Locally defined operators and a partial solution of a conjecture

We prove a weaker form of conjecture concerning the representation theorem for locally defined operators acting between some spaces of differentiable functions presented by K. Lichawski, J. Matkowski and J. Mi? K. Lichawski, J. Matkowski, J. Mi?, Locally defined operators in the space of differentiable functions, Bull. Polish Acad. Sci. Math. 37 (1989) 315-325].As a corollary we obtain the representation formula for continuous locally defined operators mapping the Banach space C^m(A) into C^k(A), for k≥2 where C^m(A) is the space of m-times Whitney continuously differentiable functions on a perfect set .