On planar domains the real part of holomorphic functions gives a class of examples of holomorphic functions. In fact, the following is true: a domain is simply connected if and only if the space of harmonic functions is equal to the space of functions which are real part of holomorphic functions. But, on a general domain, what can be said about the quotient space of harmonic functions by the subspace of real part of holomorphic functions? An interesting article that answers the above question--when the domain is finitely connected--is due to Sheldon Axler: Harmonic functions from complex analysis view-point, 1986. It would be interesting to see what happens when the domain is infinitely connected?Â
2. The following webpage has really interesting content: https://www.mathpages.com/home/index.htm