Let $\phi: \mathbb{R} \to \mathbb{R}$ and $\lambda > 0$.
This question concerns a statement made in the book Elias Stein, Harmonic Analysis: Real-variable Methods, Orthogonality and Oscillatory Integrals. Princeton University Press, 1993.
The statement is in Chapter 8, Section 1.2, on page 332.
Here is the statement:
Suppose we know only that $$|\phi^{(k)}(x)| \geq 1$$ for all $x \in [a,b]$, where $k$ is some fixed positive integer. We wish to obtain an estimate for $$ \int_{a}^{b} e^{i \lambda \phi(x)} dx $$ that is independent of $a$ and $b$. The change of variable $x \mapsto \lambda^{-1/k} x'$ shows that the only possible estimate for the integral is $$ O(\lambda^{-1/k}). $$
My question is: How? How does this change of variable show that?
Attempt:
With the change of variable, the integral becomes $$ \int_{\lambda^{1/k}a}^{\lambda^{1/k}b} e^{i \lambda \phi(\lambda^{-1/k} x')} \lambda^{-1/k}dx' = \lambda^{-1/k} \int_{\lambda^{1/k}a}^{\lambda^{1/k}b} e^{i \lambda \phi(\lambda^{-1/k} x')} dx' $$ Let $c = \lambda^{1/k}a$ and $d = \lambda^{1/k}b$. Let $f(x) = \lambda \phi(\lambda^{-1/k} x)$. Then $f^{(k)}(x) = \phi^{(k)}(\lambda^{-1/k}x)$, and so $$ |f^{(k)}(x)| \geq 1 $$ for all $x \in [c,d]$. And the integral becomes $$ \int_{c}^{d} e^{i f(x')} dx'. $$ But what I am supposed to do with this?