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By developing a unified approach based on integral representations, we establish sharp quantitative stability estimates of the fractional and higher-order Sobolev inequalities, induced by the embedding   $\dot{H}^{s}(\mathbb{R}^{n}) \hookrightarrow L^{2n \over n-2s}(\mathbb{R}^{n})$   for any   $s \in (0,\frac{n}{2})$  , in the critical point setting.

Sharp Quantitative Stability Estimates for Critical Points of Fractional Sobolev Inequalities