The central image of a strongly lensed background source places constraints on the foreground lens galaxys inner mass profile slope, core radius and mass of its nuclear supermassive black hole. Using high-resolution long-baseline Atacama Large Millimeter/submillimeter Array (ALMA) observations and archival $Hubble~Space~Telescope$ ($HST$) imaging, we model the gravitational lens H-ATLAS J090311.6+003906 (also known as SDP.81) and search for the demagnified central image. There is central continuum emission from the lens galaxys active galactic nucleus (AGN) but no evidence of the central lensed image in any molecular line. We use the CO maps to determine the flux limit of the central image excluding the AGN continuum. We predict the flux density of the central image and use the limits from the ALMA data to constrain the innermost mass distribution of the lens. For a power-law profile with a core radius of $0.15^{primeprime}$ measured from $HST$ photometry of the lens galaxy assuming that the central flux is attributed to the AGN, we find that a black hole mass of $mathrm{log(M_{BH}/M_{odot})} gtrsim 8.5$ is preferred. Deeper observations with a detection of the central image will significantly improve the constraints of the innermost mass distribution of the lens galaxy.
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