For an arbitrary open, nonempty, bounded set 816305412&_mathId=si1.gif&_user=111111111&_pii=S0001870816305412&_rdoc=1&_issn=00018708&md5=6b9e949dd851938f0fc4478d11304a01" title="Click to view the MathML source">Ω⊂Rn, 816305412&_mathId=si2.gif&_user=111111111&_pii=S0001870816305412&_rdoc=1&_issn=00018708&md5=69c9ef5221267c27b13c5eb2e8213826" title="Click to view the MathML source">n∈N, and sufficiently smooth coefficients 816305412&_mathId=si3.gif&_user=111111111&_pii=S0001870816305412&_rdoc=1&_issn=00018708&md5=1cb6493e7e787bf1213b235b01262b51" title="Click to view the MathML source">a,b,q, we consider the closed, strictly positive, higher-order differential operator 816305412&_mathId=si258.gif&_user=111111111&_pii=S0001870816305412&_rdoc=1&_issn=00018708&md5=0929aaa775cb1ee552fc970147260c5d" title="Click to view the MathML source">AΩ,2m(a,b,q) in 816305412&_mathId=si369.gif&_user=111111111&_pii=S0001870816305412&_rdoc=1&_issn=00018708&md5=d121ec289aba44d21c895447b153ac28" title="Click to view the MathML source">L2(Ω) defined on 816305412&_mathId=si6.gif&_user=111111111&_pii=S0001870816305412&_rdoc=1&_issn=00018708&md5=b42d7a0e6590734e640a190aac9f6c74">816305412-si6.gif">, associated with the differential expression
and its Krein–von Neumann extension 816305412&_mathId=si11.gif&_user=111111111&_pii=S0001870816305412&_rdoc=1&_issn=00018708&md5=e2b56797399b5ae4f9caee6dda0d791f" title="Click to view the MathML source">AK,Ω,2m(a,b,q) in 816305412&_mathId=si369.gif&_user=111111111&_pii=S0001870816305412&_rdoc=1&_issn=00018708&md5=d121ec289aba44d21c895447b153ac28" title="Click to view the MathML source">L2(Ω). Denoting by 816305412&_mathId=si62.gif&_user=111111111&_pii=S0001870816305412&_rdoc=1&_issn=00018708&md5=cdc449ee02c64da54ea89b5ac8934320" title="Click to view the MathML source">N(λ;AK,Ω,2m(a,b,q)), 816305412&_mathId=si10.gif&_user=111111111&_pii=S0001870816305412&_rdoc=1&_issn=00018708&md5=54b878bd15ec619da6e5887571a78720" title="Click to view the MathML source">λ>0, the eigenvalue counting function corresponding to the strictly positive eigenvalues of 816305412&_mathId=si11.gif&_user=111111111&_pii=S0001870816305412&_rdoc=1&_issn=00018708&md5=e2b56797399b5ae4f9caee6dda0d791f" title="Click to view the MathML source">AK,Ω,2m(a,b,q), we derive the bound
where 816305412&_mathId=si13.gif&_user=111111111&_pii=S0001870816305412&_rdoc=1&_issn=00018708&md5=06a37e118520b7bf93583079aad096e5" title="Click to view the MathML source">C=C(a,b,q,Ω)>0 (with 816305412&_mathId=si14.gif&_user=111111111&_pii=S0001870816305412&_rdoc=1&_issn=00018708&md5=69e16cf5a79976c0bdc5bc08b8e48a3a" title="Click to view the MathML source">C(In,0,0,Ω)=|Ω|) is connected to the eigenfunction expansion of the self-adjoint operator 816305412&_mathId=si15.gif&_user=111111111&_pii=S0001870816305412&_rdoc=1&_issn=00018708&md5=1c56f34c60fab11ca22431b52517af80">816305412-si15.gif"> in 816305412&_mathId=si16.gif&_user=111111111&_pii=S0001870816305412&_rdoc=1&_issn=00018708&md5=5bd6d975b045fb410d215a3c8a761dd6" title="Click to view the MathML source">L2(Rn) defined on 816305412&_mathId=si17.gif&_user=111111111&_pii=S0001870816305412&_rdoc=1&_issn=00018708&md5=f8d3fa08ca5c2aa4866bc01139f4d46b" title="Click to view the MathML source">W2m,2(Rn), corresponding to 816305412&_mathId=si18.gif&_user=111111111&_pii=S0001870816305412&_rdoc=1&_issn=00018708&md5=2e600e305f4eb19886d204c2c0a318fe" title="Click to view the MathML source">τ2m(a,b,q). Here 816305412&_mathId=si19.gif&_user=111111111&_pii=S0001870816305412&_rdoc=1&_issn=00018708&md5=a1a8e064040393224564798b3223ed71" title="Click to view the MathML source">vn:=πn/2/Γ((n+2)/2) denotes the (Euclidean) volume of the unit ball in 816305412&_mathId=si20.gif&_user=111111111&_pii=S0001870816305412&_rdoc=1&_issn=00018708&md5=c87582d62e5ee4c2a0c93f5c7dd08d47" title="Click to view the MathML source">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 816305412&_mathId=si21.gif&_user=111111111&_pii=S0001870816305412&_rdoc=1&_issn=00018708&md5=1762617bddf196643438789af1b5a5fb">816305412-si21.gif"> in 816305412&_mathId=si16.gif&_user=111111111&_pii=S0001870816305412&_rdoc=1&_issn=00018708&md5=5bd6d975b045fb410d215a3c8a761dd6" title="Click to view the MathML source">L2(Rn).
We also consider the analogous bound for the eigenvalue counting function for the Friedrichs extension 816305412&_mathId=si22.gif&_user=111111111&_pii=S0001870816305412&_rdoc=1&_issn=00018708&md5=509aa1004e0773e4317b0b20c7f7e4b3" title="Click to view the MathML source">AF,Ω,2m(a,b,q) in 816305412&_mathId=si369.gif&_user=111111111&_pii=S0001870816305412&_rdoc=1&_issn=00018708&md5=d121ec289aba44d21c895447b153ac28" title="Click to view the MathML source">L2(Ω) of 816305412&_mathId=si258.gif&_user=111111111&_pii=S0001870816305412&_rdoc=1&_issn=00018708&md5=0929aaa775cb1ee552fc970147260c5d" title="Click to view the MathML source">AΩ,2m(a,b,q).