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