  The eigenstate thermalization hypothesis (ETH) describes the properties of
diagonal and off-diagonal matrix elements of local operators in the eigenenergy
basis. In this work, we propose a relation between (i) the singular behaviour
of the off-diagonal part of ETH at small energy differences, and (ii) the
smooth profile of the diagonal part of ETH as a function of the energy density.
We establish this connection from the decay of the autocorrelation functions of
local operators, which is constrained by the presence of local conserved
quantities whose evolution is described by hydrodynamics. We corroborate our
predictions with numerical simulations of two non-integrable spin-1 Ising
models, one diffusive and one super-diffusive, which we perform using dynamical
quantum typicality up to 18 spins.
