Uniformly bounded composition operators in the Banach space of absolutely continuous functions

Let I,J⊂RI,J⊂R be intervals and let h:I×J→Rh:I×J→R be an arbitrary function. Our main result says that if the Nemytskij composition operator HH of the generator hh mapping the set AC(I,J)AC(I,J) into the Banach space AC(I,R)AC(I,R) is uniformly bounded (or equidistantly uniformly bounded), then h(x,y)=a(x)y+b(x),x∈I,y∈R, for some functions a,b∈AC(I,R)a,b∈AC(I,R).