Approximation by harmonic functions in the \(C^m\)-norm and harmonic \(C^m\)-capacity of compact sets in \(\mathbb{R}^n\) (Q1277533)

The author studies the function \(\Lambda^m(X)\), \(0< m<1\), of compact sets \(X\) in \(\mathbb{R}^n\), \(n\geq 2\), defined as the distance in the space \(C^m(X)\equiv \text{lip}^m(X)\) 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 \(X\). Among other results, it is proven that the conditions \(\Lambda^m(X)= 0\) and \(C^m(X)= H^m(X)\) are equivalent.