Let $Z\left(\mathrm{Ann}\right(r,R\left)\right)$ be the class of all continuous functions $f$ on the annulus $\mathrm{Ann}(r,R)$ in ${ℂ}^n$ with twisted spherical mean $f\times {\mu }_s\left(z\right)=0,$ whenever $z\in {ℂ}^n$ and $s\>0$ satisfy the condition that the sphere $S_s\left(z\right)\subseteq \mathrm{Ann}(r,R)$ and ball $B_r\left(0\right)\subseteq B_s\left(z\right).$ In this paper, we give a characterization for functions in $Z\left(\mathrm{Ann}\right(r,R\left)\right)$ in terms of their spherical harmonic coefficients. We also prove support theorems for the twisted spherical means in ${ℂ}^n$ which improve some of the earlier results.

By a Fourier multiplier technique on Cantor-like Abelian groups with characters of finite order, the norms from L² into $L^{2l}$ of certain embeddings of character sums are computed. It turns out that the orders of the characters are immaterial as soon as they are at least four.

We study the relationship between the growth rate of an integer sequence and harmonic and functional properties of the corresponding sequence of characters. In particular we show that every polynomial sequence contains a set that is $\Lambda \left(p\right)$ for all $p$ but is not a Rosenthal set. This holds also for the sequence of primes.

We give a characterization of the pairs of weights (v,w), with w in the class $A_{\infty }$ of Muckenhoupt, for which the fractional maximal function is a bounded operator from $L^p(X,vd\mu )$ to $L^q(X,wd\mu )$ when 1 < p ≤ q < ∞ and X is a space of homogeneous type.