An Integration-by-Parts Formula in $L_1$-Spaces

Abstract

Let $d \in \mathbb {N}$ and $\theta \in [0,\frac{\pi }{2})$. Let $a_{ij} \in W^{1,\infty }(\mathbb {R}^d, \mathbb {R})$ for all $i,j \in \{1, \ldots , d\}$. Assume $C = (a_{ij})_{1 \le i,j \le d}$ satisfies $(C(x) \, \xi , \xi ) \in \Sigma _\theta$ for all $x \in \mathbb {R}^d$ and $\xi \in \mathbb {C}^d$, where $\Sigma _\theta$ is the closed sector with vertex 0 and semi-angle $\theta$ in the complex plane. Consider the operator $A_1$ in $L_1(\mathbb {R}^d)$ formally given by

$$\begin{aligned} A_1 u = - \sum _{i,j=1}^d \partial _i (a_{ij} \, \partial _j u). \end{aligned}$$

We prove that $A_1$ is accretive on $W^{3,1}(\mathbb {R}^d)$.