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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

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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When establishing Littlewood-Paley theory on the real line $\mathbb{R}$, passing from $L^2(\mathbb{R})$ to $L^p(\mathbb{R})$ causes the orthogonality relations to fail ($L^p$ is not a Hilbert space). In order to compensate for the lack of orthogonality, we provide a substitute. Set $\hat{P_jf}=\chi \hat{f}$ to be the Littlewood-Paley projections of $f$. Then the square function \begin{equation*} (\sum |P_j f|^2)^{1/2} \end{equation*} followed by an application of Plancheral's theorem allow us to make a vector-valued extension to $L^p(\mathbb{R})$. A similar principle applies with the Littlewood-Paley $g-$function. Define the Poisson integral \begin{equation*} u(x,y)=\int_{\mathbb{R}^n}P_y(t)f(x-t)dt,~f\in L^p(\mathbb{R}^n). \end{equation*} Establishing the square function \begin{equation*} g(f)(x)=(\int^{\infty}_{0}|\nabla u(x,y)|^2ydy)^{1/2} \end{equation*} allows us to perform another vector-valued extension to compare $L^p$ norms. The importance of Littlewood-Paley theory is that LP techniques have been used to establish Strichartz estimates to prove global existence results in dispersive PDE theory (I can send you some references if you like).

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