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I have been reading through chapter 1 of E.M. Stein's textbook Harmonic Analysis: Real Variable Methods, Orthogonality, and Oscillatory Integrals. In chapter 1, Stein discusses the relationship between singular integrals, maximal functions, and square functions. I have taken an introductory course on harmonic analysis and therefore understand the importance of singular integrals and maximal functions. However, I haven't encountered square functions yet, and I don't have any intuition about them or any context to understand why they are important. Can someone explain to me why square functions are important so that I can understand them a bit better?

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What do you mean by square functions? Do you mean $L^2$ functions? –  Thomas Andrews Jul 15 '13 at 1:01
I expect OP means the following: given a martingale $(f_n)$, its difference sequence is $d_n = f_n - f_{n-1}$ and its square function is $S(f) = (\sum_n d_n^2)^{1/2}$. –  Ryan O'Donnell Jul 15 '13 at 1:49
@Thomas I'm referring to the operator $s_{\phi}$ that sends a function $f$ to the function $\left( \int_0^{\infty} |f * \phi_t(x)|^2 \frac{dt}{t} \right)^{1/2}$, where $\phi$ is an $L^1$ function with average value $0$, and $\phi_t$ is the rescaling $\phi_t(x) = t^{-n} \phi(x/t)$ where $n$ is the number of dimensions. Stein says that this is called a "square function." –  Rob F. Jul 15 '13 at 3:14
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One of the nice things about square functions is that they can be used to dominate the norm (in many spaces, but primarily $L^p$) of oscillatory functions.

Take for example the Littlewood-Paley square function, which can be seen as a discretization of the square function you mention in your comment. The basic idea is to write a function $f$ as a sum of frequency-localized "projections" $$ f = \sum_n P_n f, $$ where $P_n$ is seen in frequency space as $$ \widehat{P_n f } = \psi_n \widehat{f} $$ and $\psi_n$ is a bump function which is basically $1$ at scale $2^n$ and $0$ at every other scale.

The square function in this case is $$ Sf = \Bigl( \sum_n |P_n f|^2 \Bigr)^{1/2}, $$ and we have (for $1 < p < \infty$) that $$ \|f\|_{L^p} \approx \|Sf\|_{L^p}. $$ This allows us to estimate $$ \|\sum_n P_n f\|_{L^p}, $$ whose parts could provide a lot of cancellation, by $$ \Bigl\|\Bigl( \sum_n |P_n f|^2 \Bigr)^{1/2}\Bigr\|_{L^p}, $$ which patently has no possibility for cancellations. It is possible now (and indeed quite common) that these "projections" $P_n f$ are simpler to estimate than $f$, but this is hard to make precise.

In the case of your square function, note that you are basically writing $f$ as a "sum" (an integral) of strictly positive parts (since the integrand is squared, and hence the name). So it is more or less the same idea.

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