Malaysian Journal of Mathematical Sciences, June 2025, Vol. 19, No. 2
Bessel-Riesz Operators on Lebesgue Spaces with Lebesgue Measures
Mehmood, S., Eridani, Fatmawati, and Raza, W.
Corresponding Email: fatmawati@fst.unair.ac.id
Received date: 6 March 2024
Accepted date: 26 December 2024
Abstract:
This study investigates a class of mathematical operators known as the Bessel-Riesz operators, defined in Euclidean space $\mathbb{R}^{n}$, given by,
\begin{align}\label{eq:1}
T_{\mu,\nu}f(z)=\int_{\mathbb{R}^{n}}K_{\mu,\nu}(|z-w|)f(w)d\nu(w),\;\;\; \text{for}\; z\in \mathbb{R}^{n}.
\end{align}
Here, \( K_{\mu,\nu} \) is called the Bessel-Riesz kernel. It can be expressed as a multiple of the Bessel kernel \( J_{\nu} \) and the Riesz kernel \( K_{\mu} \). These operators originated from the Schr\"{o}dinger 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 \( \mathbb{R}^{n} \) with the Lebesgue measure, as well as in spaces with other measure types like \( d\rho(w) \).
Keywords: Bessel-Riesz operators; doubling measure; Young inequality; Lebesgue measure; Companato spaces
