Riemann localisation on the sphere

Abstract: 

We study Riemann localisation on the sphere. Upper and lower bounds on the local Fourier convolution, which is the Fourier convolution with a neighborhood omitted, are obtained for the unit sphere in three or more dimensional Euclidean spaces. The estimates show that the local Fourier convolution has the Riemann localisation property for sufficiently smooth functions on the unit sphere in three dimensions, but not in higher dimensions.