Sharp \(L^ p\)-Hodge decompositions for Lipschitz domains in \(\mathbb R^ 2\). (Q5954516)

The author studies some aspects of the Hodge decomposition in two variables when the domain is not necessarily smooth. In fact, it is well known that, when the domain is smooth, \(L^p\) can be decomposed in the following way: NEWLINE\[NEWLINE L^p(\Omega,\mathbb{R}^2)= \nabla H^{1,p}_0(\Omega) \oplus \nabla^t H^{1,p}(\Omega) \oplus {\mathcal H}^p_{\text{nor}}(\Omega,\mathbb{R}^2) NEWLINE\]NEWLINE NEWLINE\[NEWLINE L^p(\Omega, \mathbb{R}^2) = \nabla H^{1,p}_0(\Omega) \oplus \nabla^t H^{1,p}(\Omega) \oplus {\mathcal H}^p_{\text{tan}}(\Omega,\mathbb{R}^2) NEWLINE\]NEWLINE for \(1<p<\infty\). However, if the boundary of the domain is allowed to have irregularities, the situation changes a lot. The main result of the paper clarifies the sharp range of \(p\)'s for which the Hodge decomposition holds true when \(\Omega\) is a Lipschitz domain.