Context : E+A galaxies are post-starburst systems that are identified based on their optical spectra .
These galaxies contain a substantial young A-type stellar component but display no emission lines , which indicates only little ongoing star formation ( SF ) .
HI 21 cm line emission is found in approximately half of the nearby E+A galaxies , indicating that they contain a reservoir of gas that could fuel active SF .

Aims : We study the distribution and kinematics of atomic and molecular gas in two HI-rich galaxies , which show a typical E+A spectrum at the centre and SF at larger radii .
From our results we aim to infer whether the SF activity of these galaxies is consistent with the activity seen in disc galaxies , or if it indicates a transition towards another evolutionary phase .

Methods : We present newly obtained high spatial resolution radio interferometric observations of the HI 21 cm emission line using the Karl Jansky Very Large Array ( VLA ) and of the CO ( 1-0 ) emission line using the Atacama Large Millimeter/submillimeter Array ( ALMA ) .
We combine these data sets to predict the star formation rate ( SFR ) using a pressure-based SF relation and show that it does not correlate well with the SFR derived from H \alpha on sub-kpc scales .
We apply a recently developed statistical model for the small-scale behaviour of the SF relation to predict and interpret the observed scatter .

Results : We find regularly rotating HI gas that is smoothly distributed across the entire disc .
The CO ( 1-0 ) emission line is not detected for either of the two galaxies .
The derived upper limit on the CO mass implies a molecular gas depletion time of t _ { depl } \lesssim 20 { Myr } .
However , because of 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 a substantial scatter we demonstrate results from small-number statistics of independent SF regions on sub-kpc scales .

Conclusions : We confirm the HI-richness of ESO534-G001 and 2dFRS S833Z022 , and find that the scale dependence of the atomic SF relation in these galaxies is consistent with the predicted increase in the scatter towards small size scales .
This is due to the incomplete sampling of independent HI clouds and SF regions .
This finding adds to the existing literature , which has reported a scale dependence of the molecular SF relation , showing that the atomic and molecular phases are both susceptible to the evolutionary cycling of individual regions .
This suggests that the atomic gas reservoirs host substantial substructure , which should be observable with future high-resolution observations .