Fourier transforms of Lipschitz functions on the hyperbolic plane \(H^2\)
The purpose of the present work is to study the order of magnitude of the Fourier transforms \(\widehat f(\lambda)\) for large \(\lambda\) of complex-valued functions \(f(z)\) satisfying certain Lipschitz conditions in the non-Euclidean hyperbolic plane \(H^2\). The main result is stated as Theorem. Let \(f(z)\) be a complex-valued function on the unit disc \(D\) and \(B=\partial D\). Then the conditions \[ \| f(z+ h)- f(z)\|_2= O(| h|^\alpha)\quad\text{as }| h|\to 0,\;0<\alpha< 1, \] and \[ \int^\infty_\lambda \int_B|\widehat f|^2 d\lambda db= O(\lambda^{-2\alpha- 1})\quad\text{as }\lambda\to \infty \] are equivalent.
scientific article; zbMATH DE number 1140285