Estimates for a uniform modulus of continuity of functions from symmetric spaces

We prove a multidimensional “correctability” theorem of the Oskolkov type for a function given in $\mathbb R^n$ whereby a sharp quantitative estimate for the uniform modulus of continuity of a function on “large” sets is given if an estimate of the modulus of continuity of this function in a symmetric space is known. We show that an estimate of a uniform modulus of continuity depends only on the eigenfunction of the symmetric space.