Dominated Uryson operators on lattice-normed spaces

The aim of this note is to introduce the space D_{U}(V,W) of the dominated Uryson operators on lattice-normed spaces. We prove an "Up-and-down" type theorem for a positive abstract Uryson operator defined on a vector lattice and taking values in a Dedekind complete vector lattice. This result we apply to prove the decomposability of the lattice valued norm of the space D_{U}(V,W) of all dominated Uryson operators. We obtain that, for a lattice-normed space V and a Banach-Kantorovich space W the space D_{U}(V,W) is also a Banach-Kantorovich space. We prove that under some mild conditions, a dominated Uryson operator has an exact dominant. We obtain formulas for calculating the exact dominant of a dominated Uryson operator. We also consider laterally continuous and completely additive dominated Uryson operators and prove that a dominated Uryson operator is laterally continuous (completely additive) if and only if so is its exact dominant.