Investigation of the behavior of some integral operators with weaksingularity in lebesgue spaces

In this thesis, the integral operators that generalize classical Riesz and Bessel potentials are defined and their behavior in Lebesgue spaces is investigated. These operators play an important role in the examination of Sobolev spaces, Bessel potential spaces and their various generalizations. First of all, the concept of "admissible bundle of operators" is introduced, and various examples are given to the families of operators that make up a "admissible bundle". Then, one-dimensional integral representations of classical Riesz and Bessel potentials are given with the help of "admissible bundles". Inspired by these integral representations, a new family of integral operators generalizing classical Riesz potentials is introduced and analogous of the famous Hardy Littlewood-Sobolev inequality is proved for this family of operators. Another important result of the thesis is to investigate of bi-parametric potential-type operators which generalize both Bessel and Flett potentials, in Lebesgue spaces.