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We have conducted interferometric observations with CARMA and an OTF mappingwith 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 archivalSINGS H$\alpha$, 24$\mu$m, THINGS H I and GALEX FUV data for both galaxies, weinvestigate the empirical scaling law that connects the surface density of SFRand cold gas (the Schmidt-Kennicutt law) on a spatially-resolved basis. Weargue that plausibly deriving SFR maps of nearby galaxies requires the diffusestellar/dust background emission to be carefully subtracted. An approach tocomplete this task is presented and applied in our pixel-by-pixel analysis onboth galaxies, showing that the controversial results whether the molecular S-Klaw is super-linear or basically linear is a result of removing or preservingthe local background. In both galaxies, the power index of the molecular S-Klaw is 1.5-1.9 at the highest available resolution (230 pc), and decreasesmonotonically for decreasing resolution; while the scatter (mainly intrinsic)increases as the resolution becomes higher, indicating a trend for which theS-K law breaks down below some scale. Both quantities are systematically largerin 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 galaxiesdepends 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 linearrelation \sigma_{H_2}=-0.2 [\delta_{dp}/kpc]+0.7 are obtained through fittingto the M51a data, which describes both galaxies impressively well on sub-kpcscales. A larger sample of galaxies with better quality data are required totest the general applicability of these relations.

The Super-linear Slope of the Spatially-Resolved Star Formation Law in NGC 3521 and NGC 5194 (M51a)