A priori estimates for bisingular operators with continuous coefficients (Q1814576)

The main result proved by the author is the following: Theorem. The bisingular operator \(A: L_ p(\Gamma_ 1\times\Gamma_ 2)\to L_ p(\Gamma_ 1\times\Gamma_ 2)\) is a Fredholm operator if and only if the following a priori estimates are valid: \[ \| f\|_ p\leq\hbox{const}(\| Af\|_ p+\| f\|_{p_ 1}), \qquad \| g\|_ q\leq\hbox{const}(\| A^*g\|_ q+\| g\|_{q_ 1}), \] where \(f\in L_ p(\Gamma_ 1\times\Gamma_ 2)\), \(g\in L_ q(\Gamma_ 1\times\Gamma2)\), \(A^*\) is adjoint operator, \(p^{-1}+q^{-1}=1\), \(1<p_ 1<p<\infty\), \(1<q_ 1<q<\infty\).