Let $f$ be the function defined on $\mathbb{R}$ as $$f(x)=\frac{e^{ix^3}}{1+x^2}$$ for $x\in\mathbb{R}$.
It is clear that $f\in L^1(\mathbb{R})$. So the Fourier transform of $f$ exists and $\hat{f}\in C_0(\mathbb{R})$.
Since $\hat{f}$ is a bounded function on $\mathbb{R}$, it is a tempered distribution. So, as a distribution, its derivatives of all order exist.
I am interested in calculating $(1-\Delta)\hat{f},$ where $\Delta$ is the Laplace operator (i.e. $\Delta=\frac{d^2}{dx^2}$).
$\textbf{My Question : }$ What kind of object is $(1-\Delta)\hat{f}$? More precisely, I would like to know whether the expression $(1-\Delta)\hat{f}$ is a bounded function.
I have done some calculations but I am not getting anywhere.