Uniformly elliptic operators with measurable coefficients (Q1897289)

The author considers higher-order elliptic operators \(H\), a typical example being \[ Hf= \Delta(a(x)\Delta f),\quad\text{with}\quad {1\over c}\leq a(x)\leq c,\;x\in \mathbb{R}^N,\text{ and } c> 0. \] He obtains pointwise upper bounds for the fundamental solution \(K(x, y, t)\) of the associated evolution equation \(df/dt= -Hf\). These bounds allow him to prove that the operators \(e^{- Ht}\) extend to strongly continuous holomorphic one-parameter semigroups on \(L^p\).