  In global fits of parton distribution functions (PDFs) a large fraction of
data points, mostly from the HERA collider, lies in a region of $x$ and $Q^2$
that is sensitive to small-$x$ logarithmic enhancements. Thus, the proper
theoretical description of these data requires the inclusion of small-$x$
resummation. In this work we provide all the necessary ingredients to perform a
PDF fit to deep-inelastic scattering (DIS) data which includes small-$x$
resummation in the evolution of PDFs and in the computation of DIS structure
functions. To this purpose, not only we include the resummation of DIS massless
structure functions, but we also consider the production of a massive final
state (e.g. a charm quark), and the consistent resummation of mass collinear
logarithms through the implementation of a variable flavour number scheme at
small $x$. As a result, we perform the small-$x$ resummation of the matching
conditions in PDF evolution at heavy flavour thresholds. The resummed results
are accurate at next-to-leading logarithmic (NLL) accuracy and matched, for the
first time, to next-to-next-to-leading order (NNLO). Furthermore, we improve on
our previous work by considering a novel all-order treatment of running
coupling contributions. These results, which are implemented in a new release
of HELL, version 2.0, will allow to fit PDFs from DIS data at the highest
possible theoretical accuracy, NNLO+NLL, thus providing an important step
forward towards precision determination of PDFs and consequently precision
phenomenology at the LHC and beyond.
