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

For an arbitrary open, nonempty, bounded set Ω⊂Rn, n∈N, and sufficiently smooth coefficients a,b,q, we consider the closed, strictly positive, higher-order differential operator AΩ,2m(a,b,q) in L2(Ω) defined on W02m,2(Ω), associated with the differential expressionτ2m(a,b,q):=(∑j,k=1n(−i∂j−bj)aj,k(−i∂k−bk)+q)m,m∈N, and its Krein-von Neumann extension AK,Ω,2m(a,b,q) in L2(Ω). Denoting by N(λ;AK,Ω,2m(a,b,q)), λ>0, the eigenvalue counting function corresponding to the strictly positive eigenvalues of AK,Ω,2m(a,b,q), we derive the boundN(λ;AK,Ω,2m(a,b,q))≤Cvn(2π)−n(1+2m2m+n)n/(2m)λn/(2m),λ>0, where C=C(a,b,q,Ω)>0 (with C(In,0,0,Ω)=|Ω|) is connected to the eigenfunction expansion of the self-adjoint operator A˜2m(a,b,q) in L2(Rn) defined on W2m,2(Rn), corresponding to τ2m(a,b,q). Here vn:=πn/2/Γ((n+2)/2) denotes the (Euclidean) volume of the unit ball in Rn.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 A˜2(a,b,q) in L2(Rn).We also consider the analogous bound for the eigenvalue counting function for the Friedrichs extension AF,Ω,2m(a,b,q) in L2(Ω) of AΩ,2m(a,b,q).No assumptions on the boundary ∂Ω of Ω are made.