Interpolation of nonlinear integral Urysohn operators in net spaces

Abstract

In this paper, we study the interpolation properties of the net spaces Np;q(M), in the case when M is a sufficiently general arbitrary system of measurable subsets from Rn: The integral Urysohn operator is considered. This operator generalizes all linear, integral operators, and non-linear integral operators. The Urysohn operator is not a quasilinear or subadditive operator. Therefore, the classical interpolation theorems for these operators do not hold. A certain analogue of the Marcinkiewicz-type interpolation theorem for this class of operators is obtained. This theorem allows to obtain, in a sense, a strong estimate for Urysohn operators in net spaces from weak estimates for these operators in net spaces with local nets. For example, in order for the Urysohn integral operator in a net space, where the net is the set of all balls in Rn, it is sufficient for it to be of weak type for net spaces, where the net is concentric balls.