On norms of trigonometric polynomials on \(SU(2)\) (Q1359206)

Let \(M_p(G)\) denote the space of \(L^p\)-multipliers on a topological group \(G\). For \(G\) compact abelian and \((p,q)\) conjugate indices it is elementary to check that \(M_p (G) =M_q(G)\). For certain \(G\) non-abelian and totally disconnected \textit{D. M. Oberlin} [Israel J. Math. 22, 175-179 (1975; Zbl 0314.43005)] has shown inequality of \(M_p(G)\) and \(M_q (G)\). For compact connected groups, e.g. \(SU(2)\), the situation is unknown. Motivated by this problem the present paper gives partial results toward settling a conjecture on the growth of \(L^4\)-norms of trigonometric polynomials on \(SU(2)\). The truth of this conjecture implies the inequality of \(M_p(G)\) and \(M_q(G)\) for \(G=SU(2)\) and for \(1<p<2\).