Sharpness of the radial Schrödinger system-exponent range beyond the diagonal threshold

Determine whether the Bessel-family system-exponent range for the frequency-localized radial Schrödinger Strichartz estimate is sharp when $q_*<q<q_c$, where $q_*=(2d+1)/(2d-1)$ and $q_c=(2d-1)/(2d-3)$.

Background

The paper proves an improved Bessel-family Strichartz estimate for radial initial data under the boundary relation $1/p+(2d-1)/(2q)=d-1/2$. For $1\le q\le q_*$, the permitted system exponent is shown to be sharp using a necessary condition derived later in the paper.

For the extended radial range $q_*<q<q_c$, the paper proves the sufficient condition $\beta<2p/(p+1)$, but the necessary condition does not determine whether this range can be improved or whether it is optimal. The unresolved region corresponds to the segment $(D,C)$ in the paper’s admissible-region figure.