Regularity conditions in the CLT for linear eigenvalue statistics of Wigner matrices

We show that the variance of centred linear statistics of eigenvalues of GUE matrices remains bounded for large $n$ for some classes of test functions less regular than Lipschitz functions. This observation is suggested by the limiting form of the variance (which has previously been computed explicitly), but it does not seem to appear in the literature. We combine this fact with comparison techniques following Tao-Vu and Erd"os, Yau, et al. and a Littlewood-Paley type decomposition to extend the central limit theorem for linear eigenvalue statistics to functions in the H"older class $C^{1\mathrm{/}2+ϵ}$ in the case of matrices of Gaussian convolution type. We also give a variance bound which implies the CLT for test functions in the Sobolev space $H^{1+ϵ}$ and $C^{1-ϵ}$ for general Wigner matrices satisfying moment conditions. Previous results on the CLT impose the existence and continuity of at least one classical derivative.