Orthogonal decomposition related to magnetic field, and Grunsky inequality (Q1916070)

For \(D\) a bounded domain in \(\mathbb{R}^3\) with smooth boundary \(\Sigma\), \(\sigma\) a \(C^\infty\) closed 1-form on \(\overline D=D\cup\Sigma\), \(\widetilde\sigma=\sigma\) in \(\overline D\) and \(=0\) outside \(\overline D\), the Weyl orthogonal decomposition \(\widetilde\sigma=*\omega+ dF\) in \(\mathbb{R}^3\) is considered. Integral formulas for \(F\) and \(p\) with \(\omega=dp\) are given. Magnetic fields induced by a surface current density on \(\Sigma\) are studied. The integral formulas in \(\mathbb{R}^3\) are extended, and formulas involving the complex \(z\)-plane are produced. Finally, a proof of the Grunsky inequality is provided, with a necessary and sufficient condition for the case of equality.