Some \(L_{p}\)-estimates for elliptic and parabolic operators with measurable coefficients
occupation measures for diffusion processes
The very interesting paper under review deals with linear elliptic and parabolic operators with measurable coefficients of the form NEWLINE\[NEWLINE {\mathcal L}=\partial_t+a^{ij}(t,x)D_{ij}+b^i(t,x)d_i - c(t,x),\quad x\in\mathbb R^d NEWLINE\]NEWLINE with a uniformly nondegenerate and bounded matrix \(\{a^{ij}\}.\) The author proves two types of \(L_{p}\)-results for the strong solutions to the equation \({\mathcal L}u,\) used recently in the strong solvability theory of fully nonlinear elliptic and parabolic second order equations developed in [\textit{H. Dong, N. V. Krylov} and \textit{X. Li}, ``On fully nonlinear elliptic and parabolic equations with VMO coefficients in domains'', Algebra Anal. 24, No. 1, 53--94 (2012; \url{http://mi.mathnet.ru/aa1269})]. The first one is an estimate for the \(L_{\gamma}\)-norm of the second order derivatives in terms of \(|{\mathcal L}u|_{L_{d+1}}\) and \(\sup|u|,\) where \(\gamma\in(0,1).\) The second result deals with estimates of the resolvent operators in \(L_{p}\) when the first-order coefficients are summable to an appropriate power.
scientific article; zbMATH DE number 6081735