A note on the Fourier transform of fractal measures
The paper is concerned with the decay of the averages of the Fourier transform of measures with finite energy. In the first part of the paper, the author generalizes a result of \textit{P. Sjölin} in [Ann. Acad. Sci. Fenn., Math. 22, No. 1, 227--236 (1997; Zbl 0865.42007)]. The author also extends and gives a different proof to \textit{T. Wolff}'s result [Int. Math. Res. Not. 1999, No. 10, 547--567 (1999); addendum ibid. 88, No. 1, 35--39 (2002; Zbl 0930.42006)] which may be regarded as a weighted version of the Stein-Tomas restriction theorem dealing with estimates for the \(L^q\)-norm of the inverse Fourier transforms of certain \(L^2\) functions. In the second part of the paper, the author considers the case of normalized surface measures on the cones in the 3-dimensional Euclidean space.
scientific article; zbMATH DE number 2123930
scientific article