)dy ?was obtained. After that, the commutators of the Riesz potential was worked on and boundedness of this commutators can be characterized BMO(R^n) was shown by the examples. Finally, the weighted (L^p,L^q )-boundedness of commutators generated by fractional integral operators with rough kernels and BMO functions were discussed.