A way to check that a $L^2(\mathbb{R})$ function is in $H^{3/2}(\mathbb{R})$, is to check that $(1+|\xi|^2)^{3/4}{\mathscr{F}}(f)$ is in $L^2(\mathbb{R})$.
What about to check that a function in $L^2([0,\infty))$ is in $H^{3/2}([0,\infty))$? One compute $(1+|\xi|^2)^{3/4}{\mathscr{F}}(f)$ with the Fourier transform computed on the interval $[0,\infty)$?
I am using $k=3/2$, $p=2$, and the half-line because that is what I am interested in. But my question can be applied to $H^{k,p}(I)$ for other values of $k$ and p, and other open intervals of $\mathbb{R}$.
The definition of $H^{k,p}([0,\infty))$ given on Wikipedia is that $f \in H^{k,p}([0,\infty))$ if and only if there exists a function $g \in H^{k,p}(\mathbb R)$ with $f = g|_{[0,\infty)}$.
So if $(1+|\xi|^2)^{3/4}{\mathscr{F}}(f)$ is in $L^2(\mathbb{R})$ where the Fourier transform is computed on $f \chi_{[0,\infty)}$, that definitely implies $f \in H^{3/2}([0,\infty))$. But the converse doesn't necessarily hold.
There should be a way to extend $f$ to $g$ in a systematic manner to perform the Fourier transform test. It is just that I am not enough of an expert to tell you precisely how to do that. So don't upvote this answer. I would make this a comment, but it is too long.