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 $L^2(\mathbb{R})$.
I am interested on problems on the half-line. To check that a function in $L^2([0,\infty))$ is in $H^{3/2}([0,\infty))$ if there is an extension of that function that is in $H^{3/2}(\mathbb{R})$. What about proving that something is not in $H^{3/2}([0,\infty))$? Is there a criterion, or a way to determine if a given function is not in $H^{3/2}([0,\infty))$ other then, for each case, proving there is no extension.
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 $\mathbb{R}$. This question is related to my other question, Sobolev space on the half-line.
One criteria follows from the following.
Theorem: If for $u \in H^k([0,\infty))$ with $k \leq 2,$ if we define $$ Eu(x) = \begin{cases} u(x) & \text{ if } x \geq 0, \\ u(-x)+3u(-x/3)-3u(-2x/3) & \text{ if } x < 0, \end{cases}$$ then $Eu \in H^k(\Bbb R),$ with comparable norms.
It may be possible to use a simpler extension here, but I would need to check the details. This can be shown by showing these fractional spaces can be equivalently defined using the Gagliardo seminorm, and establishing a suitable extension theorem. The details are essentially covered in the following reference:
This gives an explicit means to check if an element lies in $H^k([0,\infty));$ you compute its extension, and check if that lies in $H^k(\Bbb R).$ If $Eu \not\in H^k(\Bbb R),$ then you can conclude that $u \in H^k([0,\infty)).$
Note: If you are trying to show explicit functions don't lie in this Sobolev space, there are many necessary conditions for $u \in H^k([0,\infty))$ that might be easier to verify; local $H^k$ regularity ($\eta u \in H^k(\Bbb R)$ for all $\eta \in C^{\infty}_c((0,\infty))$), whether $u$ has the correct Hölder regularity, etc.