An endpoint Littlewood-Paley inequality for BVP associated with the Laplacian on Lipschitz domains
It was considered the heat equation \[ \frac {\partial u}{\partial t} - \Delta u=0\text{ in } \Omega \times \left(0,+\infty \right), \] \[ u\left(\cdot ,0\right) =0 \text{ in } \Omega , \] where \(\Delta \) denotes the Laplacian in \(\Omega \) with Dirichlet or Neumann boundary condition, \(u\) vanishes as \(t\) goes to \(\infty \) and \(\Omega \) is Lipschitz domain in \({\mathbb{R}}^n\). It was proved a commutator inequality of Littlewood-Paley type between partial derivatives and functions of the Laplacian on the Lipschitz domain, which gives interior energy estimates for some BVP: \[ \int\limits_{0}^{\infty }\int\limits_{\Omega }\left|c_{t}f\right|^2 \frac {dxdt}{t}\leq c\int\limits_{\Omega }\left|D_{x}f\right|^2dx, f\in V. \] Here on \(V=H^1_0\left(\Omega \right)\) for a Dirichlet boundary condition, \(V=H^1\left(\Omega \right)\) for a Neumann boundary condition, \(c_{t}=\left[D_{x},R_{t}\right], \left[A,B\right]=AB-BA, R_{t}=\left(1-t^2\Delta \right)^{-1}\). It can be seem as an endpoint inequality for a family of energy estimates. Besides, it was explained the result in the case of upper half-space. Then it was proved for a special Lipschitz domain and for bounded Lipschitz domains.
commutator inequality of Littlewood-Paley type
scientific article; zbMATH DE number 1418173