Abstract | Aims: We present an analysis of the relation between the star formation rate (SFR) surface density (ΣSFR) and mass surface density of molecular gas (ΣH2), commonly referred to as the Kennicutt-Schmidt (K-S) relation, on its intrinsic spatial scale, i.e. the size of giant molecular clouds (~10-150 pc), in the central, high-density regions of four nearby low-luminosity active galactic nuclei (AGN). These are AGN extracted from the NUclei of GAlaxies (NUGA) survey. This study investigates the correlations and slopes of the K-S relation, as a function of spatial resolution and of the different 12CO emission lines used to trace ΣH2, and tests its validity in the high-density central regions of spiral galaxies. Methods: We used interferometric IRAM 12CO(1-0) and 12CO(2-1) and SMA 12CO(3-2) emission line maps to derive ΣH2 and HST-Hα images to estimate ΣSFR. Results: Each galaxy is characterized by a distinct molecular SF relation on spatial scales between 20 to 200 pc. The K-S relations can be sublinear, but also superlinear, with slopes ranging from ~0.5 to ~1.3; slopes are generally superlinear on spatial scales >100 pc and sublinear on smaller scales. Depletion times range from ~1 and 2 Gyr, which is compatible with results for nearby normal galaxies. These findings are valid independently of which transition - 12CO(1-0), 12CO(2-1), or 12CO(3-2) - is used to derive ΣH2. Because of either star-formation feedback, the lifetime of clouds, turbulent cascade, or magnetic fields, the K-S relation might be expected to degrade on small spatial scales (<100 pc). However, we find no clear evidence of this, even on scales as small as ~20 pc, and this might be because of the higher density of GMCs in galaxy centers that have to resist higher shear forces. The proportionality between ΣH2 and ΣSFR found between 10 and 100 M☉ pc-2 is valid even at high densities, ~103 M☉ pc-2. However, by adopting a common CO-to-H2 conversion factor (αCO), the central regions of the NUGA galaxies have higher ΣSFR for a given gas column than those expected from the models, with a behavior that lies between the mergers or high-redshift starburst systems and the more quiescent star-forming galaxies, assuming that the first ones require a lower value of αCO. The maps are only available at the CDS via anonymous ftp to http://cdsarc.u-strasbg.fr (ftp://130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/577/A135 |