Equivalence between the $L^2$ norm and the $L^2$ norm of Laplace transform

Question

It is well-known that the Laplace transform, defined by $$\mathcal{L} \colon f(x) \in L^2(\mathbb{R}_+) \to \hat{f}(\xi) \in L^2(\mathbb{R}_+)$$ via $$\hat{f}(\xi) = \int_{\mathbb{R}_+} f(x)\,\mathrm{e}^{-\xi x} \mathrm{d}x,$$ is a linear and bounded operator with $\|\hat{f}(\xi)\|^2_{L^2(\mathbb{R}_+)} \leq \pi\,\|f(x)\|^2_{L^2(\mathbb{R}_+)}$ (actually, it can be shown that the $L^2$ operator norm of $\mathcal{L}$ is $\|\mathcal{L}\| = \sqrt{\pi}$). Now I am wondering if a reversed inequality can be shown, namely, can we find a generic positive constant $C > 0$ such that $$\|\hat{f}(\xi)\|^2_{L^2(\mathbb{R}_+)} \geq C\,\|f(x)\|^2_{L^2(\mathbb{R}_+)} \tag{1}\label{1}$$ for all $f \in L^2(\mathbb{R}_+) $? I dig harder into the literature and it seems that I can not find relevant materials.

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Edit: I forgot to mention that I am only interested in the validity of \eqref{1} when $f$ is a real-valued function.

Answer 1

A comment but I amn’t entitled. This is well trod territory. The one-sided Laplace transform as in your example is an equivalence between the $L^2$-space on the positive axis and the Hardy space on the right half plane, i.e., a suitable Hilbert space of analytic functions there which satisfy appropriate growth conditions.

A precise formulation can be found on p. 131 of the 1962 classic “Banach Spaces of Analytic Functions” by K. Hoffman, which is readily available online.