Hörmander multiplier theorem on \(SU(2)\) (Q1842121)

Let \(G\) denote a connected, simply connected, compact semisimple Lie group and let \(g\) denote its Lie algebra. The Hardy space \(H^p (G)\) is defined as the space of distributions \(f\) of the form \(f = \sum c_k a_k\) with \(\sum |c_k |^p < \infty\), where \(a_k (x)\) is a regular atom (defined in terms of the dimension of \(G)\) or an exceptional atom. The Hardy norm is defined to be \(|f |_{H^p} = \inf (\sum |c_k |^p)^{ 1/p}\) where the infimum is taken over all possible representations of \(f\). If a function \(K(x)\) on \(G\) satisfies certain boundedness criteria, then the authors show that the convolution operator \(Tf = K^*f\) is \(L^2\) bounded and can be extended to a bounded operator on the Hardy space. This result is used to prove a multiplier theorem for \(H^p (G)\) for the case \(G = SU(2)\). The theorem states that for \(f\) in \(H^p (G)\) having the representation \(f(y) \approx \sum^\infty_{n = 1} f*n \chi_n (y)\) where \(\chi_n (y) = \sin n \theta/\sin n \theta\) for \(y\) conjugate to the diagonal matrix \(\left( \begin{smallmatrix} e^{i \theta} & 0 \\ 0 & e^{- i \theta} \end{smallmatrix} \right)\), the convolution operator \(T_\lambda\) defined by \(T_\lambda f(y) \approx \sum^\infty_{n = 1} \lambda (n)n \chi_n*f\) can be extended to a bounded operator on Hardy space when \(\lambda\) satisfies a certain boundedness criterion (belongs to the Hörmander multiplier class).