Cauchy transformation of analytic functionals on the sphere (Q1893931)

This is a survey, without proofs, of recent work by the author. The classical, one-dimensional case, is as follows. Let \(D[r]= \{z\in \mathbb{C}: |z|\leq r\}\). Let the space of holomorphic germs on \(D[r]\) with the usual inductive limit topology be denoted by \({\mathcal O}(D[r])\). Any function \(f\in {\mathcal O}(D(\rho))\) (the holomorphic functions in a \(\rho\)-neighborhood of \(0\)) \(r< \rho\), has a Cauchy integral representation as \[ f(w)= \int_{S^1} f(\rho z) {1\over 1- w\overline z/\rho} d\mu(z), \] for all \(|w|< \rho\). For an analytic functional \(T\in {\mathcal O}'(D[r])\), we have \[ \langle T, f\rangle= \int_{S^1} f(\rho z) \langle T_w,\;{1\over 1- w\overline z/\rho}\rangle d\mu(z). \] The Cauchy transform \(\check T\) of \(T\) is defined to be the element of \({\mathcal O}(D(1/r))\) given by \[ \check T(z)= \langle T_w, {1\over 1- wz}\rangle. \] Proposition: The Cauchy transformation establishes a topological isomorphism between \({\mathcal O}'(D[r])\) and \({\mathcal O}(D(1/r))\). The extension of this result to various settings in several variables, such as the spherical and conical cases, is described.