  We have conducted interferometric observations with CARMA and an OTF mapping
with the 45-m telescope at NRO in the CO (J=1-0) emission line of NGC 3521.
Combining these new data, together with CARMA+NRO45 data for M51a and archival
SINGS H$\alpha$, 24$\mu$m, THINGS H I and GALEX FUV data for both galaxies, we
investigate the empirical scaling law that connects the surface density of SFR
and cold gas (the Schmidt-Kennicutt law) on a spatially-resolved basis. We
argue that plausibly deriving SFR maps of nearby galaxies requires the diffuse
stellar/dust background emission to be carefully subtracted. An approach to
complete this task is presented and applied in our pixel-by-pixel analysis on
both galaxies, showing that the controversial results whether the molecular S-K
law is super-linear or basically linear is a result of removing or preserving
the local background. In both galaxies, the power index of the molecular S-K
law is 1.5-1.9 at the highest available resolution (230 pc), and decreases
monotonically for decreasing resolution; while the scatter (mainly intrinsic)
increases as the resolution becomes higher, indicating a trend for which the
S-K law breaks down below some scale. Both quantities are systematically larger
in M51a than in NGC 3521, but when plotted against the de-projected scale
(\delta_{dp}), they become highly consistent between the two galaxies,
tentatively suggesting that the sub-kpc molecular S-K law in spiral galaxies
depends only on the considered scale, without varying amongst spiral galaxies.
A logarithmic function \gamma_{H_2}=-1.1 log[\delta_{dp}/kpc]+1.4 and a linear
relation \sigma_{H_2}=-0.2 [\delta_{dp}/kpc]+0.7 are obtained through fitting
to the M51a data, which describes both galaxies impressively well on sub-kpc
scales. A larger sample of galaxies with better quality data are required to
test the general applicability of these relations.
