Constructive description of Hölder classes on some multidimensional compact sets

We give a constructive description of Hölder classes of functions on certain compacts in Rm (m > 3) in terms of a rate of approximation by harmonic functions in shrinking neighborhoods of these compacts. The considered compacts are a generalization to the higher dimensions of compacts that are subsets of a chord-arc curve in R3. The size of the neighborhood is directly related to the rate of approximation it shrinks when the approximation becomes more accurate. In addition to being harmonic in the neighborhood of the compact the approximation functions have a property that looks similar to Hölder condition. It consists in the fact that the difference in values at two points is estimated in terms of the size of the neighborhood, if the distance between these points is commensurate with the size of the neighborhood (and therefore it is estimated in terms of the distance between the points).

A Counterexample in Lp Approximation by Harmonic Functions

For we show that the conditions for all open sets G, C2,q denoting Bessel capacity, are not sufficient to characterize the compact sets X with the property that each function harmonic on and in Lp(X) is the limit in the Lp norm of a sequence of functions which are harmonic on neighbourhoods of X.