BOUNDEDNESS OF INTEGRAL OPERATOR WITH HOMOGENEOUS KERNEL IN HERZ SPACES

• Main Author: Zanu, Pebrudal
• Format: Dissertations
• Language: Indonesia
• Online Access: https://digilib.itb.ac.id/gdl/view/82980
• Institution: Institut Teknologi Bandung

Herz introduced a class of function spaces to identify the boundedness of Fourier transform on Lipschitz spaces. Later on, Lu and Yang rewrote these spaces of Herz based on two types of spatial decomposition of Rn \ {0}, and of Rn. They attributed these spaces as homogeneous Herz and non–homogeneous Herz spaces, respectively. In a different stream than that of Herz, Samko in 2020 investigated the boundedness of integral operators on the local Morrey spaces, having kernel of homogeneous degree ?n, which is invariant with respect to rotations. The operators such as Hardy, Hilbert, fractional integral, are a particular type of interest. Samko proved some sufficient conditions for the boundedness of these operators in local Morrey spaces. The purpose of this doctoral thesis is to extend integral operator with homogeneous kernels of degree ?n and ?n + ?. The first is to establish necessary conditions for boundedness integral operators on Lebesgue spaces and Herz spaces. Secondly, we also calculate the exact norm of integral operators in Lebesgue spaces and Herz spaces. The first result in this research is necessary and sufficient conditions for the boundedness of operators on Lebesgue spaces, with homogeneous kernels of degree ?n and ?n + ?. Secondly, we also calculate the exact norm of the Hardy operator and its dual in Lebesgue space. The other results are considered necessary and sufficient conditions for boundedness integral operators in Herz spaces. The necessary condition is also sufficient conditions in Herz space. The sufficient conditions contain parameters condition which generalizes of Lu and Yang’s. It turns out that our sufficient condition is also necessary for boundedness of integral operators on homogeneous, and nonhomogeneous Herz spaces. Furthermore, we also successfully obtained the exact norms of the integral operators in a more specific space, power weighted Lebesgue space |x|?