Locally defined operators and a partial solution of a conjecture Original Research Article

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. Miś [K. Lichawski, J. Matkowski, J. Miś, 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 Cm(A)Cm(A) into Ck(A)Ck(A), for k≥2k≥2 where Cm(A)Cm(A) is the space of mm-times Whitney continuously differentiable functions on a perfect set View the MathML sourceA⊂Rn