The question comes from E. M. Stein. Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, we concerned about the highly oscillatory distribution $$ D(x)=\mathrm{p.v.} \frac{\mathrm{e}^{\mathrm{i}/x}}{x^\gamma},\quad x\in(0,\infty),\quad \gamma\in(0,2) $$ and if $x>1$, we cant modified $D(x)$ to be smooth compact support hence we’ll omit it.
Now we deal with the Fourier transform of $D$: $$ \hat{D}(\xi) = \lim_{\varepsilon\to0}\int_\varepsilon^1 \frac{\mathrm{e}^{\mathrm{i}/x+\mathrm{i}x\xi}}{x^\gamma},\,\mathrm{d}x. $$ By integration by parts, $\hat{D}(\xi)$ exists, let $\phi(x)$ be the phase function $\frac{1}{x}+x\xi$, since $\phi$ contains a nondegenerate critical points $x_0=\frac{1}{\sqrt\xi}$, we can use the method of stationary phase: let $a = x_0/2$, $b = 3x_0/2$, so: $$ \hat{D}(\xi) = \lim_{\varepsilon\to0}\int_\varepsilon^a +\int_a^b+\int_b^1 = I_1+I_2+I_3. $$ By $|\phi'(x)|\gtrsim\xi$, hence by van der Corput lemma, $\int_b^1=\mathcal{O}(\xi^{\gamma/2-1})$, similarly, $\int_a^b = \mathcal{O}(\xi^{-3/4+\gamma/2})$. Hence the question is the oscillatory integral near the Riemann singularity $0$. Stein claimed “$\lim_{\varepsilon\to0}\int_\varepsilon^a = \mathcal{O}(\xi^{\gamma/2-1})$”:
But $$ I_1 = \int \frac{(\mathrm{e}^{\mathrm{i}\phi(x)})' }{\mathrm{i}x^\gamma\phi'(x)} = \frac{\mathrm{e}^{\mathrm{i}\phi(x)}}{\mathrm{i}x^\gamma\phi'(x)}\biggr|^a_0 + \int \mathrm{e}^{\mathrm{i}\phi(x)}\Bigl(\frac{1}{\mathrm{i}x^\gamma\phi'(x)}\Bigr)'\,\mathrm{d}x $$ the first one is $\mathcal{O}(\xi^{\gamma/2-1})$, but the magnitude function of the second is unbounded: $\frac{\mathrm{i} x^{1-\gamma } (-\gamma +\gamma \xi x^2+2)}{(\xi x^2-1)^2}$, hence I can’t use method of stationary phase or van der Corput lemma, but what Stein said made me think it was easy to estimate. Since I can’t deal with oscillatory integral near a singularity, please help.