We present new measurements for the nuclear stellar velocity dispersion $sigma_{ast}$ within sub-arcsecond apertures for 28 nearby galaxies. Our data consist of Space Telescope Imaging Spectrograph (STIS) long-slit spectra obtained with the G750M grating centred on the H$alpha$ spectral range. We fit the spectra using a library of single stellar population models and Gaussian emission lines, while constraining in most cases the stellar-population content from an initial fit to G430L STIS spectra. We illustrate how these $sigma_{ast}$ measurements can be useful for constraining the mass $M_{bullet}$ of supermassive black holes (SBHs) by concentrating on the cases of the lenticular galaxies NGC4435 and NGC4459. These are characterized by similar ground-based half-light radii stellar velocity dispersion $sigma_{rm e}$ values but remarkably different $M_{bullet}$ as obtained from modelling their central ionised-gas kinematics, where NGC4435 appears to host a significantly undermassive SBH compared to what is expected from the $M_{bullet}-sigma_{rm e}$ relation. For both galaxies, we build Jeans axisymmetric dynamical models to match the ground-based stellar kinematics obtained with SAURON integral-field spectrograph, including a SBH with $M_{bullet}$ value as predicted by the $M_{bullet}-sigma_{rm e}$ relation and using high-resolution HST images taken with the Advanced Camera for Surveys to construct the stellar-mass model. By mimicking the HST observing conditions we use such reference models to make a prediction for the nuclear $sigma_{ast}$ value. Whereas this was found to agree with our nuclear $sigma_{ast}$ measurement for NGC4459, for NGC4435 the observed $sigma_{ast}$ is remarkably smaller than the predicted one, which further suggests that this galaxy could host an undermassive SBH.
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