Analysis, Dynamics and Applications Seminar

Abstract:     The theorem of Hausdorff-Young states that when p is between 1 and 2, the Fourier transform of a function in L^p belongs to the dual Lebesgue space L^p’. One of the most famous (partly) open problems in classical analysis is the restriction conjecture. The latter asks when the Fourier transform of a function in L^p admits a restriction to L^q of the unit sphere. Since such manifold has measure zero, and since by Hausdorff-Young the Fourier transform is in L^p’, it is all but clear that such restriction be possible. A celebrated theorem of Tomas and Stein states that the restriction conjecture is true, at least when the target space is L^2 of the sphere. In this talk I identify a one-parameter family of inequalities for the Fourier transform whose limiting case is the restriction conjecture. Using Stein's method of complex interpolation I prove the conjectured inequalities when the target space is $L^2$,  and show that this recovers in the limit the Tomas-Stein theorem.