How do I open up part of a photo that underexposed?

In retrospect I like Sean’s answer. Start with the Shadows slider, because that is a quick adjustment that should restrict itself to the darker parts. If that takes care of it, you are done.

If Shadows affects lighter areas and you don’t want it to, that’s when you might try a luminance mask like the one I showed. That is a more manual way to restrict the adjustment to darker tones as defined by you.

It should not be necessary to take the time to paint a mask with a brush; the reason to do that would be if the luminance mask is still not precise enough (if it lightens areas you don’t want to be lightened).

Just add a local correction using a brush and increase the values using any of the sliders.

I'd literally just increase Shadows on this. Alternatively use Tone Curve Shadows, or create a point curve that pushes up about 1/4 in, and then flatten out the top again. Another option is to use the Luminance HSL targeted adjustment tool and click and drag a point up. @Conrad_C 's method is for more precise ranges if you need it that precise. 

A quick Luminance Mask will isolate the dark areas. You can then adjust Exposure or Shadows for the masked area, as shown in the demo below.

What software are you going to use?