
> 
> **Question:** Given exponents $0<\alpha<\beta$ and an interval
> $[a,b]\subset(0,\infty)$ are there constants $C,d>0$ such that for any
> $\lambda\_1,\lambda\_2\in\mathbb{R}$,
> $$\left|\int\_a^be(\lambda\_1x^\alpha+\lambda\_2x^\beta)dx\right|\leq C\max\left(\frac1{|\lambda\_1|^d},\frac1{|\lambda\_2|^d}\right)?$$
> (where I'm using $e(x):=e^{2\pi ix}$.)
> 
> 
> 


The reason I hope the answer may be positive is a [lemma of van der Corput](https://en.wikipedia.org/wiki/Van_der_Corput_lemma_(harmonic_analysis)) which implies that for any $[a,b]\subset\mathbb{R}$ and any smooth $f:[a,b]\to\mathbb{R}$ with $|f''(x)|\geq1$ on $[a,b]$,
$$\left|\int\_a^be\big(\lambda f(x)\big)dx\right|\leq 4|\lambda|^{-1/2}.$$


I would (perhaps naively) expect a generalization for several smooth functions $f\_1,\dots,f\_k:[a,b]\to\mathbb{R}$ satisfying *some conditions*; namely that
$$\left|\int\_a^be\big(\lambda\_1 f\_1(x)+\cdots+\lambda\_kf\_k(x)\big)dx\right|\leq C\max|\lambda\_i|^{-d},$$
where $C$ and $d$ depend on the $f\_i$ and $[a,b]$ but not on the $\lambda\_i$. It's not clear to me which conditions to impose, but it seems reasonable to ask that such an extension holds for $f\_i(x)=x^{\alpha\_i}$ for distinct positive $\alpha\_i$.


I would also appreciate any references treating this kind of extension of the van der Corput's lemma.



Generalization of van der Corput's estimate on oscillatory integrals
For convenience of notation write $\Phi(x) = \lambda\_1 x^\alpha + \lambda\_2 x^\beta$. The general argument is based on the integration by parts argument
$$ \int\_a^b e\circ\Phi = \int\_a^b \frac{1}{2\pi i \Phi'} \frac{d}{dx} e\circ \Phi = \frac{1}{2\pi i \Phi'(b)} e(\Phi(b)) - \frac{1}{2\pi i \Phi'(a)} e(\Phi(a)) + \int\_a^b \frac{\Phi''}{2\pi i (\Phi')^2} e\circ\Phi $$


If $\lambda\_1 = \lambda\_2 = 0$, then there is nothing to prove, since the integral is exactly $b-a$.


### Case 1: $\lambda\_1 \lambda\_2 \geq 0$ and not both zero.


In this case the key observation is that
$\Phi'(x) = \alpha \lambda\_1 x^{\alpha - 1} + \beta \lambda\_2 x^{\beta - 1}$ is signed, and has the lower bound
$$ |\Phi'(x)| \geq |\lambda\_1| \alpha \min(a^{\alpha - 1}, b^{\alpha - 1}) + |\lambda\_2| \beta \min(a^{\beta-1}, b^{\beta -1}). $$
Additionally, $\Phi''(x)$ has the bound
$$ |\Phi''(x)| \leq |\lambda\_1 \alpha (\alpha - 1) \max(a^{\alpha - 2}, b^{\alpha - 2})| + |\lambda\_2 \beta (\beta - 2) \max(a^{\beta - 2}, b^{\beta - 2})| $$
So there is some constant $C$ depending on $a, b, \alpha, \beta$ such that
$$ |\int\_a^b e\circ \Phi| \leq C ( \max( |\lambda\_1|, |\lambda\_2| )^{-1} $$


*(Note that it is the inverse of the max, not the max of the inverses.)*


### Case 2: $\lambda\_1 \lambda\_2 < 0$


At issue here is whether $\Phi'$ has a root. Its unique positive root is at the point
$$ c = \left( \frac{\alpha |\lambda\_1|}{\beta |\lambda\_2|} \right)^{\frac{1}{\beta - \alpha}} $$
Note that for $|\lambda\_1| \gg |\lambda\_2|$ or $|\lambda\_2| \gg |\lambda\_1|$, necessarily $c \not\in [a,b]$, and in these cases we have $|\Phi'(x)| > \tilde{C} \max(|\lambda\_1|, |\lambda\_2|)$ again, and the analyses as in Case 1 will show the same decay rate.


So the only issue is when $|\lambda\_1| \approx |\lambda\_2|$.


Computing
$$ \Phi''(c) = c^{\alpha - 2} \lambda\_1 \alpha (\alpha - \beta) $$
Let $\delta$ be a number of size $1/\sqrt{\lambda\_1}$. Split the integral into
$$ \int\_a^{c-\delta} + \int\_{c-\delta}^{c+\delta} + \int\_{c+\delta}^b $$


On the interval $[a,c-\delta]$, it is not too hard to check that the minimum of $|\Phi'(x)|$ occurs at one of the end points $a$ or $c-\delta$, at $a$ one sees $\Phi'(x)$ is of size $\lambda\_1$ (recall that $|\lambda\_1| \approx |\lambda\_2|$), and at $c-\delta$ we have $\Phi'(x)$ is of size $\Phi''(c) \delta \approx \sqrt{|\lambda\_1|}$.


We could **almost** apply the method of Case 1: however, we still have $\Phi''$ is size $\lambda\_1$ and now $(\Phi')^2$ may only be lower bounded by $\lambda\_1$ also, which is not good enough. So we need a trick.
Looking at $\Phi''$ we see that on $[a,b]$ it has at most one zero. Which means we can decompose $[a, c-\delta]$ and $[c+\delta,b]$ into finitely many disjoint intervals on which $\Phi''$ is signed. On such an interval, we have
$$ \left| \int\_a^b \frac{\Phi''}{2\pi i(\Phi')^2} e\circ \Phi \right| \leq \int\_a^b \frac{|\Phi''|}{2\pi (\Phi')^2} = \left| \int\_a^b \frac{|\Phi''|}{2\pi(\Phi')^2} \right| = \left| \frac{1}{2\pi \Phi'(b)} - \frac{1}{2\pi \Phi'(a)} \right| $$
Using this trick in addition to the analyses of Case 1, we see that the integrals over $[a,c-\delta]$ and $[c+\delta,b]$ contribute terms of size $\frac{1}{\sqrt{|\lambda\_1|}}$ asymptotically.


The integral over $[c-\delta, c+\delta]$ is bounded in absolute values by $2\delta$ which is also size $\frac{1}{\sqrt{|\lambda\_1|}}$.




---


Summary
-------


Skipping some details tidying up the analysis, you find, in the end, that there exists a constant $C$ depending on $a, b, \alpha, \beta$, such that for any $\lambda\_1, \lambda\_2\in \mathbb{R}$, you have


$$ \left| \int\_a^b e\circ \Phi \right| \leq C \min( 1, |\alpha \lambda\_1 + \beta \lambda\_2|^{-1}, |\lambda\_1|^{-1/2} ) $$


### Remark


Case 1 directly generalizes to multiple sums of the form $\lambda\_1 f\_1 + \cdots + \lambda\_n f\_n$, when all the $\lambda\_i$ have the same sign and all of the $f\_i$ have first derivatives bounded below by some positive number, and second derivatives bounded.


Case 2 is more delicate in multiple sums. In the analysis here we used that for $\Phi$ of the form specified, $\Phi^{(k)}(x)$ has at most one positive root for any derivative. When the sum is of three or more terms, one has to worry about not just roots but "approximate roots" of $\Phi'$. But what you would generally expect is that the "decay rate" is directional (similar to how in Case 2 above, when $\lambda\_1 \approx - \lambda\_2$ you can only get $|\lambda\_1|^{-\frac12}$ decay, but you get better decay outside of this zone).


I should note that in many cases such asymptotics have been computed.
For constant coefficient PDEs, you have a representation formula for solutions via the Fourier transform. And these kinds of computations are exactly used to establish decay rates of the solution in different directions. You can find some examples in Rauch's PDE textbook, for Stein's Functional Analysis textbook (in the Princeton Analysis series).


