We present $NuSTAR$ observations of four active galactic nuclei (AGN) located within 15 Mpc. These AGN, namely ESO 121-G6, NGC 660, NGC 3486 and NGC 5195, have observed X-ray luminosities of $L_{rm 2-10 keV, obs} lesssim$ 10$^{39}$ erg s$^{-1}$, classifying them as low luminosity AGN (LLAGN). We perform broadband X-ray spectral analysis for the AGN by combining our $NuSTAR$ data with $Chandra$ or $XMM-Newton$ observations to directly measure their column densities ($N_{rm H}$) and infer their intrinsic power. We complement our X-ray data with archival and new high angular resolution mid-infrared (mid-IR) data for all objects, except NGC 5195. Based on our X-ray spectral analysis, we found that both ESO 121-G6 and NGC 660 are heavily obscured ($N_{rm H}$ > 10$^{23}$ cm$^{-2}$; $L_{rm 2-10 keV, int} sim$ 10$^{41}$ erg s$^{-1}$), and NGC 660 may be Compton-thick. We also note that the X-ray flux and spectral slope for ESO 121-G6 have significantly changed over the last decade, indicating significant changes in the obscuration and potentially accretion rate. On the other hand, NGC 3486 and NGC 5195 appear to be unobscured and just mildly obscured, respectively, with $L_{rm 2-10 keV, int} <$ 10$^{39}$ erg s$^{-1}$; i.e., genuine LLAGN. Both of the heavily obscured AGN have $L_{rm bol} >$ 10$^{41}$ erg s$^{-1}$ and $lambda_{rm Edd} gtrsim$ 10$^{-3}$, and are detected in high angular resolution mid-IR imaging, indicating the presence of obscuring dust on nuclear scale. NGC 3486 however, is undetected in high-resolution mid-IR imaging, and the current data do not provide stringent constraints on the presence or absence of obscuring nuclear dust in the AGN.
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