The inhomogeneous dyadic frequency partition and its Littlewood-Paley operators

Assume Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Fix a function $\psi$ and its partition $({\phi }_j{)}_{j\ge 0}$ as in Existence of a smooth inhomogeneous dyadic frequency partition. Define the companion sequence by ${\tilde{\phi }}_j:={\phi }_{j-1}+{\phi }_j+{\phi }_{j+1}(j\ge 0),$ with ${\phi }_{-1}:=0$. For $f\in {\mathcal{S}}^{\mathrm{\prime }}({\mathbb{R}}^n)$ define ${\mathrm{\Delta }}_{j}f:={\mathcal{F}}^{-1}({\phi }_j\cdot \mathcal{F}f),{\tilde{\mathrm{\Delta }}}_{j}f:={\mathcal{F}}^{-1}({\tilde{\phi }}_j\cdot \mathcal{F}f),$ where ${\phi }_j\cdot \mathcal{F}f$ and ${\tilde{\phi }}_j\cdot \mathcal{F}f$ are the products of the tempered distribution $\mathcal{F}f$ with the smooth polynomially bounded symbols ${\phi }_j,{\tilde{\phi }}_j$ (transposition, Smooth polynomially bounded multipliers on schwartz space); for $f\in \mathcal{S}$ these operators are the translation-invariant Fourier multipliers $T_{{\phi }_j}f$, $T_{{\tilde{\phi }}_j}f$ of Translation-invariant Fourier multiplier on the Schwartz core; for $f\in L^p({\mathbb{R}}^n;\mathbb{C})$, $1\le p<\mathrm{\infty }$, define ${\mathrm{\Delta }}_{j}f:=f\ast K_j$ and ${\tilde{\mathrm{\Delta }}}_{j}f:=f\ast {\tilde{K}}_j$, where $K_j:={\mathcal{F}}^{-1}{\phi }_j,{\tilde{K}}_j:={\mathcal{F}}^{-1}{\tilde{\phi }}_j$ are the inverse transforms of the symbols (Fourier transform of a tempered distribution). The following well-definedness and compatibility facts are part of the definition and are recorded with their cited suppliers.

1. Domains. Each ${\phi }_j$ and each ${\tilde{\phi }}_j$ is smooth, compactly supported and bounded together with all its derivatives, hence a smooth polynomially bounded multiplier: for $f\in \mathcal{S}({\mathbb{R}}^n)$ the products ${\phi }_j\widehat{f}$ and ${\tilde{\phi }}_j\widehat{f}$ lie in $\mathcal{S}({\mathbb{R}}^n)$ (Smooth polynomially bounded multipliers on schwartz space), so $\mathcal{S}\subset D_{{\phi }_j}\cap D_{{\tilde{\phi }}_j}$ and $T_{{\phi }_j}f$, $T_{{\tilde{\phi }}_j}f$ are well-defined tempered distributions. Because ${\phi }_0=\psi$ and ${\phi }_j(\xi )=\psi (2^{-j}\xi )-\psi (2^{-(j-1)}\xi )$ for $j\ge 1$, the pieces satisfy the rescaling law ${\phi }_{k+j-1}(\xi )={\phi }_k(2^{-(j-1)}\xi )$ for every $k\ge 1$ and $j\ge 1$ (so ${\phi }_j(\xi )={\phi }_1(2^{-(j-1)}\xi )$ for $j\ge 1$); the companion symbols are sums of neighbouring pieces, and no single rescaling law for all $j\ge 0$ is used. They have the recorded supports $\mathrm{supp}{\tilde{\phi }}_j\subset \{\mathrm{\mid }\xi \mathrm{\mid }\le 2^{j+2}\}$ for every $j\ge 0$, with $\mathrm{supp}{\tilde{\phi }}_j\subset \{2^{j-2}\le \mathrm{\mid }\xi \mathrm{\mid }\le 2^{j+2}\}$ for $j\ge 2$, together with $\mathrm{supp}{\tilde{\phi }}_0\subset \{\mathrm{\mid }\xi \mathrm{\mid }\le 4\}$. The vanishing is strict: for $j\ge 1$, since $\psi =1$ on the unit ball, ${\phi }_j(\xi )=0$ whenever $\mathrm{\mid }\xi \mathrm{\mid }\le 2^{j-1}$ (both arguments of $\psi$ have modulus at most $1$), and since $\psi =0$ for $\mathrm{\mid }\xi \mathrm{\mid }\ge 2$, ${\phi }_j(\xi )=0$ whenever $\mathrm{\mid }\xi \mathrm{\mid }\ge 2^{j+1}$; thus for $j\ge 1$ its nonzero set is contained in the open annulus $2^{j-1}<\mathrm{\mid }\xi \mathrm{\mid }<2^{j+1}$, while its closed support lies in $2^{j-1}\le \mathrm{\mid }\xi \mathrm{\mid }\le 2^{j+1}$. Boundary points can belong to the support even though the function vanishes there, and ${\phi }_0=0$ for $\mathrm{\mid }\xi \mathrm{\mid }\ge 2$.
2. The $L^p$ definition. $K_j,{\tilde{K}}_j\in \mathcal{S}({\mathbb{R}}^n)$ (Fourier transform acts continuously on Schwartz space), hence $K_j,{\tilde{K}}_j\in L^1({\mathbb{R}}^n)$, and Young's inequality (Young's convolution inequality under Countable Choice) shows that for $f\in L^p$, $1\le p<\mathrm{\infty }$, the convolutions $f\ast K_j$, $f\ast {\tilde{K}}_j$ are defined almost everywhere, lie in $L^p$ and satisfy $\mathrm{\parallel }f\ast K_j{\mathrm{\parallel }}_p\le \mathrm{\parallel }{K}_j{\mathrm{\parallel }}_1\mathrm{\parallel }f{\mathrm{\parallel }}_p$, $\mathrm{\parallel }f\ast {\tilde{K}}_j{\mathrm{\parallel }}_p\le \mathrm{\parallel }{\tilde{K}}_j{\mathrm{\parallel }}_1\mathrm{\parallel }f{\mathrm{\parallel }}_p$; the convolution is the one of Convolution of two functions on ${\mathbb{R}}^n$.
3. Agreement on $\mathcal{S}$ and composition. For $f\in \mathcal{S}$ the convolution $K_j\ast f$ is Schwartz by Schwartz convolution and product laws, hence lies in $L^1$, its integral transform is $\widehat{K_j\ast f}={\widehat{K}}_j\widehat{f}={\phi }_j\widehat{f}$ by the convolution theorem (Fourier transform turns L1 convolution into multiplication), and Fourier inversion (Fourier inversion on Schwartz space) gives $K_j\ast f={\mathcal{F}}^{-1}({\phi }_j\widehat{f})$ as functions; since $T_{{\phi }_j}f={\mathcal{F}}^{-1}(u_{{\phi }_j\widehat{f}})$ and the right side is the regular distribution of ${\mathcal{F}}^{-1}({\phi }_j\widehat{f})$, the two definitions of ${\mathrm{\Delta }}_{j}f$ agree on $\mathcal{S}$, and $\widehat{{\mathrm{\Delta }}_{j}f}={\phi }_j\widehat{f}$ as tempered distributions (using that the distributional transform agrees with the integral transform on $L^1$ functions, Fourier transform agrees with l one and plancherel transforms); the same holds for the companions. Moreover, if $m,n$ are smooth polynomially bounded symbols with $\mathcal{S}\subset D_m\cap D_n$, then $T_m{T}_n=T_{mn}$ on $\mathcal{S}$: for $f\in \mathcal{S}$ the density $n\widehat{f}$ lies in $\mathcal{S}$, so $T_{n}f$ is the regular distribution of ${\mathcal{F}}^{-1}(n\widehat{f})\in \mathcal{S}$ and $T_m(T_{n}f)=T_{mn}f$ by the same computation.
4. The companion identity and the low-frequency block. Because ${\phi }_j{\phi }_k=0$ pointwise whenever $\mathrm{\mid }j-k\mathrm{\mid }\ge 2$ (the annular supports meet at most at the endpoint spheres, where both factors vanish), expanding $1=({\sum }_j{\phi }_j{)}^2$ gives $1={\sum }_j{\phi }_j^2+2{\sum }_j{\phi }_j{\phi }_{j+1}={\sum }_j({\phi }_{j-1}+{\phi }_j+{\phi }_{j+1}){\phi }_j={\sum }_j{\tilde{\phi }}_j{\phi }_j$, the sums being locally finite. Neither the low-frequency block ${\mathrm{\Delta }}_0$ nor its companion ${\tilde{\mathrm{\Delta }}}_0$ is assigned mean zero: indeed ${\int }_{{\mathbb{R}}^n}{K}_j={\phi }_j(0)$ and ${\int }_{{\mathbb{R}}^n}{\tilde{K}}_j={\tilde{\phi }}_j(0)$, so $\int K_0=\psi (0)=1$ and $\int {\tilde{K}}_0={\phi }_0(0)+{\phi }_1(0)=1$, and no cancellation is claimed for these blocks.

This partition is fixed once and for all on this page. The operator norms of ${\mathrm{\Delta }}_j$ on $L^2$ are at most $\mathrm{\parallel }{\phi }_j{\mathrm{\parallel }}_{\mathrm{\infty }}\le 1$ (Exact L2 Fourier multiplier norm), and all constants below refer to this fixed partition.