Littlewood-Paley square-function equivalence on Lp for 1<p<infinity

Upper bound for Schwartz functions. Let $f\in \mathcal{S}$ and $N\ge 0$. The lower pointwise Khintchine bound of [F1] with $q=p$ gives $S_{N}f(x)\le c_p^{-1}({\int }_0^1\mathrm{\mid }{\sum }_{j<N}{\epsilon }_j(t){\mathrm{\Delta }}_{j}f(x){\mathrm{\mid }}^{p}dt{)}^{1\mathrm{/}p}$; raising to the $p$-th power, integrating in $x$, and applying Tonelli (the integrand is nonnegative and product-measurable by [F1]) gives $\mathrm{\parallel }{S}_{N}f{\mathrm{\parallel }}_p^p\le c_p^{-p}{\int }_0^1\mathrm{\parallel }{T}_{m_{t,N}}f{\mathrm{\parallel }}_p^{p}dt\le c_p^{-p}(C_{n,\psi }\max (p,(p-1{)}^{-1}){)}^p\mathrm{\parallel }f{\mathrm{\parallel }}_p^p$, where the last inequality is the uniform signed-sum bound [F2] applied at each $t$ to the truncation symbols. Since $S_{N}f\uparrow Sf$ pointwise, monotone convergence [F5] gives $\mathrm{\parallel }Sf{\mathrm{\parallel }}_p\le C_p\mathrm{\parallel }f{\mathrm{\parallel }}_p$ with $C_p:=c_p^{-1}{C}_{n,\psi }\max (p,(p-1{)}^{-1})$; in particular $Sf\in L^p$. The same computation for any finite set $F$ in place of $\{0,\dots ,N-1\}$ gives $\mathrm{\parallel }{S}_{F}f{\mathrm{\parallel }}_p\le C_p\mathrm{\parallel }f{\mathrm{\parallel }}_p$.

Companion upper bound. Since ${\tilde{\phi }}_j={\phi }_{j-1}+{\phi }_j+{\phi }_{j+1}$ and the multiplier map is linear in the symbol, ${\tilde{\mathrm{\Delta }}}_{j}f={\mathrm{\Delta }}_{j-1}f+{\mathrm{\Delta }}_{j}f+{\mathrm{\Delta }}_{j+1}f$ (with ${\mathrm{\Delta }}_{-1}:=0$); hence $\mathrm{\mid }{\tilde{\mathrm{\Delta }}}_{j}f{\mathrm{\mid }}^2\le 3(\mathrm{\mid }{\mathrm{\Delta }}_{j-1}f{\mathrm{\mid }}^2+\mathrm{\mid }{\mathrm{\Delta }}_{j}f{\mathrm{\mid }}^2+\mathrm{\mid }{\mathrm{\Delta }}_{j+1}f{\mathrm{\mid }}^2)$ pointwise, and summing over $j\in F$ gives ${\tilde{S}}_{F}f\le 3S_{F^{\mathrm{\prime }}}f$ with the finite set $F^{\mathrm{\prime }}=\{j-1,j,j+1:j\in F\}\cap \{0,1,2,\dots \}$. By step 1.1 applied to $F^{\mathrm{\prime }}$, $\mathrm{\parallel }{\tilde{S}}_{F}f{\mathrm{\parallel }}_p\le 3C_p\mathrm{\parallel }f{\mathrm{\parallel }}_p$; letting $F$ increase to all indices and using monotone convergence for ${\tilde{S}}_{F}f\uparrow \tilde{S}f$ gives $\mathrm{\parallel }\tilde{S}f{\mathrm{\parallel }}_p\le 3C_p\mathrm{\parallel }f{\mathrm{\parallel }}_p$.

Extension to $L^p$. For $u,v\in \mathcal{S}$ the pointwise inequality $\mathrm{\mid }Su-Sv\mathrm{\mid }\le S(u-v)$ of [F4] and step 1.1 give $\mathrm{\parallel }Su-Sv{\mathrm{\parallel }}_p\le C_p\mathrm{\parallel }u-v{\mathrm{\parallel }}_p$; hence $u\mapsto Su$ is Lipschitz on the dense subspace $\mathcal{S}$ of $L^p$, and for any $f\in L^p$ and any $f_k\in \mathcal{S}$ with $f_k\to f$ the sequence $S{f}_k$ is Cauchy in $L^p$. By completeness of $L^p$ [F5] it has a limit $F$, which is independent of the approximating sequence: if ${\tilde{f}}_k$ is another such sequence, the interleaved sequence $f_1,{\tilde{f}}_1,f_2,{\tilde{f}}_2,\dots$ also converges to $f$ in $L^p$ and its image is Cauchy, so the two limits agree. This assignment is the unique continuous extension of the Schwartz square function, by the same Lipschitz bound.

Lower bound for Schwartz functions. Let $f\in \mathcal{S}$. By steps 1.1 and 2.1, $\mathrm{\parallel }Sf{\mathrm{\parallel }}_p<\mathrm{\infty }$ and, for every $g\in \mathcal{S}$, $\mathrm{\parallel }\tilde{S}g{\mathrm{\parallel }}_{p^{\mathrm{\prime }}}<\mathrm{\infty }$; the reproducing formula [F3] therefore gives $⟨f,g⟩={\sum }_j⟨{\mathrm{\Delta }}_{j}f,{\tilde{\mathrm{\Delta }}}_{j}g⟩$ with ${\sum }_j\mathrm{\mid }⟨{\mathrm{\Delta }}_{j}f,{\tilde{\mathrm{\Delta }}}_{j}g⟩\mathrm{\mid }\le \int Sf\tilde{S}g$. By the pointwise Cauchy-Schwarz inequality and Holder [F4], $\int Sf\tilde{S}g\le \mathrm{\parallel }Sf{\mathrm{\parallel }}_p\mathrm{\parallel }\tilde{S}g{\mathrm{\parallel }}_{p^{\mathrm{\prime }}}\le 3C_{p^{\mathrm{\prime }}}\mathrm{\parallel }Sf{\mathrm{\parallel }}_p\mathrm{\parallel }g{\mathrm{\parallel }}_{p^{\mathrm{\prime }}}$. Hence $\mathrm{\mid }⟨f,g⟩\mathrm{\mid }\le 3C_{p^{\mathrm{\prime }}}\mathrm{\parallel }Sf{\mathrm{\parallel }}_p\mathrm{\parallel }g{\mathrm{\parallel }}_{p^{\mathrm{\prime }}}$ first for all $g\in \mathcal{S}$ and then, by density of $\mathcal{S}$ in $L^{p^{\mathrm{\prime }}}$ and continuity of the pairing, for all $g\in L^{p^{\mathrm{\prime }}}$; taking the supremum over $\mathrm{\parallel }g{\mathrm{\parallel }}_{p^{\mathrm{\prime }}}\le 1$ and using the duality formula [F5] gives $\mathrm{\parallel }f{\mathrm{\parallel }}_p\le 3C_{p^{\mathrm{\prime }}}\mathrm{\parallel }Sf{\mathrm{\parallel }}_p$. This is the lower bound of part 1 with $c_p:=(3C_{p^{\mathrm{\prime }}}{)}^{-1}$.

Identification with the pointwise square function. For fixed $N$, [F2] and the finite reverse-triangle inequality give $\mathrm{\parallel }{S}_{N}u-S_{N}v{\mathrm{\parallel }}_p\le {\sum }_{j<N}\mathrm{\parallel }{\mathrm{\Delta }}_j(u-v){\mathrm{\parallel }}_p\le N{B}_p\mathrm{\parallel }u-v{\mathrm{\parallel }}_p$ for all $u,v\in L^p$, where $B_p$ is a uniform bound for the pieces. Therefore $S_N$ is continuous on $L^p$, and approximation by Schwartz functions passes the bound of step 1.1 to $\mathrm{\parallel }{S}_{N}f{\mathrm{\parallel }}_p\le C_p\mathrm{\parallel }f{\mathrm{\parallel }}_p$ for every $f\in L^p$, uniformly in $N$. Monotone convergence [F5] gives $\mathrm{\parallel }Sf{\mathrm{\parallel }}_p\le C_p\mathrm{\parallel }f{\mathrm{\parallel }}_p$ for the increasing pointwise limit, so $Sf$ is finite almost everywhere. Since $\mathrm{\mid }Sf-S_{N}f{\mathrm{\mid }}^p\le (Sf{)}^p$, dominated convergence [F5] gives $S_{N}f\to Sf$ in $L^p$, as well as pointwise. Passing the finite reverse-triangle inequality to the limit gives $\mathrm{\mid }Su-Sv\mathrm{\mid }\le S(u-v)$ almost everywhere and hence $\mathrm{\parallel }Su-Sv{\mathrm{\parallel }}_p\le C_p\mathrm{\parallel }u-v{\mathrm{\parallel }}_p$ on all of $L^p$. In particular, for $f_k\in \mathcal{S}$ with $f_k\to f$, $S{f}_k\to Sf$ in $L^p$, identifying this function with the extension of step 2.2. Taking limits in steps 1.1 and 3.1 gives the two-sided estimate for $f$.

Conclusion. Steps 1.1 and 3.1 prove part 1, and steps 2.2 and 4.1 prove part 2: the Cauchy property and independence of the approximating sequence, the identification of the extension with the increasing pointwise square function both in $L^p$ and almost everywhere, and the inherited two-sided estimate.