Local Noncommutative De Leeuw Theorems Beyond Reductive Lie Groups

Let Γ be a discrete subgroup of a unimodular locally compact group G. M. Caspers et al. [Local and multilinear noncommutative de Leeuw theorems, Math. Ann. 388 (2024) 4251–4305] showed that the L_p -norm of a Fourier multiplier m: G → C on Γ can be bounded locally by its L_p -norm on G, modulo a constant c(A) which depends on the support A of m|_Γ . In the context where G is a connected Lie group with Lie algebra g, we develop tools to find explicit bounds on c(A) . We show that the problem reduces to: (1) The adjoint representation of the semisimple quotient s = g/r of g by the radical r ⊆ g (which was handled in the paper of M. Caspers et al. cited above). (2) The action of s on a set of real irreducible representations that arise from quotients of the commutator series of r . In particular, we show that c(G) = 1 for unimodular connected solvable Lie groups.