Approximation by harmonic functions in the Cm-norm and harmonic Cm-capacity of compact sets in Rn

We study the function Lambda(m)(X), 0 < m < 1, of compact sets X in R-n, n greater than or equal to 2, defined as the distance in the space C-m(X) = lip(m)(X) from the function \x\(2) to the subspace H-m(X) which is the closure in C-m(X) of the class of functions harmonic in the neighborhood of X (each function in its own neighborhood). We prove the equivalence of the conditions Lambda(m)(X) = 0 and C-m(X) = H-m(X). We derive an estimate from above that depends only on the geometrical properties of the set X (on its volume).