The \(L ^{p}\) resolvents of elliptic operators with uniformly continuous coefficients.
The author considers a strongly elliptic operator \(A\) of order \(2m\) in divergence form with bounded measurable coefficients and uniformly continuous top order coefficients. For \(1< p<\infty\), \(A\) is a bounded linear operator from the \(L^p\) Sobolev space \(H^{m,p}\) into \(H^{-m,p}\). The Mihlin multiplier theorem is used to prove that for appropriate complex \(\lambda\), the resolvent \((A-\lambda)^{-1}: H^{-m,p}\to H^{m,p}\) exists and satisfies \[ \|(A- \lambda)^{-1}\|_{H^{-i,p}\to H^{j,p}}\leq c|\lambda|^{-1+(i+ j)/2m} \] for \(0\leq i\), \(j\leq m\). A key component of the proof is the construction of a parametrix for \(A-\lambda\).
scientific article; zbMATH DE number 1902860
scientific article