Spectral Theory of the Weighted Fourier Transform with respect to a Function in $\mathbb{R}^n$: Uncertainty Principle and Diffusion-Wave Applications

Abstract

In this paper, we generalize the weighted Fourier transform with respect to a function, originally proposed for the one-dimensional case in \cite{Dorrego}, to the $n$-dimensional Euclidean space $\mathbb{R}^{n}$. We develop a comprehensive spectral theory on a weighted Hilbert space, establishing the Plancherel identity, the inversion formula, the convolution theorem, and a Heisenberg-type uncertainty principle depending on the geometric deformation. Furthermore, we utilize this framework to rigorously define the weighted fractional Laplacian with respect to a function, denoted by $(-\Delta_{\phi,\omega})^{s}$. Finally, we apply these tools to solve the generalized time-space fractional diffusion-wave equation, demonstrating that the fundamental solution can be expressed in terms of the Fox H-function, intrinsically related to the generalized $\omega$-Mellin transform introduced in \cite{Dorrego}. In this paper, we generalize the weighted Fourier transform with respect to a function, originally proposed for the one-dimensional case, to the n-dimensional Euclidean space $\mathbb{R}^n$. We develop a comprehensive spectral theory on a weighted Hilbert space, establishing the Plancherel identity, the inversion formula, the convolution theorem, and a Heisenberg-type uncertainty principle depending on the geometric deformation. Furthermore, we utilize this framework to rigorously define the weighted fractional Laplacian with respect to a function, denoted by $(-\Delta_{\phi,\omega})^s$. Finally, we apply these tools to solve the generalized time-space fractional diffusion-wave equation involving the weighted Hilfer derivative. We demonstrate that the fundamental solution can be explicitly expressed in terms of the Fox H-function, revealing an intrinsic connection with the generalized Mellin transform.