Inverting some integral operators with degenerating and oscillating symbols
In the frames of \(L_p\)-spaces, the authors construct an inversion of multidimensional integral operators \[ K^\alpha_{P_m,b, \gamma,k} (\xi)= P_m\bigl(| \xi'-b | \bigr)e^{-i\gamma |\xi|} |\xi|^{- \alpha-m}, \quad \alpha>0 \tag{1} \] where \(P_m(z)= \sum^m_{s=0} a_sz^s\); \(a_s\in \mathbb{C}\), \(1\leq s\leq m\), \(\xi'= (\xi_1, \dots, \xi_k)\), \(1\leq k\leq n\); \(\gamma\in \mathbb{R}\). Formally, the inverse to (1) can be written in the form \[ (K^\alpha_{ P_m,b, \gamma,k})^{-1} f=F^{-1} \left({1 \over K^\alpha_{P_m,b, \gamma,k} (\xi)} \right)*f. \tag{2} \] For \(\gamma\neq 0\), the operators (1) are representable in the form of a sum of compositions of potential-type operators, singular integrals and operators of the form \[ (B^\delta_\gamma \varphi) (x)= \int_{\mathbb{R}^n} B_{\delta, \gamma} \bigl(| t|\bigr) \varphi (x-t)dt, \quad {n-1\over 2} <\delta<n; \] the case \(\gamma=0\) represents a separate interest for the authors. Also, the inverse (2) is studied.
scientific article; zbMATH DE number 1135534