A strong-form stability for a class of $L^p$ Caffarelli-Kohn-Nirenberg interpolation inequality

We study the stability of a class of Caffarelli-Kohn-Nirenberg (CKN) interpolation inequality and establish a strong-form stability as following: \begin{equation*} \inf_{v\in$\mathcal{M}${p,a,b}}\frac{ |u-v|{H_b^p} |u-v|{L^p_a}^{p-1} }{|u|{H^p_b}|u|{L^p_a}^{p-1}} \le Cδ{p,a,b}(u)^{t}, \end{equation*} where $t=1$ for $p=2$ and $t=\frac{1}{p}$ for $p>2$, and ${\delta }_{p,a,b}(u)$ is deficit of the CKN. We also note that it is impossible to establish stability results for $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_{H_b^p}$ or $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_{L_a^p}$ separately. Moreover, we consider the second-order CKN inequalities and establish similar results for radial functions.