We consider $t=1$ for simplicity and wright $K_{\alpha}=\mathcal{F}(\eta(|\xi|) e^{i|\xi|^\alpha})+\mathcal{F}((1-\eta )e^{i|\xi|^\alpha})$,where $\eta\in C^{\infty}(\mathbb{R})$,and $\eta=0$,near 0,$\eta(t)=1$,when $t\ge 1$,the second term in the RHS is smooth and has good behaviour at $\infty$,so we look at the first term,in A.Miyachi's paper "On some singular fourier multipliers"see http://repository.dl.itc.u-tokyo.ac.jp/dspace/bitstream/2261/6297/1/jfs280206.pdf it has a thoroughly analysis on it.When $0<\alpha<\frac{1}{2}$,we have $K\in C^{\infty}(\mathbb{R}^{n}{0})$ C^{\infty}(\mathbb{R}^{n}\backslash{0})$and $$K_{\alpha}=C|x|^{\frac{n(\alpha-1)}{1-2\alpha}}e^{iB|x|^{-\frac{2\alpha}{1-2\alpha}}}+o(|x|^{\frac{n(\alpha-1)}{1-2\alpha}})\quad \text{as}|x|\to 0$$ When$\alpha>\frac{1}{2}$,$K$is smooth throughout$\mathbb{R}^{n}$,and $$K_{\alpha}=C|x|^{\frac{n(\alpha-1)}{1-2\alpha}}e^{iB|x|^{-\frac{2\alpha}{1-2\alpha}}}+o(|x|^{\frac{n(\alpha-1)}{1-2\alpha}})\quad \text{as}|x|\to\infty$$ In this case we can see that unlike$\alpha=1$, for$\alpha>1$,$K_{\alpha}$has decay of$|x|^{\frac{n(\alpha-1)}{2\alpha-1}}$|x|^{-\frac{n(\alpha-1)}{2\alpha-1}}$ at $\infty$.
We wright $K_{\alpha}=\mathcal{F}(\eta(|\xi|) e^{i|\xi|^\alpha})+\mathcal{F}((1-\eta )e^{i|\xi|^\alpha})$,where $\eta\in C^{\mathbb{R}}$,and C^{\infty}(\mathbb{R})$,and$\eta=0$,near 0,$\eta(t)=1$,when$t\ge 1$,the second term in the RHS is smooth and has good behaviour at$\infty$,so we look at the first term,in A.Miyachi's paper "On some singular fourier multipliers"see http://repository.dl.itc.u-tokyo.ac.jp/dspace/bitstream/2261/6297/1/jfs280206.pdf it has a thoroughly analysis on it.When$0<\alpha<\frac{1}{2}$,we have$K\in C^{\infty}(\mathbb{R}^{n}\0)$C^{\infty}(\mathbb{R}^{n}{0})$ and $$K_{\alpha}=C|x|^{\frac{n(\alpha-1)}{(1-2\alpha)}e^{iB|x|^{-\frac{2\alpha}{1-2\alpha}}+o(|x|^{\frac{n(\alpha-1)}{1-2\alpha})\quad K_{\alpha}=C|x|^{\frac{n(\alpha-1)}{1-2\alpha}}e^{iB|x|^{-\frac{2\alpha}{1-2\alpha}}}+o(|x|^{\frac{n(\alpha-1)}{1-2\alpha}})\quad \text{as}|x|\to 0$$
When $\alpha>\frac{1}{2}$,$K$ is smooth throughout $\mathbb{R}^{n}$,and $$K_{\alpha}=C'|x|^{\frac{n(\alpha-1)}{(1-2\alpha)}e^{iB|x|^{-\frac{2\alpha}{1-2\alpha}}+o(|x|^{\frac{n(\alpha-1)}{1-2\alpha})\quad \text{as}|x|\to K_{\alpha}=C|x|^{\frac{n(\alpha-1)}{1-2\alpha}}e^{iB|x|^{-\frac{2\alpha}{1-2\alpha}}}+o(|x|^{\frac{n(\alpha-1)}{1-2\alpha}})\quad \infty$$ text{as}|x|\to\infty $$In this case we can also see that unlike \alpha=1, for \alpha>1,K_{\alpha} has decay in of |x| |x|^{\frac{n(\alpha-1)}{2\alpha-1}} at \infty or order \frac{n(\alpha-1)}{2\alpha-1}.\infty. 1 We wright K_{\alpha}=\mathcal{F}(\eta(|\xi|) e^{i|\xi|^\alpha})+\mathcal{F}((1-\eta )e^{i|\xi|^\alpha}),where \eta\in C^{\mathbb{R}},and \eta=0,near 0,\eta(t)=1,when t\ge 1,the second term in the RHS is smooth and has good behaviour at \infty,so we look at the first term,in A.Miyachi's paper "On some singular fourier multipliers"see http://repository.dl.itc.u-tokyo.ac.jp/dspace/bitstream/2261/6297/1/jfs280206.pdf it has a thoroughly analysis on it.When 0<\alpha<\frac{1}{2},we have K\in C^{\infty}(\mathbb{R}^{n}\0) and$$ K_{\alpha}=C|x|^{\frac{n(\alpha-1)}{(1-2\alpha)}e^{iB|x|^{-\frac{2\alpha}{1-2\alpha}}+o(|x|^{\frac{n(\alpha-1)}{1-2\alpha})\quad \text{as}|x|\to 0 $$When \alpha>\frac{1}{2},K is smooth throughout \mathbb{R}^{n},and$$ K_{\alpha}=C'|x|^{\frac{n(\alpha-1)}{(1-2\alpha)}e^{iB|x|^{-\frac{2\alpha}{1-2\alpha}}+o(|x|^{\frac{n(\alpha-1)}{1-2\alpha})\quad \text{as}|x|\to \infty In this case we can also see that unlike $\alpha=1$, for $\alpha>1$,$K_{\alpha}$ has decay in $|x|$ at $\infty$ or order $\frac{n(\alpha-1)}{2\alpha-1}$.