Singular integrals in several complex variables. IV: The derivative of Cauchy integral on sphere (Q1063736)

[Part I\(=Chin\). Ann. Math. 3, 483-502 (1982; Zbl 0545.32003); part II\(=ibid.\), Ser. B 4, 307-318 (1983; see the review above); part III\(=ibid.\), Ser. B 4 (1983; Zbl 0573.32006).] For the notation see the preceding review. - In this part IV the authors obtain the following Plemelj formula for the derivative of the Cauchy integral on the sphere expressed by the corresponding Hadamard principal value (see (3.7)\(\}\). Let f: \(\bar B\to {\mathbb{C}}\) be a function with three times continuously differentiable real and imaginary parts. There exists a number I such that K-\(\lim_{z\to v}\frac{\partial}{\partial z}\frac{1}{\omega_{2n- 1}}\int_{S}\frac{f(u)\dot u}{(1-\sum^{n}_{j=1}z_ j\bar u_ j)^ n} =\) \(\lim_{\epsilon \to 0}\frac{1}{\omega_{2n- 1}}\int_{\sigma_{\epsilon}(v,\alpha,\beta)}\frac{u(f(u)-f(v))\dot u}{(1-\sum^{n}_{j=1}v_ j\bar u_ j)^{n+1}}\) \(+\) \(\frac{1}{2}(\frac{2\beta}{\alpha +\beta})^{n-1}\{\frac{\partial f(v)}{\partial u}[\frac{2\beta}{\alpha +\beta}I - n\frac{\beta - \alpha}{\alpha +\beta}v'v]+\) \[ n\frac{\partial f(v)}{\partial \bar u}v'\bar v+\frac{2\beta}{\alpha +\beta}\bar v[tr(\frac{\partial^ 2f}{\partial u\partial \bar u}(v))- v\frac{\partial^ 2f(v)}{\partial u\partial \bar u}\bar v']\}. \]