  We present an analysis of the relation between star formation rate (SFR)
surface density (sigmasfr) and mass surface density of molecular gas
(sigmahtwo), commonly referred to as the Kennicutt-Schmidt (K-S) relation, at
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). We used interferometric IRAM CO(1-0) and CO(2-1), and
SMA CO(3-2) emission line maps to derive sigmahtwo and HST-Halpha images to
estimate sigmasfr. Each galaxy is characterized by a distinct molecular SF
relation at spatial scales between 20 to 200 pc. The K-S relations can be
sub-linear, but also super-linear, with slopes ranging from 0.5 to 1.3.
Depletion times range from 1 and 2Gyr, compatible with results for nearby
normal galaxies. These findings are valid independently of which transition,
CO(1-0), CO(2-1), or CO(3-2), is used to derive sigmahtwo. Because of
star-formation feedback, life-time 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 for this, even on scales as small
as 20 pc, and this might be because of the higher density of GMCs in galaxy
centers which have to resist higher shear forces. The proportionality between
sigmahtwo and sigmasfr found between 10 and 100 Msun/pc2 is valid even at high
densities, 10^3 Msun/pc2. However, by adopting a common CO-to-H2 conversion
factor (alpha_CO), the central regions of the galaxies have higher sigmasfr for
a given gas column than those expected from the models, with a behavior that
lies between the mergers/high-redshift starburst systems and the more quiescent
star-forming galaxies, assuming that the first ones require a lower value of
alpha_CO.
