We show that if the summability means in the Fourier inversion formula for a tempered distribution f ∈ S′(ℝn) converge to zero pointwise in an open set Ω, and if those means are locally bounded in L1(Ω), then Ω ⊂ ℝn\supp f. We prove this for several summability procedures, in particular for Abel summability, Cesàro summability and Gauss-Weierstrass summability.

For a function $f\in L^p(\mathbb{R}d)$, $d\ge 2$, let $A_tf(x)$ be the mean of $f$ over the sphere of radius $t$ centred at $x$. Given a set $E\subset(0,\infty)$ of dilations we prove various endpoint bounds for the maximal operator $M_E$ defined by $M_E f(x)=\sup_{t\in E}|A_tf(x)|$, under some regularity assumptions on $E$.

AMS 2000 Mathematics subject classification: Primary 42B25. Secondary 42B20

Let G/K be a compact symmetric space, and let G = KAK be a Cartan decomposition of G. For f in L1(G) we define the spherical means f(g, t) = ∫k∫k ∫(gktk′) dk dk′, g ∈ G, t ∈ A. We prove that if f is in Lp(G), 1 ≤ p ≤ 2, then for almost every g ∈ G the functions t → f(g, t) belong to certain Soblev spaces on A. From these regularity results for the spherical means we deduce, if G/K is a compact rank one symmetric space, a theorem on the almost everywhere localization of spherical harmonic expansions of functions in L2 (G/K).