The Littlewood-Paley square function

Assume Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). With the fixed partition and operators of The inhomogeneous dyadic frequency partition and its Littlewood-Paley operators, for $f\in L^p({\mathbb{R}}^n;\mathbb{C})$ with $1\le p<\mathrm{\infty }$ and $N\ge 0$ define, for $x\in {\mathbb{R}}^n$, $S_{N}f(x):=({\sum }_{j=0}^{N-1}\mathrm{\mid }{\mathrm{\Delta }}_{j}f(x){\mathrm{\mid }}^2{)}^{1\mathrm{/}2},Sf(x):=({\sum }_{j\ge 0}\mathrm{\mid }{\mathrm{\Delta }}_{j}f(x){\mathrm{\mid }}^2{)}^{1\mathrm{/}2}\in [0,\mathrm{\infty }]\mathrm{.}$ The functions ${\mathrm{\Delta }}_{j}f$ in these formulae are the convolution representatives $f\ast K_j$ of Dyadic pieces have annular Fourier support and uniformly bounded rescaled kernels. The following conventions and well-definedness facts are part of the definition.

1. Each ${\mathrm{\Delta }}_{j}f=f\ast K_j$ lies in $L^p$ by Dyadic pieces have annular Fourier support and uniformly bounded rescaled kernels. Its integral is defined at every $x$: Holder gives $\int \mathrm{\mid }f(u)K_j(x-u)\mathrm{\mid }du\le \mathrm{\parallel }f{\mathrm{\parallel }}_p\mathrm{\parallel }{K}_j{\mathrm{\parallel }}_{p^{\mathrm{\prime }}}$, with $p^{\mathrm{\prime }}=\mathrm{\infty }$ when $p=1$ (Complex Holder, Minkowski, and the quotient norm). Translations of a Schwartz kernel are continuous in $L^{p^{\mathrm{\prime }}}$: for finite $p^{\mathrm{\prime }}$ use dominated convergence with a common Schwartz majorant, and for $p^{\mathrm{\prime }}=\mathrm{\infty }$ use its bounded first derivatives. Holder therefore makes this everywhere convolution representative continuous, hence Borel measurable (Dominated convergence, Borel measurable and Lebesgue measurable functions on ${\mathbb{R}}^n$). Finite sums, products and the square root of nonnegative measurable functions preserve measurability (Arithmetic and lattice operations preserve measurability whenever they are defined), so every $S_{N}f$ is measurable and $S_{N}f\le S_{N+1}f$ pointwise.
2. $Sf={\sup }_N{S}_{N}f$ is measurable as the increasing limit of measurable functions, with values in $[0,\mathrm{\infty }]$; no finiteness is asserted, that is $Sf(x)=+\mathrm{\infty }$ is allowed a priori.
3. Replacing $f$ by an almost everywhere equal function in $L^p$ replaces every convolution representative ${\mathrm{\Delta }}_{j}f$ by an almost everywhere equal function, hence replaces $Sf$ by an almost everywhere equal function; the functional is therefore defined on almost everywhere classes, and $Sf$ is recorded as an almost everywhere function, exactly as the $L^p$ classes of Complex Lp classes and Euclidean test-function conventions are.
4. One writes $\mathrm{\parallel }Sf{\mathrm{\parallel }}_p$ for the $L^p$ norm of the class of $Sf$ when $Sf\in L^p$; the norm is that of $L^p({\mathbb{R}}^n;\mathbb{C})$, with $Sf$ interpreted as the complex-valued function $x\mapsto Sf(x)$ (it is real-valued and nonnegative). The notation $S_{F}f$ for a finite set $F\subset \{0,1,2,\dots \}$ means $({\sum }_{j\in F}\mathrm{\mid }{\mathrm{\Delta }}_{j}f{\mathrm{\mid }}^2{)}^{1\mathrm{/}2}$, so that $S_{N}f=S_{\{0,\dots ,N-1\}}f$.

The operator-theoretic counterpart of $S$ for functions on the frequency side is not asserted here: the definition names only the pointwise square function of the convolution representatives, and the strict-range equivalence with $\mathrm{\parallel }f{\mathrm{\parallel }}_p$ is a theorem proved later on this page.