We know that one complex variable methods have been largely used in the early research of (real variable) harmonic analysis. But, as is known to me, there is much difficulty when mathematicians attempt to generalize their results to several variable cases if they still consider complex variable methods.
My questions:
I don't know exactly what difficulties the mathematicians have been faced in the early times and why these can prevent them from continuing to use complex variable methods in the research of harmonic analysis. And why real variable methods can take place of complex variable methods thoroughly in today's research ? Thanks in advance.
I am absolutely no expert on those things, but the way I see it, complex variable methods tend to be more rigid than real variable ones. They tend to either work wonderfully or be completely useless.
For example, you can develop a theory of solutions to the Laplace equation $$\Delta u =0$$ in the plane by means of complex analysis: this is of great importance -among other things- in fluid mechanics. However, the same techniques are not applicable to the elliptic equation $$ \nabla \cdot \left( a(x)\nabla u\right) =0, \quad a(x)\in \mathbb{R}^{2\times 2}, $$ unless trivial cases, of course. This is despite the fact that both equations model essentially the same physical phenomena: the second one is a non-isotropic version of the first.