  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.
