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 Erdös, Yau, et al. and a Littlewood–Paley type decomposition to extend the central limit theorem for linear eigenvalue statistics to functions in the Hölder class C1/2+ϵC1/2+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 H1+ϵH1+ϵ and C1−ϵC1−ϵ for general Wigner matrices satisfying moment conditions. If the additional assumption of the test function being supported away from the edge of the spectrum is made, we prove the CLT for test functions of regularity H˙1/2∩L∞ and H1/2+H1/2+ for GUE and Johansson matrices respectively. Previous results on the CLT impose the existence and continuity of at least one classical derivative.