Inverse-closedness of the set of integral operators with L1-continuously varying kernels

Let N be an integral operator of the form(Nu)(x)=∫Rcn(x,x−y)u(y)dy acting in Lp(Rc) with a measurable kernel n satisfying the estimate|n(x,y)|≤β(y), where β∈L1. It is proved that if the function t↦n(t,⋅) is continuous in the norm of L1 and the operator 1+N has an inverse, then (1+N)−1=1+M, where M is an integral operator possessing the same properties.