  High-order harmonic generation (HHG) is combined with resonant x-ray
excitation of a core electron into the transient valence vacancy that is
created in the course of the HHG process. To describe this setting, I develop a
two-active-electron quantum theory for a single atom assuming no Coulomb
interaction among the electrons; one electron performs a typical HHG three-step
process whereas another electron is excited (or even Rabi flops) by intense x
rays from the core shell into the valence hole after the first electron has
left the atom. Depending on the amplitude to find a vacancy in the valence and
the core, the returning continuum electron recombines with the valence and the
core, respectively, emitting high-order harmonic (HH) radiation that is
characteristic of the combined process. After presenting the theory of x-ray
boosted HHG for continuous-wave light fields, I develop a description for x-ray
pulses with a time-varying amplitude and phase. My prediction offers novel
prospects for nonlinear x-ray physics, attosecond x rays, and HHG-based
time-dependent chemical imaging involving core orbitals.
