We call an $L^p$-multiplier m tame if for each complex homomorphism χ acting on the space of $L^p$ multipliers there is some ${\gamma }_0\in \Gamma$ and |a| ≤ 1 such that $\chi \left(\gamma m\right)=am\left({\gamma }_0\gamma \right)$ for all γ ∈ Γ. Examples of tame multipliers include tame measures and one-sided Riesz products. Tame multipliers show an interesting similarity to measures. Indeed we show that the only tame idempotent multipliers are measures. We obtain quantitative estimates on the size of $L^p$-improving tame multipliers which are similar to those obtained for measures, but...

Let G be a metrizable, compact abelian group and let Λ be a subset of its dual group Ĝ. We show that $C_{\Lambda }\left(G\right)$ has the almost Daugavet property if and only if Λ is an infinite set, and that $L{¹}_{\Lambda }\left(G\right)$ has the almost Daugavet property if and only if Λ is not a Λ(1) set.

Let G be an infinite, compact abelian group and let Λ be a subset of its dual group Γ. We study the question which spaces of the form $C_{\Lambda }\left(G\right)$ or $L{¹}_{\Lambda }\left(G\right)$ and which quotients of the form $C\left(G\right)/C_{\Lambda }\left(G\right)$ or $L¹\left(G\right)/L{¹}_{\Lambda }\left(G\right)$ have the Daugavet property. We show that $C_{\Lambda }\left(G\right)$ is a rich subspace of C(G) if and only if $\Gamma \setminus {\Lambda }^{-1}$ is a semi-Riesz set. If $L{¹}_{\Lambda }\left(G\right)$ is a rich subspace of L¹(G), then $C_{\Lambda }\left(G\right)$ is a rich subspace of C(G) as well. Concerning quotients, we prove that $C\left(G\right)/C_{\Lambda }\left(G\right)$ has the Daugavet property if Λ is a Rosenthal set, and that $L{¹}_{\Lambda }\left(G\right)$ is a poor subspace of L¹(G) if Λ is...

A number of recent papers have been devoted to the study of prevalence, a generalization of the property of being of full Haar measure to topological groups which need not have a Haar measure, and the dual concept of shyness. These concepts give a notion of "largeness" which often differs from the category analogue, comeagerness, and may be closer to the intuitive notion of "almost everywhere." In this paper, we consider the group of permutations of natural numbers. Here, in the sense of category,...

Pointwise upper bounds for characters of compact, connected, simple Lie groups are obtained which enable one to prove that if μ is any central, continuous measure and n exceeds half the dimension of the Lie group, then ${\mu }^n\in L^1$. When μ is a continuous, orbital measure then ${\mu }^n$ is seen to belong to $L^2$. Lower bounds on the p-norms of characters are also obtained, and are used to show that, as in the abelian case, m-fold products of Sidon sets are not p-Sidon if p < 2m/(m+1).

The class of $H^{\left(N\right)}$-sets forms an important subclass of the class of sets of uniqueness for trigonometric series. We investigate the size of this class which is reflected by the family of measures (called polar) annihilating all sets from the class. The main aim of this paper is to answer in the negative a question stated by Lyons, whether the polars of the classes of $H^{\left(N\right)}$-sets are the same for all N ∈ ℕ. To prove our result we also present a new description of $H^{\left(N\right)}$-sets.

We prove that if $E\subseteq Ĝ$ does not contain parallelepipeds of arbitrarily large dimension then for any open, non-empty $S\subseteq G$ there exists a constant c > 0 such that $\parallel f{1}_S{\parallel }_2\ge c\parallel f{\parallel }_2$ for all $f\in L^2\left(G\right)$ whose Fourier transform is supported on E. In particular, such functions cannot vanish on any open, non-empty subset of G. Examples of sets which do not contain parallelepipeds of arbitrarily large dimension include all Λ(p) sets.

We consider sets in the real line that have Littlewood-Paley properties LP(p) or LP and study the following question: How thick can these sets be?

Let $L_{\alpha }^q(R^D),\alpha \&gt;0,1\&lt;q\&lt;\infty$, denote the space of Bessel potentials $f=G_{\alpha }*g$, $g\in L^q$, with norm $\parallel f{\parallel }_{\alpha ,q}=\parallel g{\parallel }_q$. For $\alpha$ integer $L_{\alpha }^q$ can be identified with the Sobolev space $H^{\alpha ,q}$.One can associate a potential theory to these spaces much in the same way as classical potential theory is associated to the space $H^{1;2}$, and a considerable part of the theory was carried over to this more general context around 1970. There were difficulties extending the theory of thin sets, however. By means of a new inequality, which characterizes the positive cone in the space...

In this paper, we give a criterion for unconditional convergence with respect to some summability methods, dealing with the topological size of the set of choices of sign providing convergence. We obtain similar results for boundedness. In particular, quasi-sure unconditional convergence implies unconditional convergence.