  For a multidimensional It\^o semimartingale, we consider the problem of
estimating integrated volatility functionals. Jacod and Rosenbaum (2013)
studied a plug-in type of estimator based on a Riemann sum approximation of the
integrated functional and a spot volatility estimator with a forward uniform
kernel. Motivated by recent results that show that spot volatility estimators
with general two-side kernels of unbounded support are more accurate, in this
paper, an estimator using a general kernel spot volatility estimator as the
plug-in is considered. A biased central limit theorem for estimating the
integrated functional is established with an optimal convergence rate. Unbiased
central limit theorems for estimators with proper de-biasing terms are also
obtained both at the optimal convergence regime for the bandwidth and when
applying undersmoothing. Our results show that one can significantly reduce the
estimator's bias by adopting a general kernel instead of the standard uniform
kernel. Our proposed bias-corrected estimators are found to maintain remarkable
robustness against bandwidth selection in a variety of sampling frequencies and
functions.
