Cohen type inequalities and divergence of Fourier series on compact Lie groups

et G be a compact Lie group of dimension n and rank l. To each increasing sequence {Σ_n} of finite subsets of the dual Σ of G whose union is Σ (summability method), we associate the partial sums S_n f=\∑_{λ ∈ Σ_n } d_λ χ_λ*f of the Fourier series of f in L^1. It is known that, for all p≠2, there exists f L^p with S_n f not converging to f. However, if we restrict f to belong to the centre of L^p (central function) then, in the case of polyhedral summability, we do have convergence for some (rather small) interval of p's containing 2, but for p outside a somewhat larger interval convergence fails. In this paper it is proved that for p>2n/(n-l) the central convergence fails for an arbitrary summability method; and that there is an infinite set {λ_j} in Σ which is of type Λ_p for all p<p_G (where p_ G>2n/(n-l) is a number which is nearly always 3). These results are also applied to show the failure of the translation-invariant uniform approximation property for compact Lie groups.