Sharp radial--angular mixed norm Fourier extension and global stability on spheres

We study the sharp Fourier extension inequality from $L^2({\mathbb{S}}^{d-1})$ to the radial–angular mixed space $L_{\mathrm{r}\mathrm{a}\mathrm{d}}^q{L}_{\mathrm{a}\mathrm{n}\mathrm{g}}^2$. Our work is motivated by and builds upon the foundational analysis of Carneiro, Oliveira e Silva and Sousa \cite{COS}, who initiated the sharp mixed-norm spherical restriction problem from the viewpoint of spherical-harmonic decomposition. Their reduction revealed that the extremal problem is governed by a competition among the individual harmonic modes, and their results established degree-zero extremizers in several important ranges of exponents, including even integer exponents and the endpoint case. The present paper addresses this question by establishing a complete modal ordering result. Building on their spherical-harmonic reduction, the sharp-constant problem is reduced to the weighted Bessel moments $$I_{\nu ,k}(q):={\int }_0^{\mathrm{\infty }}\mathrm{\mid }{J}_{\nu +k}(r){\mathrm{\mid }}^q{r}^{2\nu +1-\nu q}dr,\nu :=\frac{d}{2}-1.$$ For every integer $d\ge 2$ and every finite admissible exponent $q>2d\mathrm{/}(d-1)$, we prove $$I_{\nu ,k}(q)<I_{\nu ,0}(q),k\ge 1,$$ and hence constants are the unique nonzero extremizers of the sharp mixed-norm extension inequality. In dimensions $d\ge 3$ we prove the stronger strict ordering $$I_{\nu ,0}(q)>I_{\nu ,1}(q)>I_{\nu ,2}(q)>\cdots \mathrm{.}$$ We resolve this difficulty by combining exact Hankel asymptotics with a finite remainder comparison. The modal ordering also yields the optimal quadratic global stability inequality. For every finite admissible $q$, $$C_{d,q}^2\mathrm{\parallel }f{\mathrm{\parallel }}_2^2-\mathrm{\parallel }\widehat{f\sigma }{\mathrm{\parallel }}_{L_{\mathrm{r}\mathrm{a}\mathrm{d}}^q{L}_{\mathrm{a}\mathrm{n}\mathrm{g}}^2}^2\ge {\mathrm{\Gamma }}_{d,q}\mathrm{\parallel }f-P_{0}f{\mathrm{\parallel }}_2^2,$$ with the best possible coefficient. When $d\ge 3$, this coefficient is explicitly $${\mathrm{\Gamma }}_{d,q}=(2\pi {)}^d[I_{\nu ,0}(q{)}^{2\mathrm{/}q}-I_{\nu ,1}(q{)}^{2\mathrm{/}q}]\mathrm{.}$$