The Definition of the Fourier Transform for Weighted Inequalities

Weighted norm inequalities for the Fourier transform have a long history, motivated in part by generalizations of the differentiation formula for Fourier transforms and by applications such as Kolmogorov′s prediction theory and restriction theory. Typically, a norm inequality, ||f̂||Lqμ ≤ ||f||Lpv, is established on a subspace of L1 contained in the weighted space Lpv, where f̂ is the Fourier transform and μ and v are weights. The problem of defining the extension of f̂ on all of Lpv is posed and solved here for several important cases. This definition sometimes results in the ordinary distributional Fourier transform, but there are other situations which are also analyzed. Precise necessary conditions and closure theorems are inextricably related to this program, and these results are established.