Area integral estimates for the biharmonic operator in Lipschitz domains

Abstract:

Let $D \subseteq {{\mathbf {R}}^n}$ be a Lipschitz domain and let $u$ be a function biharmonic in $D$, i.e., $\Delta \Delta u= 0$ in $D$. We prove that the nontangential maximal function and the square function of the gradient of $u$ have equivalent ${L^p}(d\mu )$ norms, where $d\mu \in {A^\infty } (d\sigma )$ and $d\sigma$ is surface measure on $\partial D$.