A bound for the eigenvalue counting function for Krein--von Neumann and Friedrichs extensions

For an arbitrary open, nonempty, bounded set $\Omega \subset \mathbb{R}^n$, $n \in \mathbb{N}$, and sufficiently smooth coefficients $a,b,q$, we consider the closed, strictly positive, higher-order differential operator $A_{\Omega, 2m} (a,b,q)$ in $L^2(\Omega)$ defined on $W_0^{2m,2}(\Omega)$, associated with the higher-order differential expression $$\tau_{2m} (a,b,q) := \bigg(\sum_{j,k=1}^{n} (-i \partial_j - b_j) a_{j,k} (-i \partial_k - b_k)+q\bigg)^m, \quad m \in \mathbb{N},$$ and its Krein--von Neumann extension $A_{K, \Omega, 2m} (a,b,q)$ in $L^2(\Omega)$. Denoting by $N(\lambda; A_{K, \Omega, 2m} (a,b,q))$, $\lambda > 0$, the eigenvalue counting function corresponding to the strictly positive eigenvalues of $A_{K, \Omega, 2m} (a,b,q)$, we derive the bound $$N(\lambda; A_{K, \Omega, 2m} (a,b,q)) \leq C v_n (2\pi)^{-n} \bigg(1+\frac{2m}{2m+n}\bigg)^{n/(2m)} \lambda^{n/(2m)} , \quad \lambda > 0,$$ where $C = C(a,b,q,\Omega)>0$ (with $C(I_n,0,0,\Omega) = |\Omega|$) is connected to the eigenfunction expansion of the self-adjoint operator $\widetilde A_{2m} (a,b,q)$ in $L^2(\mathbb{R}^n)$ defined on $W^{2m,2}(\mathbb{R}^n)$, corresponding to $\tau_{2m} (a,b,q)$. Here $v_n := \pi^{n/2}/\Gamma((n+2)/2)$ denotes the (Euclidean) volume of the unit ball in $\mathbb{R}^n$. Our method of proof relies on variational considerations exploiting the fundamental link between the Krein--von Neumann extension and an underlying abstract buckling problem, and on the distorted Fourier transform defined in terms of the eigenfunction transform of $\widetilde A_{2} (a,b,q)$ in $L^2(\mathbb{R}^n)$. We also consider the analogous bound for the eigenvalue counting function for the Friedrichs extension $A_{F,\Omega, 2m} (a,b,q)$ in $L^2(\Omega)$ of $A_{\Omega, 2m} (a,b,q)$. No assumptions on the boundary $\partial \Omega$ of $\Omega$ are made.