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I'm struggling to follow the "obvious" k=0 step in the proof of this theorem:

THEOREM 7.7.1. Let $K\subset\mathbb{R}^n$ be a compact set, $X$ an open neighborhood containing $K$ and $j, k$ non-negative integers. If $u\in C^{k}_c(K), f\in C^{k+1}(X)$ and $\Im[f]\geq 0$ in $X$ then for all $w > 0$, $$ w^{j+k}\left\lvert \int u(x)\left(\Im[f(x)]\right)^j e^{iwf(x)} dx \right\rvert \leq C \sum_{\lvert\alpha\rvert \leq k}\sup\lvert D^\alpha u\rvert \left(\lvert f^\prime\rvert^2 + \Im[f]\right)^{\lvert{\alpha}/2-k} $$

Proof available here

I suspect that I'm making some very basic stupid error. In my reading, the case $k=0$ reduces to:

$$ w^{j}\left\lvert \int u(x)\left(\Im[f(x)]\right)^j e^{iwf(x)} dx \right\rvert \leq C \sup\lvert u\rvert. $$

I can see how the imaginary part of $f$ in the exponential is dealt with by using the boundedness of $t^je^{-j}$. But how do I bound the leftover $$ w^{j}\left\lvert \int u(x)\left(\Im[f(x)]\right)^j e^{iw\Re f(x)} dx \right\rvert \,, $$

given that this is unbounded in $j$ e.g. for everywhere constant $f$ with nonvanishing real and imaginary parts?

Nathanael Schilling
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1 Answers1

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If the integral is over $K$ (over $X$ is the same, if $\mathcal{L}(X)<\infty$), then \begin{align*} w^{j}\left\lvert \int u(x)\left(\Im[f(x)]\right)^j e^{iwf(x)} dx \right\rvert &\leq w^{j} \int_{K}|u(x)| \mathfrak{I}[f(x)]^{j}|e^{i w f(x)}| dx \\ & \leq w^{j} \int_{K}|u(x)| \mathfrak{I}[f(x)]^{j}e^{-w \Im[f(x)]}dx \\ &\leq w^{j} \sup_{K}|u(x)| \mathcal{L}(K) \left( \sup_{K} \mathfrak{I}[f(x)]^{j}e^{-w \Im[f(x)]} \right), \end{align*} where I used that $|e^{z}|=e^{\Re (z)}$.

Sebathon
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