Let $\mathcal{H}$ be a complex Hilbert space. Let us consider the following integral: \begin{equation*} \int_{\mathcal{H}}e^{-i\cdot f(x)} dx \end{equation*} Is there an upper bound of this integral (at least in the case $\mathcal{H}=\mathbb{R}^d$)? For which condition of $f(x)$?
Remark: actually I need the convergence to write the exponential function as series and swap it with the integral.