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Sharp \(L^ p\)-Hodge decompositions for Lipschitz domains in \(\mathbb R^ 2\). (English) Zbl 1039.31004

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: \[ L^p(\Omega,\mathbb{R}^2)= \nabla H^{1,p}_0(\Omega) \oplus \nabla^t H^{1,p}(\Omega) \oplus {\mathcal H}^p_{\text{nor}}(\Omega,\mathbb{R}^2) \]
\[ L^p(\Omega, \mathbb{R}^2) = \nabla H^{1,p}_0(\Omega) \oplus \nabla^t H^{1,p}(\Omega) \oplus {\mathcal H}^p_{\text{tan}}(\Omega,\mathbb{R}^2) \] 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.

MSC:

31A10 Integral representations, integral operators, integral equations methods in two dimensions
58A14 Hodge theory in global analysis
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
45E05 Integral equations with kernels of Cauchy type
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