The multiscale primitive map is continuous into C[0,1]

For fixed $t$ and level $l\ge l_0$, a wavelet whose support lies wholly on one side of $t$ contributes zero: it is either disjoint from $[0,t]$ or its entire support is integrated and its mean is zero. Only supports meeting $t$ can contribute. Uniformly bounded overlap bounds their number by a constant, and each coefficient has magnitude at most $\mathrm{\parallel }{\psi }_{lk}{\mathrm{\parallel }}_1\le C_0{2}^{-l\mathrm{/}2}$. Thus ${\sup }_t{\sum }_k\mathrm{\mid }⟨1_{[0,t]},{\psi }_{lk}⟩\mathrm{\mid }\le C_1{2}^{-l\mathrm{/}2}$. The finitely many coarsest levels have a finite uniform sum as well.

By [F1] and step 1.1, the absolute sum of all terms at levels $l\ge l_0$ is at most $C_1\mathrm{\parallel }x{\mathrm{\parallel }}_{M(w)}{\sum }_{l\ge l_0}{w}_l{2}^{-l\mathrm{/}2}$, uniformly over $t$. Its tails tend uniformly to zero because $S_w<\mathrm{\infty }$. Each finite-level sum is continuous in $t$, since $t\mapsto {\int }_0^t{\psi }_{lk}(u)du$ is continuous for an $L^1$ function. The uniform limit is continuous. Applying the same estimate to $x-y$ proves the stated operator-norm bound.

If $x_{lk}=⟨f,{\psi }_{lk}⟩$ for $f\in L^2$, the finite orthogonal projections converge to $f$ in $L^2$ by [F2]. The map $f\mapsto {\int }_0^{t}f(u)du=⟨f,1_{[0,t]}⟩$ is a continuous $L^2$ functional because $\mathrm{\parallel }{1}_{[0,t]}{\mathrm{\parallel }}_2\le 1$. Applying it to the finite projections shows that their coefficient sums converge to ${\int }_0^{t}f$, while step 2.1 identifies the same limit with $Jx(t)$.