The two major classical interpolation theorems in analysis are Riesz-Thorin Theorem (complex method) and Marcinkiewicz Theorem (real method).
One can see the statements of the theorems and realize the differences between them (sublinearity, weak end-point estimates, and so on). I also know that these theorems can be extended to whole theories.
As far as I know, the complex method is called like so because in Riesz-Thorin Theorem it is neccesary that the scalar field is $\mathbb{C}$ (to use complex analysis tools). My questions are:
Is there any real difference if the scalar field is $\mathbb{R}$ or $\mathbb{C}$? Isn't the scalar field of all $L^p$ spaces $\mathbb{C}$? (I can also define them over $\mathbb{R}$ but I cannot see the advantage). Do I need my functions to be real-valued to use Marcinkiewicz? Can I use Riesz-Thorin to real-valued functions? What is the different between both interpolation methods in this aspect?
Thank you.
No.
Depends on context, not necessarily.
No.
Yes.
The terms complex and real interpolation don't refer to the scalar field. They refer to the nature of proof.
Marcinkiewicz's theorem is proven using real-variable methods, decomposing functions cleverly according to size, using sublevel sets, etc.
Riesz-Thorin is proven by a magic application of complex analysis, typically the proof uses Hadamard's three-lines lemma (which is a consequence of the maximum modulus principle).