Negative results concerning Fourier series on the complete product of S3

Abstract

In this talk I deal with the studies about convergence in $L^p$-norm of the Fourier series based on representative product systems on the complete product of arbitrary finite groups. Suppose that each finite group has discrete topology and normalized Haar measure. Let $G$ be the compact group formed by the complete direct product of the finite groups with the product of theirs topologies, operations and measures. The most simple example of this groups is the complete product of $\mathcal{S}_3$, i.e. the symmetric group on 3 elements. The orthonormal system with which we work, is the product system of the normalized coordinate functions $\varphi^s_k$ of the finite groups, namely \begin{equation*} \psi_n(x):=\prod_{k=0}^{\infty}\varphi^{n_k}_k(x_k)\qquad(x\in G), \end{equation*}

For the complete product of $\mathcal{S}_3$, G.~G\'at and R.~Toledo proved there exists a $1<p<2$ and a function $f$ in $L^p(G)$ for which the n-th partial sum of its Fourier series $S_nf$ does not converge to $f$ in $L^p$-norm. Now, we proved this statement for all $1<p<\infty$ and $p\ne2$.

In general, we proved

\begin{thm} Let $G$ be a bounded group and suppose all the finite groups appearing in the product of $G$ have the same system $\varphi$ at all of their occurrences. If the sequence $\Psi$ is unbounded, then the operator $S_n$ is not of type $(p,p)$ for all $p\ne2$. \end{thm}