What are $L^2$ tools?

Question

In the reseach statement from K. Smith I read about the notion of $L^2$ tools, as in "can be proved using $L^2$ tools from analysis".

What are $L^2$ tools? And what are the references to study them?

Answer 1

I suspect that $L^2$ stands for the Hilbert space $L^2(\Omega, \,d\mu)$ in general. Here $\Omega\subseteq\mathbb{R}^n$ and $\mu:\mathcal{B}(\Omega)\to[0,+\infty]$ is a measure defined on the Borels set $\mathcal{B}(\Omega)$. $L^2$ spaces are nice since they are Hilbert spaces they are endowed with an inner product that in turn generates a norm on it. It is a complete normed linear space (all Cauchy sequences converge in the space). So basically the tools of an $L^2$ space would consist: (i) the inner product and consequently the respective norm generated (ii) completeness and (iii) existence of a complete orthonormal basis closure of which is the entire space. One can use these tools to estimate in general given integrals or sums, and also convergence results. About $L^2$ space there is numerous sources where one can learn from, in particular any book on Functional Analysis would certainly help.