  This note gives a wide-ranging update on the multiplier theorems by
Akylzhanov and the second author [J. Funct. Anal., 278 (2020), 108324]. The
proofs of the latter crucially rely on $L^p$-$L^q$ norm estimates for spectral
projectors of left-invariant weighted subcoercive operators on unimodular Lie
groups, such as Laplacians, sub-Laplacians and Rockland operators. By relating
spectral projectors to heat kernels, explicit estimates of the $L^p$-$L^q$
norms can be immediately exploited for a much wider range of (connected
unimodular) Lie groups and operators than previously known. The comparison with
previously established bounds by authors show that the heat kernel estimates
are sharp. As an application, it is shown that several consequences of the
multiplier theorems, such as time asymptotics for the $L^p$-$L^q$ norms of the
heat kernels and Sobolev-type embeddings, are then automatic for the considered
operators.
