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E+A galaxies are post-starburst systems that are identified from theiroptical spectra. These galaxies contain a substantial young A-type stellarcomponent, but have only little ongoing star formation (SF). HI 21-cm lineemission is found in approximately half of the nearby E+A galaxies, indicatingthat they contain a reservoir of gas that could fuel active SF. Here, we studytwo HI-rich galaxies, which show a typical E+A spectrum at the centre and SF atlarger radii. We present new high spatial resolution radio interferometricobservations of the HI 21-cm emission line using the VLA and of the CO(1-0)emission line using ALMA. We combine these data sets to predict the SFR andshow that it does not correlate well with the SFR derived from H alpha onsub-kpc scales. We apply a recently developed statistical model for the smallscale behaviour of the SF relation to predict and interpret the observedscatter. We find smoothly distributed, regularly rotating HI gas. The CO(1-0)emission line is not detected for both galaxies. The derived upper limit on theCO mass implies a molecular gas depletion time shorter than 20 Myr. However,due to the low metallicity, the CO-to-H2 conversion factor is highly uncertain.In the relations between the H alpha-based SFR and the HI mass, we observe asubstantial scatter we demonstrate results from small-number statistics ofindependent star-forming regions on sub-kpc scales. This finding adds to theexisting literature reporting a scale dependence of the molecular SF relation,showing that the atomic and molecular phases are both susceptible to theevolutionary 'cycling' of individual regions. This suggests that the atomic gasreservoirs host substantial substructure, which should be observable withfuture high-resolution observations. (abridged)

Spatially resolved star formation relation in two HI-rich galaxies with central post-starburst signature