Given $f\in H^1(\mathbb{R}^3)$ and $t>0$, consider the following integral: $$I_f(t):=\int_{\mathbb{R}^3}\int_0^{+\infty}e^{-s+i\frac{(s+|x|)^2}{t}}f(x)dtdx$$ I need to show that $I_f(t)$ is finite, and to find how fast it grows as $t\rightarrow +\infty$.

If $f$ decays fast enough at infinity so that $f\in L^1(\mathbb{R}^3)$, we easily get:

$$|I(f)|\lesssim\int_{\mathbb{R}^3}|f(x)|<+\infty$$ so in this case $I_f(t)$ is finite, uniformily in $t$.

If $f$ doesn't dacays fast enough, we can't simply pass to absolute value. We must take into account the oscillatory term $e^{i(s+|x|)^2/t}$, which should produce some polynomial growth in $t$. How to proceed in this case?

Thank you for any suggestions.

Given $f\in H^1(\mathbb{R}^3)$ and $t>0$, consider the following integral: $$I_f(t):=\int_{\mathbb{R}^3}\int_0^{+\infty}e^{-s+i\frac{(s+|x|)^2}{t}}f(x)dsdx$$ I need to show that $I_f(t)$ is finite, and to find how fast it grows as $t\rightarrow +\infty$.

If $f$ decays fast enough at infinity so that $f\in L^1(\mathbb{R}^3)$, we easily get:

$$|I(f)|\lesssim\int_{\mathbb{R}^3}|f(x)|dx<+\infty$$ so in this case $I_f(t)$ is finite, uniformily in $t$.

If $f$ doesn't dacays fast enough, we can't simply pass to absolute value. We must take into account the oscillatory term $e^{i(s+|x|)^2/t}$, which should produce some polynomial growth in $t$. How to proceed in this case?

Thank you for any suggestions.