Let B(z,\(\zeta)\) be the Bergman kernel of a compact simply connected domain D (with the Ljapunov boundary \(\Gamma)\), defined as \(B(z,\zeta)=- (2/\pi)(\partial^ 2G(z,\zeta)/\partial z\overline{\partial \zeta})\) where G(z,\(\zeta)\) is the Green’s function for the Laplacian with the singularity at \(z=\zeta \in \Gamma\). The integral equation
(1) \(Af=a(z)f(z)+(b(z)/\pi)(z/| z|)^ n\iint_{D}\overline{f(\zeta)}/(\zeta -z)^ 2ds_{\zeta}+c(z)\iint_{D}B(z,\zeta)f(\zeta)ds_{\zeta}=g(z)\)
with continuous coefficients a(z), b(z), c(z) (z\(\in \bar D=D\cup \Gamma)\) is investigated in the space \(L^ p(D,| z|^{\beta - 2/p})\) \((1<p<\infty\), \(0<\beta <2)\), Theorem 2. If a(0)\(\neq 0\), \(| a(z)| \neq | b(z)|\), (z\(\in \bar D)\), \(a(t)+b(t)\neq 0\) (t\(\in \Gamma)\) then: 1) (1) is the Fredholm equation in \(L^ p(D,| z|^{\beta -2/p})\) and \(Ind A=-2\quad ind [a+b]_{\Gamma}+2n;\) 2) Ker A is the same in all indicated spaces. A similar theorem holds for \(a(0)=0\).