We report on our study of high-energy properties of two peculiar TeV emitters: the extreme blazar 1ES 0347-121 and the extreme blazar candidate HESS J1943+213 located near the Galactic Plane. Both objects are characterized by quiescent synchrotron emission with flat spectra extending up to the hard X-ray range, and both were reported to be missing GeV counterparts in the Fermi-LAT 2-year Source Catalog. We analyze a 4.5 year accumulation of the Fermi-LAT data, resulting in the detection of 1ES 0347-121 in the GeV band, as well as in improved upper limits for HESS J1943+213. We also present the analysis results of newly acquired Suzaku data for HESS J1943+213. The X-ray spectrum is well represented by a single power law extending up to 25 keV with photon index 2.00+/-0.02 and a moderate absorption in excess of the Galactic value, in agreement with previous X-ray observations. No short-term X-ray variability was found over the 80 ks duration of the Suzaku exposure. Under the blazar hypothesis, we modeled the spectral energy distributions of 1ES 0347-121 and HESS J1943+213, and derived constraints on the intergalactic magnetic field strength and source energetics. We conclude that although the classification of HESS J1943+213 has not yet been determined, the blazar hypothesis remains the most plausible option, since in particular the broad-band spectra of the two analyzed sources along with the source model parameters closely resemble each other, and the newly available WISE and UKIDSS data for HESS J1943+213 are consistent with the presence of an elliptical host at the distance of approximately ~600 Mpc.
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