A priori bounds for elliptic operators in weighted Sobolev spaces (Q2902024)

The authors study weighted estimates for second order elliptic equations, not necessarily in divergence form, on unbounded open sets. The precise conditions on the open sets \(\Omega\) and the weights \(m\) are given in the paper. A sufficient condition that \(m\) belongs to the class \(G(\Omega)\) for which the main results are established is that \(\log m \in \text{Lip}(\Omega)\). Examples of functions in \(G(\Omega)\) include \(m(x) = e^{t| x|}\) and \(m(x) = ( 1 + | x|^2)^t, x \in \Omega, t \in R\).NEWLINENEWLINEWhen \(m \in G(\Omega)\), the Sobolev space \(W^{k,p}_s(\Omega)\) is the set of all distributions \(u\) on \(\Omega\) such that \(m^s \partial^{\alpha}u \in L^p(\Omega),|\alpha | \leq k\) equipped with the obvious norm. \(W^{\circ, k,p}_s(\Omega)\) is the closure of the \(C^{\infty}(\Omega)\) functions with compact support in \(W^{k,p}_s(\Omega)\) and \(W^{0,p}_s(\Omega) = L^p_s(\Omega)\).NEWLINENEWLINEThe authors prove a priori estimates of the type NEWLINE\[NEWLINE\| u \|_{W^{2, p}_s(\Omega)} \leq c \left( \| Lu \|_{L^p_s(\Omega)} + \| u \|_{L^p_s(\Omega)} \right), \forall u \in W^{2, p}_s(\Omega) \cap W^{\circ, 1,p}_s(\Omega), NEWLINE\]NEWLINE \noindent when the coefficients of the elliptic operator \(L\) are bounded and locally in \(VMO\) on \(\Omega\), in addition to other technical conditions on the coefficients. The key to their results is multiplication results on the Sobolev spaces.