Sharp bounds for the spherical restriction Fourier transform in classical Lebesgue-Riesz and Grand Lebesgue spaces for ordinary and radial functions

TL;DR

This paper establishes precise bounds for the Fourier transform restriction operator on the sphere within Lebesgue-Riesz and Grand Lebesgue spaces, with special focus on radial functions, enhancing understanding of these bounds.

Contribution

It provides the first exact estimate for the restriction Fourier transform operator norm on radial functions in these spaces.

Findings

Derived bilateral estimates for restriction constants in Lebesgue-Riesz spaces.

Obtained an exact norm estimate for radial functions in Grand Lebesgue Spaces.

Enhanced the theoretical understanding of Fourier restriction phenomena for radial functions.

Abstract

We derive bilateral estimates for the constants appearing in the Fourier transform restricted theorems on the Euclidean sphere for the ordinary and especially radial functions belonging to the Lebesgue-Riesz spaces as well as belonging to the Grand Lebesgue Spaces. We obtain an exact estimate for the norm of the restriction Fourier transform operator acting on the radial functions.