Littlewood-Paley characterisation of the Hilbert-Sobolev spaces

Forward direction: identification of the pieces. Let $U=E_s(V)$ with $V\in H^s$ and let $g=J_s(V)\in L^2$, so that $\mathrm{\parallel }U{\mathrm{\parallel }}_{H^s}=\mathrm{\parallel }g{\mathrm{\parallel }}_2$ and $⟨\xi {⟩}^s\mathcal{F}U=u_g$ by [F2]. Put $h:=⟨\xi {⟩}^{-s}g$; since $⟨\xi {⟩}^{-s}$ is smooth and locally bounded, $h\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^2({\mathbb{R}}^n)$, and $u_h=⟨\xi {⟩}^{-s}{u}_g=\mathcal{F}U$ by the invertibility of the bracket multiplier [F1]. Then ${\mathrm{\Delta }}_{j}U={\mathcal{F}}^{-1}({\phi }_j\mathcal{F}U)={\mathcal{F}}^{-1}(u_{{\phi }_{j}h})=u_{({\phi }_{j}h{)}^{\vee }}$ by [F3], and ${\phi }_{j}h\in L^2$ because $h$ is locally square integrable and ${\phi }_j$ is compactly supported and bounded; Plancherel [F4] gives $\mathrm{\parallel }{\mathrm{\Delta }}_{j}U{\mathrm{\parallel }}_2^2=\mathrm{\parallel }{\phi }_{j}h{\mathrm{\parallel }}_2^2={\int }_{{\mathbb{R}}^n}{\phi }_j(\xi {)}^2\mathrm{\mid }h(\xi ){\mathrm{\mid }}^{2}d\xi$.

Converse direction: reconstruction. Assume ${\sum }_j{2}^{2js}\mathrm{\parallel }{\mathrm{\Delta }}_{j}U{\mathrm{\parallel }}_2^2<\mathrm{\infty }$; then for every $j$ the distribution ${\mathrm{\Delta }}_{j}U$ is the regular distribution $u_{H_j}$ of an $L^2$ function $H_j$ (the case $H_j=0$ included), and we may take $H_j={\mathrm{\Delta }}_{j}U$ as an $L^2$ class. Put $h_j:={\mathcal{F}}_2{H}_j$ and $g_j:=⟨\xi {⟩}^s{h}_j$. Since $\mathcal{F}({\mathrm{\Delta }}_{j}U)={\phi }_j\mathcal{F}U$ and $\mathcal{F}{u}_{H_j}=u_{{\mathcal{F}}_2{H}_j}$, the distribution $u_{h_j}={\phi }_j\mathcal{F}U$ is supported in $\mathrm{supp}{\phi }_j$, so $h_j=0$ almost everywhere off that support; by [F5] this gives $\mathrm{\parallel }{g}_j{\mathrm{\parallel }}_2^2=\int ⟨\xi {⟩}^{2s}\mathrm{\mid }{h}_j{\mathrm{\mid }}^2{\asymp }_{n,s}{2}^{2js}\mathrm{\parallel }{H}_j{\mathrm{\parallel }}_2^2=2^{2js}\mathrm{\parallel }{\mathrm{\Delta }}_{j}U{\mathrm{\parallel }}_2^2$, so ${\sum }_j\mathrm{\parallel }{g}_j{\mathrm{\parallel }}_2^2<\mathrm{\infty }$. The supports of the $g_j$ lie in the supports of the ${\phi }_j$, at most three of which meet at any point by [F3]; hence $\mid {\sum }_{j\in F}{g}_j{\mid }^2\le 3{\sum }_{j\in F}\mathrm{\mid }{g}_j{\mathrm{\mid }}^2$ pointwise for every finite $F$. Therefore the partial sums are Cauchy in $L^2$ (their tail squared norms are at most three times the tails of ${\sum }_j\mathrm{\parallel }{g}_j{\mathrm{\parallel }}_2^2$) and converge by [F8] to some $g\in L^2$, with $\mathrm{\parallel }g{\mathrm{\parallel }}_2^2\le 3{\sum }_j\mathrm{\parallel }{g}_j{\mathrm{\parallel }}_2^2{\asymp }_{n,s}{\sum }_j{2}^{2js}\mathrm{\parallel }{\mathrm{\Delta }}_{j}U{\mathrm{\parallel }}_2^2$. No lower norm estimate is used until the reconstruction identifies $g_j={\phi }_{j}g$.

Forward direction: the weighted sum. Multiplying the identity of step 1.1 by $2^{2js}$ and summing, the pointwise comparison $2^{2js}⟨\xi {⟩}^{-2s}\in [c_s,C_s]$ on $\mathrm{supp}{\phi }_j$ from [F5] gives ${\sum }_{j\ge 0}{2}^{2js}\mathrm{\parallel }{\mathrm{\Delta }}_{j}U{\mathrm{\parallel }}_2^2={\sum }_j\int 2^{2js}{\phi }_j^2\mathrm{\mid }h{\mathrm{\mid }}^2{\asymp }_{n,s}\int \left({\sum }_j{\phi }_j^2\right)\mathrm{\mid }g{\mathrm{\mid }}^2$, where [F7] interchanges the nonnegative sum and integral and the last comparison uses $\mathrm{\mid }h{\mathrm{\mid }}^2=⟨\xi {⟩}^{-2s}\mathrm{\mid }g{\mathrm{\mid }}^2$. Since ${\sum }_j{\phi }_j^2\in [1\mathrm{/}3,1]$ by [F3], this is comparable to $\int \mathrm{\mid }g{\mathrm{\mid }}^2=\mathrm{\parallel }U{\mathrm{\parallel }}_{H^s}^2$; in particular the series is finite and the right-hand inequality with $C_{n,s,\psi }$ holds, while the left-hand inequality follows from the same comparison with $c_s$.

Converse direction: $U$ lies in the range of $E_s$. With $g$ as in step 1.2, test against any $\chi \in C_c^{\mathrm{\infty }}$. Only finitely many ${\phi }_j$ meet its compact support, and ${\sum }_j{\phi }_j=1$ there by [F3], so $⟨⟨\xi {⟩}^s\mathcal{F}U,\chi ⟩={\sum }_j\int g_j\chi =\int g\chi$, the last equality following from the $L^2$ convergence in step 1.2 and Cauchy-Schwarz [F6]. Both sides are tempered distributions, and density of $C_c^{\mathrm{\infty }}$ in $\mathcal{S}$ [F8] extends this identity to every Schwartz test, giving $⟨\xi {⟩}^s\mathcal{F}U=u_g$. Multiplication by ${\phi }_j$ then shows $g_j={\phi }_{j}g$ as $L^2$ functions. Tonelli [F7] and [F3] give $\frac{1}{3}\mathrm{\parallel }g{\mathrm{\parallel }}_2^2\le {\sum }_j\mathrm{\parallel }{g}_j{\mathrm{\parallel }}_2^2=\int ({\sum }_j{\phi }_j(\xi {)}^2)\mathrm{\mid }g(\xi ){\mathrm{\mid }}^{2}d\xi \le \mathrm{\parallel }g{\mathrm{\parallel }}_2^2\mathrm{.}$ Thus $U$ lies in the range of $E_s$ by [F2], and this frame comparison, the piecewise comparison in step 1.2, and the exact norm identity $\mathrm{\parallel }U{\mathrm{\parallel }}_{H^s}=\mathrm{\parallel }g{\mathrm{\parallel }}_2$ give $c_{n,s,\psi }{\sum }_j{2}^{2js}\mathrm{\parallel }{\mathrm{\Delta }}_{j}U{\mathrm{\parallel }}_2^2\le \mathrm{\parallel }U{\mathrm{\parallel }}_{H^s}^2\le C_{n,s,\psi }{\sum }_j{2}^{2js}\mathrm{\parallel }{\mathrm{\Delta }}_{j}U{\mathrm{\parallel }}_2^2\mathrm{.}$

Conclusion. Steps 2.1 and 2.2 are the two directions of the asserted equivalence and the two-sided norm comparison (the constants of step 2.1 for the forward direction and those of step 2.2 for the converse are both of the form $c_{n,s,\psi },C_{n,s,\psi }$); the weight $2^0=1$ on the low-frequency block is part of the definition of the series, and the finiteness of the series in the forward direction is contained in step 2.1.