Unit 3, continued: short-block comparison and maximal averages on whole ancestor intervals

This part develops D §6.5.2, pp.55–57, equations (6.20)–(6.22). The preceding part constructed fixed Hilbert-space-valued families on the two sides, with total square energies $P G_0|R|$ and $G_0|R|$ . Retaining the difference matching, we combine randomization, short-block comparison, and maximal averages on whole ancestor intervals to obtain the near-difference gain $P^{-1/2}$ .

The short-block comparison itself is the source result H Lemma 7.2. We state its function classes and quantifiers fully and develop the randomization and maximal summation. Randomized functions with only $L^2$ control are not inserted directly into the fixed- $L^q$ local theorem.

1. The fixed-function constant before a uniform bound is known

Fix $2<q<3$ and set

M_qg=(M(|g|^q))^{1/q},

where $M$ is the uncentered Hardy–Littlewood maximal operator.

For every integer $N\ge1$ , let $C_N(q)$ be the least constant such that the following holds for every dyadic grid, every finite family of original intervals from at most $N$ consecutive depths, every permitted kernel family, and four compactly supported scalar $L^q$ functions fixed across the entire interval family:

\sum_{I\in\mathcal A}|I|\, |\mathcal H_I(g_0,g_1,g_2,g_3)| \le C_N(q)\int_{\mathbb R}\prod_{j=0}^{3}M_qg_j. \tag{Fixed-local}

This constant is finite even before depth independence is established. The local absolute estimate uses four $L^q$ norms since $4/q<2$ . Intervals at one depth are disjoint, and for $x\in I$ we have $M_qg_j(x)\ge\|g_j\|_{q,I}$ . Thus each depth's sum is bounded by a fixed constant times the integral on the right. At most $N$ depths give

C_N(q)\le CN.

No use has yet been made of $\sup_NC_N(q)<\infty$ .

In D, take $N=d+1$ , where $d$ is the original tree's maximal depth. Reassigning local constructions only changes the assignment of original intervals; it creates no new summation intervals for the form. Ancestors and added boundary leaves, even when deeper, supply only geometric and leaf-norm conditions. They do not count among these $N$ original summation depths.

2. The precise scope of the short-block comparison

Take a finite tree fragment in one block, with root $Q$ . Its internal summation intervals lie within the original $N$ -depth range and occupy at most $K$ consecutive depths; leaves lie within the block or one depth below it.

On $Q$ , let four supported deterministic Hilbert-space-valued functions $z_j$ satisfy on every leaf $L$

\|z_j\|_{2,L}\le s_j.

Their scalar randomizations $z_j^\sigma$ satisfy, for every finite real $a\ge1$ ,

\left(\mathbb E_\sigma|z_j^\sigma(y)|^a\right)^{1/a} \le C_a\|z_j(y)\| \quad\text{almost everywhere}, \tag{Moment}

with $C_a$ independent of the tree and $K$ . Randomizations in the four positions may share some signs to preserve the required matching.

The short-block conclusion cited in D equation (6.21) applies to every subset of internal intervals and all measurable complex coefficients $|\varepsilon_I(\sigma)|\le1$ :

\begin{aligned} \mathbb E_\sigma\left| \sum_I|I|\varepsilon_I(\sigma) \mathcal H_I(z_0^\sigma,z_1^\sigma,z_2^\sigma,z_3^\sigma) \right| \le{}&(1+C_N(q))C_q(1+\log(K+2))^C\\ &\cdot(K+2)^{C_{19}(q-2)}|Q|\prod_js_j. \end{aligned} \tag{Short-block}

The exponent $C_{19}$ is absolute. The absolute value encloses the entire interval sum, followed by expectation over the random parameters. Individual randomized interval absolute values are not summed first.

3. Randomization selects interval coordinates and retains every matched difference

Fix the group, block, and auxiliary choice $\nu$ , and a near offset $\ell$ . Write the first-side coordinates as $W_{i,p}$ and the second-side coordinates as $V_j$ . Relabel the frozen complementary differences $C^{p+\ell}$ by $p$ as $C_p$ . Denote the other frozen complementary function by $D$ .

Take mutually independent Rademacher families

(\epsilon_i)_i,\qquad(\eta_j)_j,\qquad(\zeta_p)_p.

Define scalar randomizations by summing physical components:

g_a^\sigma=\sum_{i,p}\epsilon_i\zeta_p(W_{i,p})_{\mathrm{phys}}, \quad g_b^\sigma=\sum_j\eta_j(V_j)_{\mathrm{phys}},

g_c^\sigma=\sum_p\zeta_p(C_p)_{\mathrm{phys}}, \quad g_d^\sigma=D_{\mathrm{phys}}.

If the current interval selects coordinates $i_0,j_0$ , include $\epsilon_{i_0}\eta_{j_0}$ in its coefficient. Independence gives

\mathbb E[\epsilon_{i_0}\eta_{j_0} \epsilon_i\eta_j\zeta_p\zeta_{p'}] =\mathbf1_{i=i_0}\mathbf1_{j=j_0}\mathbf1_{p=p'}.

Quadrilinearity and finiteness therefore give

\mathbb E_\sigma\bigl[ \epsilon_{i_0}\eta_{j_0} \mathcal H_I(g_a^\sigma,g_b^\sigma,g_c^\sigma,g_d^\sigma)\bigr] =\sum_{p\text{ valid}} \mathcal H_I(W_{i_0,p},V_{j_0},C_p,D). \tag{Matching}

The right-hand side is the complete matched sum at the fixed offset. Randomization has not selected just one $p$ .

3.1 Linearizing the absolute value on each original interval

Denote the right-hand side of (Matching) by $H_I^{\mathrm{match}}$ . Choose the deterministic phase

\tau_I= \begin{cases} \overline{H_I^{\mathrm{match}}}/|H_I^{\mathrm{match}}|, &H_I^{\mathrm{match}}\ne0,\\ 0,&H_I^{\mathrm{match}}=0. \end{cases}

The length-weighted absolute sum is then at most

\mathbb E_\sigma\left| \sum_I|I|\tau_I\epsilon_{i_0(I)}\eta_{j_0(I)} \mathcal H_I(g_a^\sigma,g_b^\sigma,g_c^\sigma,g_d^\sigma) \right|.

These are precisely the measurable coefficients allowed by (Short-block). Take the transfer coefficient $-\ell/P$ outside for fixed $\ell$ ; its absolute value is at most $2/P$ .

3.2 Dimension-free moment bounds

Khintchine's inequality handles single-sign sums. For double-sign sums, apply it first to one sign family, then to the other together with Minkowski's inequality. For $a\ge2$ , this gives

\left(\mathbb E\left| \sum_{i,p}\epsilon_i\zeta_p w_{i,p}\right|^a\right)^{1/a} \le C_a\left(\sum_{i,p}|w_{i,p}|^2\right)^{1/2}.

For $1\le a<2$ , use the $L^2$ moment. The squared sum of physical components is at most the squared Hilbert-vector norm, so (Moment) holds in every position. Sharing $\zeta_p$ between positions does not affect these individual moment estimates.

4. The distinction between randomized $L^2$ and $L^q$ functions

Once the signs are fixed, the functions above are fixed scalar $L^2$ functions. This alone does not permit (Fixed-local), since $q>2$ .

The short-block proof in H contains a separate reconstruction of a continuous form. With the auxiliary random and fast parameters fixed, its burst functions are finite scalar combinations of indicators of bounded intervals. The functions actually inserted into (Fixed-local) are therefore compactly supported $L^q$ functions, fixed across the original summation intervals. The auxiliary parameters are integrated out afterward.

The truncation and mixed maximal-operator calculation in that step give the block factor

(K+2)^{11(q-2)/(q-1)} \le(K+2)^{11(q-2)};

The remaining block costs are logarithmic. We invoke the full comparison in H Lemma 7.2 without reproving its burst construction here. D checks (Moment) and the leaf-norm hypotheses to apply that result.

5. Whole-ancestor maximal averages for the fixed families

For a full index $\nu$ , let $\Omega_\nu$ be the union of its original intervals of use. The maximal intervals of use are disjoint. All dyadic ancestors between them and the actual uses form a finite complete ancestor collection, called the hull below.

For the already fixed vector function, define

m_\nu^{\mathrm{full}}(x) =\sup_{\substack{I\text{ in the full hull}\\x\in I}} \frac1{|I|}\int_I\|\operatorname{stack}_\nu(y)\|^2\,\mathrm{d}y,

and set it to zero outside the hull. Every candidate interval is used in full, including descendant portions that may later be cut out.

Suppose one side has total energy

\sum_\nu\|\operatorname{stack}_\nu\|_2^2=E|R|,

and

\sum_\nu|\Omega_\nu|\le M_0|R|, \qquad M_0\le\exp(C_qk^C).

This support count comes from the original active group/interval multiplicity and auxiliary choices; all groups and blocks are already included. If $E=0$ , the corresponding contribution vanishes. Assume $E>0$ below.

6. A weak bound and truncated integral for the full maximal function

For each $\nu$ , select maximal hull intervals whose average exceeds $T$ . They are disjoint, and each satisfies

T|I|<\int_I\|\operatorname{stack}_\nu\|^2.

Summation gives

\sum_\nu|\{m_\nu^{\mathrm{full}}>T\}| \le\min\{M_0|R|,CE|R|/T\}. \tag{Weak-max}

For fixed $C_1\ge1$ , integration of the distribution function gives

\begin{aligned} \sum_\nu\int_R\min(m_\nu^{\mathrm{full}},C_1E) &=\int_0^{C_1E}\sum_\nu|\{m_\nu^{\mathrm{full}}>t\}|\,\mathrm{d}t\\ &\le CE(1+\log(2+C_1M_0))|R|. \end{aligned} \tag{Truncated-max}

If the transition point $CE/M_0$ lies below the upper integration limit, split there into a constant part and a $1/t$ part. Otherwise the constant part alone is at most $CE|R|$ . Since the truncation height is proportional to $E$ , no additional energy size enters the final logarithm.

7. Two first-crossing stops give a common pointwise bound

For each family member, first stop at the hull intervals where its average first exceeds $C_1E$ . By (Weak-max), the union of all such intervals has length at most $C|R|/C_1$ .

Stop before adding that interval's new value to the retained maximal function. Retained ancestor averages stay defined on their whole intervals. At a point entering a stopped interval, keep only the earlier ancestor contributions. The retained maximal function is therefore everywhere bounded by

\min(m_\nu^{\mathrm{full}},C_1E).

Next make one common first-crossing stop for the sum over all $\nu$ of the retained ancestor suprema. Equation (Truncated-max) and Markov's inequality allow common threshold

CE(1+\log(2+C_1M_0))

times a sufficiently large fixed constant, making the new stopping length smaller than a prescribed fixed fraction.

When a common interval is about to cross the threshold, remove its new and future descendant values first and retain the old ancestor values. The final bound consequently holds at every point, including points inside later cut intervals that still belong to a retained ancestor.

Apply the preceding energy bounds $E_a\le PG_0$ and $E_b\le G_0$ on the two sides, and absorb $\log(2+M_0)$ into an allowed enlargement of $G_0$ . This yields

\sum_\nu m_{a,\nu}(x)\le PG_0, \qquad \sum_\nu m_{b,\nu}(x)\le G_0. \tag{Retained-max}

Every potentially needed within-block ancestor interval must be included among the candidates before these common stops.

8. A sparse decomposition supplies the leaf bounds

In each retained hull, start at a root $Q$ and stop when either side's squared average first exceeds sixteen times its root average. If one root average is zero, that side's fixed vector vanishes on $Q$ and the corresponding contribution is zero.

Otherwise each side's bad subintervals have total length at most $|Q|/16$ , and their union has length at most $|Q|/8$ . Set

E_Q=Q\setminus\bigcup\{\text{these stopped subintervals}\},

Then $|E_Q|\ge7|Q|/8$ . For one $\nu$ , the sets $E_Q$ belonging to different sparse roots are disjoint.

Complete each tree fragment with the immediate children of its included intervals as boundary leaves. A child's nonnegative squared average is at most twice its parent's. Thus both leaf averages are at most thirty-two times the root averages, giving the leaf $L^2$ hypotheses in (Short-block).

The frozen complementary functions are controlled on whole ancestors and boundary leaves by the preceding treatment of near-difference uses after removing large-value indicators. Added leaves supply only norm conditions, not new summands of the form.

9. Why maximal averages must remain on whole ancestor intervals

Write the sizes on a sparse root as

a_Q^2=\frac1{|Q|}\int_Q\|\operatorname{stack}_{a,\nu}\|^2, \qquad b_Q^2=\frac1{|Q|}\int_Q\|\operatorname{stack}_{b,\nu}\|^2.

For each $x\in E_Q$ , the whole interval $Q$ remains a candidate for both maximal functions. Hence

m_{a,\nu}(x)\ge a_Q^2,\qquad m_{b,\nu}(x)\ge b_Q^2.

This remains true even if $x$ lies in a later cut interval: descendant candidates are removed, but no part is cut from a retained ancestor interval.

Using $|E_Q|\ge7|Q|/8$ and disjointness of the remaining sets,

\begin{aligned} \sum_{\nu,Q}|Q|a_Qb_Q &\le C\int_R\sum_\nu\sqrt{m_{a,\nu}(x)m_{b,\nu}(x)}\,\mathrm{d}x\\ &\le C\int_R \left(\sum_\nu m_{a,\nu}(x)\right)^{1/2} \left(\sum_\nu m_{b,\nu}(x)\right)^{1/2}\,\mathrm{d}x\\ &\le P^{1/2}G_0|R|. \end{aligned} \tag{Root-size}

Setting retained ancestor values to zero below cut intervals would invalidate the first line. This is the information preserved by the source's whole-hull convention.

10. Combining short-block comparison with the coefficient $P^{-1}$

Apply (Short-block) and (Matching) on each sparse tree fragment. The two non-core leaf sizes are controlled by $a_Q,b_Q$ , and the frozen complementary leaf sizes are uniformly bounded. Summing over $\nu,Q$ and applying (Root-size) gives $P^{1/2}G_0|R|$ .

The transfer coefficient for each fixed near offset is $O(P^{-1})$ . Absorb the finitely many near offsets, the numbers of fixed search branches and transfer steps, and the short-block logarithms into an allowed enlargement of $G_0$ . This gives D equation (6.22):

(1+C_N(q))C_q P^{-1/2}G_0 (K+2)^{C_{19}(q-2)}|R|. \tag{Near-bound}

It controls the absolute sum of contributions over original intervals, retaining the prescribed fixed-offset matched difference sum inside each interval.

The constant $C_N(q)$ still depends on original tree depth. D §7 must use the fixed-function proof in H to choose a common $q$ , obtain $\sup_NC_N(q)<\infty$ , and absorb the block power. Equation (Near-bound) alone does not yet establish a depth-independent constant for the local theorem.

Source coverage and the next step

D p.55, equation (6.20): the fixed scalar $L^q$ constant and its initial finite-depth bound.
D p.56, equation (6.21): Hilbert-function moment assumptions, matched randomization, and the source short-block comparison.
D pp.56–57: the whole-ancestor weak maximal bound, two first-crossing stops, sparse leaf conditions, and root-size products.
D p.57, equation (6.22): the $P^{-1/2}$ gain with matching retained.
H Lemma 7.2: the short-block comparison remains an explicit input, including its passage from scalar $L^2$ data to the fixed- $L^q$ local estimate.

The next part, D §6.6, treats at least three core positions. Each group's complete tuple count $N_s^G$ must stay inside its power. Take the minimum on each original interval first, then use subadditivity across groups and configurations. Do not split into individual tuples before taking powers.