Inversion of singular integrals with multiplicative Cauchy kernel and infinite integration domain (Q647683)

The authors consider \(K(x, \sigma) = \frac{1}{(x_1-\sigma_1)(x_2-\sigma_2)(x_3-\sigma_3)}\) as kernel in a domain \(D\times D\), \(D\subset \mathbb{R}^3\), and study the integral equation \[ \int_D K(x, \sigma)\varphi(\sigma) d\sigma = f(x) \] in Hölder classes. They give the solution \[ \varphi(\sigma) = \frac{1}{\sqrt{\sigma_1\sigma_2\sigma_3}}\quad R_1(f,\sigma) +\frac{c_1(\sigma_2, \sigma_3)}{\sqrt{\sigma_1}}+ \frac{c_2(\sigma_1, \sigma_3)}{\sqrt{\sigma_2}}+ \frac{c_3(\sigma_1, \sigma_2)}{\sqrt{\sigma_3}}, \] where \[ R_1(f,\sigma)=\frac{1}{\pi^3} \int_D \sqrt{x_1x_2x_3} \frac{(\sigma_1+1)(\sigma_2+1)(\sigma_3+1)}{(x_1+1)(x_2+1)(x_3+1)} K(x,\sigma)f(x)dx \] and \(c_k\) are arbitrary functions in special class \(h(\infty)\) in the case \(D= \mathbb{R}_+^1\times \mathbb{R}_+^1 \times \mathbb{R}_+^1.\) The authors give additional conditions for \(\varphi\) to guarantee uniqueness of solution. They study the case \(D=\mathbb{R}^3\) as well. Some examples are given.