Uniformly continuous set-valued composition operators in the space of total $\varphi$-bidimensional variation in the sense of Riesz

In this paper we prove that if a Nemytskij composition operator, generated by a function of three variables in which the third variable is a function one, maps a suitable large subset of the space of functions of bounded total $\varphi$-bidimensional variation in the sense of Riesz, into another such space, and is uniformly continuous, then its generator is an affine function in the function variable. This extends some previous results in the one-dimensional setting.