Bessel-Riesz Operators on Lebesgue Spaces with Lebesgue Measures

This study investigates a class of mathematical operators known as the Bessel-Riesz operators, defined in Euclidean space Rn, given by, Tμ,νf(z) =Z Rn Kμ,ν (|z − w|)f(w)dν(w), for z ∈ Rn. (1) Here, Kμ,ν is called the Bessel-Riesz kernel. It can be expressed as a multiple of the Bessel kernel Jν and the Riesz kernel Kμ. These operators originated from the Schrödinger equation, which describes particle behavior in quantum mechanics. The primary goal of this research is to explore the behavior of these operators when applied to Lebesgue spaces with different measures, focusing on their boundedness and the conditions under which these operators act predictably. The research aims to establish foundational results for how these operators behave in spaces such as Rn with the Lebesgue measure, as well as in spaces with other measure types like dρ(w).