More on hypersingular integrals and embeddings into hölder spaces

In this chapter we present results on hypersingular operators of order α < 1 acting on some Sobolev type variable exponent spaces, where the underlying space is a quasimetric measure space. The proofs are based on some pointwise estimations of differences of Sobolev functions. These estimates lead also to embeddings of variable exponent Hajlasz-Sobolev spaces into variable order Hölder spaces. In the Euclidean case we prove denseness of C^∞₀-functions in W^1,p(·)(Rⁿ). Note that in this chapter we consider quasimetric measure spaces with symmetric distance: d(x, y) = d(y, x).