Interpolation of nonlinear integral Urysohn operators in net spaces

Abstract

In this paper, we study the interpolation properties of the net spaces N_p,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.