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

For an arbitrary open, nonempty, bounded set Ω⊂RnΩ⊂Rn, n∈Nn∈N, and sufficiently smooth coefficients a,b,qa,b,q, we consider the closed, strictly positive, higher-order differential operator AΩ,2m(a,b,q)AΩ,2m(a,b,q) in L2(Ω)L2(Ω) defined on W02m,2(Ω), associated with the differential expression τ2m(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)AK,Ω,2m(a,b,q) in L2(Ω)L2(Ω). Denoting by N(λ;AK,Ω,2m(a,b,q))N(λ;AK,Ω,2m(a,b,q)), λ>0λ>0, the eigenvalue counting function corresponding to the strictly positive eigenvalues of AK,Ω,2m(a,b,q)AK,Ω,2m(a,b,q), we derive the bound N(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)>0C=C(a,b,q,Ω)>0 (with C(In,0,0,Ω)=|Ω|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)L2(Rn) defined on W2m,2(Rn)W2m,2(Rn), corresponding to τ2m(a,b,q)τ2m(a,b,q). Here vn:=πn/2/Γ((n+2)/2)vn:=πn/2/Γ((n+2)/2) denotes the (Euclidean) volume of the unit ball in RnRn. 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)L2(Rn). We also consider the analogous bound for the eigenvalue counting function for the Friedrichs extension AF,Ω,2m(a,b,q)AF,Ω,2m(a,b,q) in L2(Ω)L2(Ω) of AΩ,2m(a,b,q)AΩ,2m(a,b,q). No assumptions on the boundary ∂Ω of Ω are made.