  We consider linear spectral statistics of the form $\mathrm{tr} ( \varphi
(H))$ for test functions $\varphi$ of low regularity and Wigner matrices $H$
with smooth entry distribution. We show that for functions $\varphi$ in the
Sobolev space $H^{1/2+\varepsilon}$ or the space $C^{1/2+\varepsilon}$, that
are supported within the spectral bulk of the semicircle distribution, these
linear spectral statistics have asymptotic Gaussian fluctuations with the same
variance as in the CLT for functions of higher regularity, for any $\varepsilon
>0$.
