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The low-frequency peaked BL Lac object (LBL) AP Librae exhibitsvery-high-energy (VHE, $E>100$GeV) $\gamma$-ray emission and hosts an extendedjet, which has been detected in radio and X-rays. The jet X-ray spectral indeximplies an inverse Compton origin. These observations are unusual for LBLscalling for a consistent explanation of this extraordinary source. Theobservationally constrained parameters necessary to describe the core emissionwithin the standard one-zone model are unable to explain the broad-bandspectrum, even if observationally unconstrained external photon fields aretaken into account. We demonstrate that the addition of the extended jetemission successfully reproduces the total spectral energy distribution. Inparticular, the VHE radiation is produced in the $>100\,$kpc long extended jetvia inverse Compton scattering of cosmic microwave background photons by highlyrelativistic electrons. We present several ways to test this theory. Theextended jet is weakly magnetized ($B_0 = 2.5\,\mu$G), while its minimum andmaximum electron Lorentz factors are $\gamma_{min}=60$ and$\gamma_{max}=5\times 10^6$, respectively. The electron spectral index is$s=2.6$. These parameters are comparable to parameters of other blazars withextended X-ray jets dominated by inverse Compton scattering.

The extended jet of AP Librae: Origin of the very high-energy gamma-ray emission?