PDE Seminar Abstracts

The interplay of smoothness and regularity in case of radial functions

Winfried Sickel
Friedrich-Schiller-Universit├Ąt Jena, Germany
Wed 10th Jul 2019, 2-3pm, Carslaw Room 829 (AGR)


At the end of the seventies Strauss was the first who observed that there is an interplay between the regularity and decay properties of radial functions. We recall his

Radial Lemma: Let d 2. Every radial function f H1(Rd) is almost everywhere equal to a function f̃, continuous for x0, such that

|f̃(x)| c|x|(1-d)2fH1(Rd),x0, (1)

where c depends only on d.

Strauss stated (1) with the extra condition |x| 1, but this restriction is not needed.

The Radial Lemma contains three different assertions:

(a) the existence of a representative of f , which is continuous outside the origin;

(b) the decay of f near infinity;

(c) the limited unboundedness near the origin.

These three properties do not extend to all functions in H1(d), of course. In particular, H1(d)L(Rd), d 2, and consequently, functions in H1(d) can be unbounded in the neigborhood of any fixed point x d. The decay properties of radial functions can be used to prove compactness of embeddings of radial subspaces into Lebesgue spaces. Let RH1(d) denote the subspace of H1(d) consisting of all radial functions in H1(d). Then


holds, if 2 < q < q* , where q* := if d = 2 and q* = 2d(d - 2) if d 3.

We will give a survey how these classical results extend to functions spaces with fractional order of smoothness like Besov and Lizorkin-Triebel spaces.