The interplay of smoothness and regularity in case of radial functions

Abstract

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 $d\ge 2$. Every radial function $f\in H^1\left(R^d\right)$ is almost everywhere equal to a function $\tilde{f}$, continuous for $x\ne 0$, such that

where $c$ depends only on $d$.

Strauss stated (1) with the extra condition $\left|x\right|\ge 1$, 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 $H^1\left({ℝ}^d\right)$, of course. In particular, $H^1\left({ℝ}^d\right)⊈L^{\infty }\left(R^d\right)$, $d\ge 2$, and consequently, functions in $H^1\left({ℝ}^d\right)$ can be unbounded in the neigborhood of any fixed point $x\in {ℝ}^d$. The decay properties of radial functions can be used to prove compactness of embeddings of radial subspaces into Lebesgue spaces. Let $R{H}^1\left({ℝ}^d\right)$ denote the subspace of $H^1\left({ℝ}^d\right)$ consisting of all radial functions in $H^1\left({ℝ}^d\right)$. Then

holds, if $2<q<q^{*}$ , where $q^{*}:=\infty$ if $d=2$ and $q^{*}=2d∕\left(d-2\right)$ if $d\ge 3$.

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