Optimal boundedness of central oscillating multipliers on compact Lie groups (Q443798)

The authors study the boundedness of a family of central oscillating multipliers, indexed by parameters \(\gamma,\) \(\beta\) and defined on compact, connected semi-simple Lie groups \(G\). Their multipliers can be viewed as analogues of the classical multipliers given by \(m(\xi)=\frac{e^{i|\xi|^\beta}}{|\xi|^\gamma}\) for \(\xi\) away from \(0.\) It is known that these classical multipliers are bounded on \(H^p(\mathbb{R}^d)\) if and only if \(|\frac{1}{2}-\frac{1}{p}|\leq\frac{\gamma}{d\beta}.\) Here the operators are shown to be bounded on \(H^p(G)\) if and only if \(|\frac{1}{2}-\frac{1}{p}|\leq\frac{\gamma}{n\beta}\) where \(n\) is the dimension of \(G.\)NEWLINENEWLINEThe proof is quite technical and relies heavily on Lie theory.