The Littlewood-Paley reproducing formula in tempered distributions

The partial symbols. For every $N\ge 0$ the function ${\sigma }_N={\sum }_{j<N}{\tilde{\phi }}_j{\phi }_j$ lies in $C_c^{\mathrm{\infty }}({\mathbb{R}}^n)$ by [F1] and satisfies $0\le {\sigma }_N\le 1$ with $1-{\sigma }_N={\sum }_{j\ge N}{\tilde{\phi }}_j{\phi }_j\ge 0$ by [F2]. For $N\ge 1$ it equals $1$ on $\{\mathrm{\mid }\xi \mathrm{\mid }<2^{N-2}\}$: a term ${\tilde{\phi }}_j{\phi }_j$ with $j\ge N$ vanishes wherever ${\phi }_j$ does, and ${\phi }_j=0$ on $\{\mathrm{\mid }\xi \mathrm{\mid }\le 2^{j-1}\}$, so every tail term vanishes on this ball. (For $N=0$, ${\sigma }_0=0$ and no plateau assertion is made.) Moreover, for every multi-index $\alpha$ there is $C_{\alpha }<\mathrm{\infty }$, depending only on $n,\psi ,\alpha$, with $\mathrm{\mid }{\mathrm{\partial }}^{\alpha }(1-{\sigma }_N)(\xi )\mathrm{\mid }\le C_{\alpha }{2}^{-N\mathrm{\mid }\alpha \mathrm{\mid }}$ for all $\xi$ and $N$: by the Leibniz rule and [F1] each summand satisfies $\mathrm{\mid }{\mathrm{\partial }}^{\alpha }({\tilde{\phi }}_j{\phi }_j)(\xi )\mathrm{\mid }\le C_{\alpha }{2}^{-j\mathrm{\mid }\alpha \mathrm{\mid }}$, and ${\sum }_{j\ge N}{2}^{-j\mathrm{\mid }\alpha \mathrm{\mid }}\le 2\cdot 2^{-N\mathrm{\mid }\alpha \mathrm{\mid }}$ for $\mathrm{\mid }\alpha \mathrm{\mid }\ge 1$, while for $\alpha =0$ the bound is just $0\le 1-{\sigma }_N\le 1$.

The finite duality identity and absolute convergence. For $f,g\in \mathcal{S}$ and $N$, [F4] applied to the real symbol ${\tilde{\phi }}_j$ gives $⟨{\tilde{\mathrm{\Delta }}}_j{\mathrm{\Delta }}_{j}f,g⟩=⟨{\mathrm{\Delta }}_{j}f,{\tilde{\mathrm{\Delta }}}_{j}g⟩$ for each $j<N$, and summing yields the first identity of part 2. For the series bound, the pointwise Cauchy-Schwarz inequality in ${\mathrm{\ell }}^2$ gives ${\sum }_{j<N}\mathrm{\mid }{\mathrm{\Delta }}_{j}f(x)\mathrm{\mid }\mathrm{\mid }{\tilde{\mathrm{\Delta }}}_{j}g(x)\mathrm{\mid }\le Sf(x)\tilde{S}g(x)$ for every $x$, and integrating the finite sum, which is legitimate termwise by [F6], gives ${\sum }_{j<N}\int \mathrm{\mid }{\mathrm{\Delta }}_{j}f\mathrm{\mid }\mathrm{\mid }{\tilde{\mathrm{\Delta }}}_{j}g\mathrm{\mid }\le \int Sf\tilde{S}g\le \mathrm{\parallel }Sf{\mathrm{\parallel }}_p\mathrm{\parallel }\tilde{S}g{\mathrm{\parallel }}_{p^{\mathrm{\prime }}}<\mathrm{\infty }$ by Holder; hence the nonnegative series ${\sum }_j\int \mathrm{\mid }{\mathrm{\Delta }}_{j}f\mathrm{\mid }\mathrm{\mid }{\tilde{\mathrm{\Delta }}}_{j}g\mathrm{\mid }$ converges (its partial sums are increasing and bounded) and dominates ${\sum }_j\mathrm{\mid }⟨{\mathrm{\Delta }}_{j}f,{\tilde{\mathrm{\Delta }}}_{j}g⟩\mathrm{\mid }$, so that series converges absolutely with the stated bound.

The tail vanishes in $\mathcal{S}$. For every $h\in \mathcal{S}$ the products $(1-{\sigma }_N)h$ tend to $0$ in the Schwartz topology: for multi-indices $\alpha ,\beta$, the Leibniz rule writes ${\mathrm{\partial }}^{\beta }((1-{\sigma }_N)h)={\sum }_{\gamma \le \beta }\left(\genfrac{}{}{0px}{}{\beta }{\gamma }\right){\mathrm{\partial }}^{\beta -\gamma }(1-{\sigma }_N){\mathrm{\partial }}^{\gamma }h$, and every term with $\gamma \mathrm{\ne }\beta$ is bounded by $C_{\beta -\gamma }{2}^{-N\mathrm{\mid }\beta -\gamma \mathrm{\mid }}\mathrm{\mid }{\xi }^{\alpha }{\mathrm{\partial }}^{\gamma }h(\xi )\mathrm{\mid }\le C{2}^{-N}{p}_{\alpha \gamma }(h)\to 0$ uniformly in $\xi$, while, for $N\ge 1$, the term with $\gamma =\beta$ satisfies $\mathrm{\mid }{\xi }^{\alpha }(1-{\sigma }_N){\mathrm{\partial }}^{\beta }h(\xi )\mathrm{\mid }\le {\sup }_{\mathrm{\mid }\xi \mathrm{\mid }\ge 2^{N-2}}\mathrm{\mid }{\xi }^{\alpha }{\mathrm{\partial }}^{\beta }h(\xi )\mathrm{\mid }\to 0$ by [F5], since $1-{\sigma }_N$ vanishes for $\mathrm{\mid }\xi \mathrm{\mid }<2^{N-2}$. Hence $p_{\alpha \beta }((1-{\sigma }_N)h)\to 0$ for every pair $\alpha ,\beta$, which is convergence to $0$ in $\mathcal{S}$ by [F5], uniformly on bounded subsets because the finitely many seminorms in these estimates are uniformly bounded.

Convergence in ${\mathcal{S}}^{\mathrm{\prime }}$ (part 1). The multiplier composition in [F2] gives ${\sum }_{j<N}{\tilde{\mathrm{\Delta }}}_j{\mathrm{\Delta }}_{j}f={\mathcal{F}}^{-1}({\sigma }_N\mathcal{F}f)$ for every $f\in {\mathcal{S}}^{\mathrm{\prime }}$. For a Schwartz test $g$, the bilinear transposition convention [F3] gives $⟨{\mathcal{F}}^{-1}({\sigma }_N\mathcal{F}f)-f,g⟩=⟨\mathcal{F}f,({\sigma }_N-1){\mathcal{F}}^{-1}g⟩$. By step 2.1 the test on the right tends to zero in $\mathcal{S}$, and continuity of $\mathcal{F}f$ makes the pairing tend to zero. The same estimates are uniform for $g$ in any bounded subset: ${\mathcal{F}}^{-1}$ maps bounded sets to bounded sets, step 2.1 is uniform there, and a continuous functional is bounded by finitely many Schwartz seminorms. Hence the partial sums converge to $f$ in both the weak and strong dual topologies.

Identification of the sum (part 2). Apply step 3.1 to the Schwartz test $\bar{g}$; the resulting distributional pairing is the Hermitian integral pairing of part 2. Thus $⟨{\sum }_{j<N}{\tilde{\mathrm{\Delta }}}_j{\mathrm{\Delta }}_{j}f,g⟩\to ⟨f,g⟩$ in that convention. Step 1.2 identifies each partial sum with ${\sum }_{j<N}⟨{\mathrm{\Delta }}_{j}f,{\tilde{\mathrm{\Delta }}}_{j}g⟩$ and proves absolute convergence with the stated bound, so the sum is $⟨f,g⟩$.