2509.09565Sep 11, 202527 pages. Comments are welcome! This version improves the readability and presentation of the paper. In particular, we add a discussion on sharp bilinear eigenfunction estimates on spheres of general dimension, where our method remains applicable (see Remark 4.2), and update the bibliography, including Ref. [15]: Sharp $L^4$ Strichartz estimate for the hyperbolic Schrödinger equation
Sharp bilinear eigenfunction estimate, Lx2∞Lt,x1p-type Strichartz estimate, and energy-critical NLS
We establish sharp bilinear eigenfunction estimates for the Laplace-Beltrami operator on the standard three-sphere S3, eliminating the logarithmic loss that has persisted in the literature since the pioneering work of Burq, Gérard, and Tzvetkov over twenty years ago. This completes the theory of multilinear eigenfunction estimates on the standard spheres. Our approach relies on viewing S3 as the compact Lie group SU(2) and exploiting its representation theory. Motivated by applications to the energy-critical nonlinear Schrödinger equation (NLS) on
Nearby in the stack
R×S3
, we also prove a refined anisotropic Strichartz estimate on the cylindrical space
Rx1×Tx2
of
Lx2∞Lt,x14
-type, adapted to certain spectrally localized functions. The argument relies on multiple sharp measure estimates and a robust kernel decomposition method. Combining these two key ingredients, we derive a refined bilinear Strichartz estimate on
R×S3
, which in turn yields small-data global well-posedness for the above mentioned NLS in the energy space.