Let us consider a real interval $[0, L]$, with $a\in (0, L)$, and let $I_1=(0, a)$ and $I_2=(a, L)$. We denote by $H^k(I_1)$ and $H^k(I_2)$ the usual Sobolev spaces, defined for $k\in \mathbb{N}$. Now, consider the direct sum of these spaces $H_a^k(I):=H^k(I_1)\bigoplus H^k(I_2)$, which has a Hilbert space structure. Furthermore, via isomorphism, it can be viewed as a closed subspace of $L^2(0, L)$. Let us fix $k\geq 1$. By Morrey's inequality, $H^k(I_k)\hookrightarrow C^{k-1, 1/2}(\overline{I_k})$, meaning that any $f\in H_a^k(I)$ has well-defined traces when restricted to $I_1$ or $I_2$. Thus, can consider a family of linear continuous functionals $\Gamma^j: H_a^k(I)\to \mathbb{R}$ given by \begin{align*} \Gamma^j(f)=\alpha_j f^{(j)}(a^-)-\beta_j f^{(j)}(a^+), \end{align*} where $\alpha_j$, $\beta_j>0$ are real numbers, not necessarily equal, and $f^{(j)}$ denotes de $j$-th derivative with $j$ ranging from $0$ to $k-1$.
Now the problem: I would like to characterize the interpolation space \begin{align*} [L^2(0, L), H_a^k(0, L)\cap \ker\Gamma^0\cap\ldots\cap \ker\Gamma^j]_\theta, \end{align*} where we consider, for instance, the $K$-method of real interpolation. My guess is that this space consists of functions in $[L^2(0, L), H_a^k(0, L)]_\theta$ intersected with the kernels up to order $i\in \{0,\ldots, j\}$ so that $k\theta>i+1/2$. Namely, every time we drop below a half-integer, we lose the corresponding constraint. In the half-integer case, some intricate condition might be satisfied (I am not interested in such case though).
I am aware of Löfstrom's article Real interpolation with constraints, where he states an abstract result that may fit perfectly to the aforementioned situation. Moreover, he exemplifies his results by characterizing the interpolation spaces of Sobolev functions on the torus under similar constraints given by differential operators. However, it is not clear to me how to tackle the situation I described with these ideas, I am a bit lost. I thought on going through some extension/restriction argument to grab some results from Löfstrom and use them, but since the extension operator and (weak) derivatives do not commute, it does not seem to be the right approach.
I would greatly appreciate any insight or some good reference that might be of help. So far, it only seems to me that my case should be treated 'by hand', but it is a bit beyond my scope.
