On $L^p$ extremals for Fourier extension estimate to fractional surface

This article investigates the Fourier extension operator associated with the fractional surface $(\xi ,\mathrm{\mid }\xi {\mathrm{\mid }}^{\alpha })$ for $\alpha \ge 2$. We show that the relevant $L^p\to L^q$ Fourier extension inequality possesses extremals for all exponents $p\in [1,2]$. Moreover, for all $p\in (1,2]$, the corresponding $L^p$-extremal sequences are precompact up to symmetries.