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 $$ tau_{2m} (a,b,q) := bigg(sum_{j,k=1}^{n} (-i partial_j - b_j) a_{j,k} (-i partial_k - b_k)+qbigg)^m, quad m in mathbb{N}, $$ 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 $$ N(lambda; A_{K, Omega, 2m} (a,b,q)) leq C v_n (2pi)^{-n} bigg(1+frac{2m}{2m+n}bigg)^{n/(2m)} lambda^{n/(2m)} , quad lambda > 0, $$ 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.