How does the luminosity of a red dwarf ($0.1 M_\odot$) compare to the Sun's luminosity ($1 M_\odot$)?

It drops drastically, perhaps to only 0.001 times the solar luminosity.

The mass-luminosity relationship shows a steep fall-off on the low-mass end of the main sequence spectrum. When comparing the Sun ($1 M_\odot$) to the smallest true hydrogen-burning stars, known as red dwarfs, which might possess only $0.1$ times the Sun's mass, the difference in output is immense. The red dwarf's luminosity plummets drastically, often reaching only $0.001$ times the Sun's output. This substantial dimming is a direct consequence of having significantly less mass, leading to much lower core temperatures, densities, and consequently, a vastly reduced rate of nuclear fusion compared to solar-type stars.