I am linearizing a PDE about a solution to obtain linear stability or instability. The space in which my perturbations are plays an important role.
The solution $w$ of the linear problem I am interested in is analytic for $x<0$. Also, it is such that $w, w'$, and $w''$ behave as $e^{x}$ as $x\rightarrow -\infty$, $w\sim |x|^p$ (for some positive $p$) as $x\rightarrow 0^-$, and $w=0$ for $x>0$.
In other words, $w, w'$ and $w''$ go to zero exponentially fast at $x=-\infty$, goes to zero as $|x|^p$ at $x=0$ from the left, and is zero for $x>0$.
I do not have an explicit expression for the solution.
For what value of $p$ is $w$ in $H^{s}(\mathbb{R})$, for some $s>3/2$?
A toy example would be $w=e^x-e^{2x}$ for $x\leq 0$ and $w=0$ for $x>0$. One can check that this is in $H^s(\mathbb{R})$ for $s<3/2$. I thus supect that in the general case, $w\in H^s(\mathbb{R})$ for $s\geq 3/2$ only for $p>1$.
Please pardon me for putting this all onto the positive real axis... since otherwise I would commit sign errors every few seconds.
To my mind, the function $f_s(x)$ which is $x^se^-x$ on $x>0$ and is $0$ to the left, decides this issue. Namely, you can subtract a multiple of this from your function, to have something whose derivatives decay very well and vanish to one higher order at $0$. It's Fourier transform is explicit (by various means).