Spectral Theory of the Weighted Fourier Transform With respect to a Function in ℝn: Uncertainty Principle and Diffusion‐Wave Applications

This article introduces an ‐dimensional weighted Fourier transform with respect to a function, extending the one‐dimensional framework proposed in [8]. A spectral theory is developed on weighted Hilbert spaces, establishing the Plancherel identity, the inversion formula, the convolution theorem, and a Heisenberg‐type uncertainty principle under geometric deformations. This framework is then used to define a weighted fractional Laplacian and to characterize its purely continuous spectrum. Finally, these tools are applied to solve the space‐time fractional diffusion‐wave equation. The fundamental solutions are expressed explicitly in terms of the Fox ‐function, establishing a direct connection with the weighted Mellin transform.