Unit 3: Extending row estimates from fixed to depth-varying inputs

This unit studies §§4.1–4.2 of D, pp.17–22, principally Lemmas 4.1–4.4. Linearizing the maximum in Unit 2 produced test functions that vary with scale. Here we examine an abstract operator-theoretic step used in the paper to control such variation.

The discussion ends at Lemma 4.4. It assumes the persistent-column structure, the refinement relations between two kinds of partitions, and the original joint Bessel estimate. Verifying these assumptions for the paper's particular projections, local modes, and smooth rows is a separate task.

1. Fixed-input and varying-input estimates

Suppose a family of bounded linear operators satisfies

All rows here act on the same $z$. If row $\nu$ instead acts on $F_{s_\nu}$, the estimate does not permit a separate replacement of $z$ in each row. The paper controls total variation within deterministic input blocks and imposes a spatial structure on the adjoint rows.

There are two different indices in this unit: the index $s$ at which a function is sampled, and the depth of a spatial dyadic interval. They need not coincide. The rows are arranged in rounds $n$; every row in one round acts on the same $F_{s_n}$, and the indices $s_n$ must be nondecreasing.

2. Amplitude and variation of the depth-indexed functions

Source: D §4.1, Lemma 4.1, pp.17–18, equations (4.1)–(4.2).

2.1 The full assumptions

Take a bounded dyadic interval $R$ of positive length. Unless stated otherwise, $L^2(R)$ uses unnormalized Lebesgue measure. Let $\mathcal H$ be a complex Hilbert space with inner product linear in the first variable.

Let $\mathcal S=\{s_-,s_-+1,\ldots,s_+\}$, and partition it into $W\ge1$ deterministic, nonempty consecutive blocks $\mathcal S_1,\ldots,\mathcal S_W$. Suppose the strongly measurable functions $F_s:R\to\mathcal H$ satisfy

almost everywhere. There are only finitely many data, so a common exceptional null set may be used for all blocks. Jumps between blocks do not enter (V).

For measurable $E\subseteq R$ with $|E|>0$, it follows immediately that

where the first norm uses normalized measure on $E$, and $1\le q\le\infty$.

2.2 A square sum of differences

Let $\mathcal J$ be a finite family of index intervals $[r,t]$, with $r\le t$ and endpoints in $\mathcal S$. Define the edge overlap number by

If there are no such edges, set the maximum to zero. Lemma 4.1 asserts that

Fix $y$ outside the exceptional null set, and write

If $r,t$ lie in the same block $\mathcal S_b$, then

Consequently,

The last step uses the fact that each edge belongs to at most $m$ intervals. Summing over all intervals whose endpoints lie in the same block yields at most $m\sum_bv_b^2\le m$.

If the endpoints lie in different blocks, assign the interval to the first block boundary it crosses. Each boundary is crossed by at most $m$ intervals, so there are at most $m(W-1)$ such intervals. The amplitude bounds at the two endpoints bound each difference by $2W^{-1/2}$. Their square sum is therefore at most $4m(W-1)/W$.

Add the two estimates pointwise, then integrate over $E$, to obtain (D). The constant does not grow with the number of depths or blocks.

2.3 Nondecreasing sampling permits skipped and repeated indices

Later, only indices $s_1\le\cdots\le s_N$ are sampled. Within an original block, each difference between consecutive sampled values is bounded by the sum of the differences across the intervening original edges. Nondecreasing order ensures that each original edge is used at most once; repeated indices contribute zero. Thus (V) remains valid along the sampled subsequence, with the original normalization $W$. Unsampled blocks contribute zero.

The same restriction applies to (D). Suppose intervals in the sampled list have edge overlap number $m$. Any original edge $(j,j+1)$ corresponds to the unique cut in the list between indices at most $j$ and indices greater than $j$. An interval crossing that original edge must cross this cut. Its overlap count is therefore still at most $m$.

3. Persistent columns and private coordinates

Source: D §4.2, Lemma 4.2, pp.18–19, equations (4.3)–(4.4).

3.1 Defining the columns

Take a finite label set $\mathcal V$. Each label $v$ has a dyadic birth interval $J_v\subseteq R$, a strongly measurable function $\phi_v:J_v\to\mathcal H_0$, and a nonzero scalar $\gamma_v$. In

define

The vector $e_v$ is the standard orthonormal coordinate reserved for label $v$. Even if two physical functions $\phi_v$ coincide, their second components distinguish them.

On a descendant interval $Q$, the available labels are

The column on $Q$ is the restriction of the same $\Phi_v$. Passing to a descendant does not redefine the function or its private direction, and existing coefficients remain unchanged. This is the paper's persistent catalog: the same fixed index set and already defined columns are used at every scale. An application may use only some labels, but unused labels remain in the full family.

3.2 Three quantitative conditions

Assume there is $D\ge2$ such that

The last condition counts every birth label containing a point, not just the labels currently selected. The cardinality of the full finite set $\mathcal V$ need not be bounded by $D$; different spatial locations may carry different labels.

3.3 Two-sided comparison of coefficients and function values

For $a=(a_v)_{v\in\mathcal V(Q)}$, Lemma 4.2 states that

The lower bound follows from the private component:

For the upper bound, apply Cauchy–Schwarz:

This uses both the bound $D$ on each column and the bound $D$ on the number of labels along a path. Since the label set is finite, a common exceptional null set can be chosen.

The lower bound implies uniqueness of the coefficient representation. If two coefficient vectors represent the same function, their difference has $L^2(Q)$ norm zero, and (P2) forces the coefficient difference to vanish. Extend the coefficient vector by zero to the full space $\ell^2(\mathcal V)$. That vector remains unchanged when the function is restricted to a descendant interval.

3.4 Restriction to a subroot does not increase $D$

Take a dyadic subroot $R'\subseteq R$. Remove birth intervals disjoint from $R'$. If $R'\subseteq J_v$, regard $R'$ as the column's birth interval in the restricted system. Otherwise retain the columns with $J_v\subseteq R'$. Intersecting dyadic intervals are nested, so these cases are exhaustive.

All retained columns keep their values and private coordinates. At each point of $R'$, the new label count is the restriction of the old one. Thus none of the three bounds in (P) increases. The same $D$ can be used when the lemmas are reapplied on a subroot.

4. The maximal estimate for adjoint partial sums

Source: D Lemma 4.3, p.19; its proof is on pp.20–22, equations (4.5)–(4.9).

4.1 Two spatial partitions

For each round $n=1,\ldots,N$, there are two finite dyadic partitions $\pi_n,\sigma_n$, satisfying

The partition $\pi_n$ specifies the supports of the whole row operators in that round. The finer partition $\sigma_n$ specifies the intervals on which their adjoints have constant-coefficient representations. The construction first assigns each whole row to a group, then makes stopping decisions on the finer partition.

4.2 Row supports, coefficient representations, and the joint Bessel estimate

Each round has a finite row-index set $\mathcal R_n$. These sets are disjoint, with union $\mathcal R$. For $\nu\in\mathcal R_n$, give a Hilbert space $Y_\nu$, an interval $Q_\nu\in\pi_n$, and a bounded linear operator

The whole row must be supported on $Q_\nu$: $A_\nu=A_\nu\mathbf1_{Q_\nu}$. For every $K\in\sigma_n$ and every $c\in Y_\nu$, there must be coefficients independent of $y\in K$ such that

No separate uniform bound on these coefficients is assumed.

In addition, suppose there is $B\ge0$ such that all the original rows satisfy

Empty rounds are allowed, and one support interval may carry several rows. The estimate must be a genuine operator bound for every $z$, rather than a bound valid only for the function used to construct the rows.

4.3 The conclusion and its constants

Take any consecutive rounds $p\le n\le q$ and test vectors $c_\nu\in Y_\nu$. Write

Set

Lemma 4.3 asserts that, for every $\lambda>0$,

The constant depends on the column-family bound $D$, but not on the number of rounds $N$ or the total number of labels $|\mathcal V|$. The proof below explains the factor $\log D$.

5. Lemma 4.3: Grouping whole rows and proving the maximal estimate

5.1 Fix the test vectors before choosing common representatives

Source: D p.20, the first paragraph of the proof.

Retain only rounds $p,\ldots,q$, and relabel them $1,\ldots,N$. Fix the test vector $c=(c_\nu)$ and threshold $\lambda>0$. Each tested adjoint $A_\nu^*c_\nu$ has a prescribed column representation on finitely many observation intervals. Remove the exceptional null sets of these representations and of the column family. All subsequent identities then hold on one common full-measure set.

This does not take a union of null sets over every possible test vector. The proof first fixes $c$, and the resulting norm estimate holds for every $c$. That is sufficient for duality.

On each $K\in\sigma_n$, the function $g_n$ has a unique constant coefficient vector $a_n(K)\in\ell^2(\mathcal V)$. Taking adjoints in the support identity yields

Thus, on $K$, only rows supported on the interval that contains $K$ in the partition $\pi_n$ contribute.

5.2 Number of groups and stopping threshold

Set

The integer $T$ is a power of two, with $T\ge D^5$. We will assign the original whole rows to at most $T$ groups. At each spatial location, the groups are used in order. When the current coefficient sum exceeds $\tau$ in norm, the current group ends and the next begins.

A single row may cover several observation intervals. It is therefore not permissible to decide when to stop separately at each point and then cut a row's output into arbitrary pieces. The two partitions prevent this problem.

5.3 Order of construction within each round

At the beginning of the first round, every support interval has group number $1$ and current coefficient vector zero. The induction assumption is that, at the beginning of round $n$, the group number and current coefficient vector are constant on each $Q\in\pi_n$, and the vector uses only labels in $\mathcal V(Q)$.

First assign all rows supported on $Q$ to that interval's current group. If its group number has already reached $T+1$, do not assign those rows.

Next, restrict attention to current group numbers at most $T$. On each corresponding finer observation interval $K\in\sigma_n$, add the round's coefficients $a_n(K)$ to the current vector. If its $\ell^2$ norm exceeds $\tau$, record a threshold crossing, increase the group number by one, and reset the current vector to zero. Otherwise retain the vector and group number. After the $T$-th crossing, freeze the state at $(T+1,0)$.

The new state is constant on each $\sigma_n$-interval. Since $\pi_{n+1}$ refines $\sigma_n$, every support interval of the next round inherits a definite state. Old columns remain unchanged on restriction to descendants, and their labels remain available. Old and new coefficient vectors can therefore be added in the same space $\ell^2(\mathcal V)$.

The induction preserves the whole-row assignment: each original row belongs to at most one group, with a single assignment over its entire support. Different stopping decisions on observation intervals never split a previously assigned row. All states are determined by finitely many intervals and operations on constant vectors, so the crossing sets are measurable.

5.4 Why test-dependent grouping still permits the original Bessel bound

Let $\mathcal B_i$ be the set of rows assigned to group $i$, and set

Although the sets $\mathcal B_i$ depend on the fixed $c$ and $\lambda$, they remain disjoint subsets of the original row-index set.

Combine all rows into the operator $A:z\mapsto(A_\nu z)_\nu$. Estimate (B) says that $\|A\|\le B$, hence $\|A^*\|\le B$. For any $J\subseteq\{1,\ldots,T\}$, set the test-vector coordinates outside $\bigcup_{i\in J}\mathcal B_i$ to zero. Then

This is D (4.6). It is simply the original joint operator inequality applied to a subset of coordinates; the dependence of the grouping on the test vector does not affect that fact. If an adjoint output were multiplied by an arbitrary spatial cutoff and then treated as part of a row, this application of (B) would no longer be automatic.

In particular, applying (Adj) to one group at a time and summing yields

5.5 The set with $T$ threshold crossings is small

Let $\Omega_T$ be the set of points at which all $T$ crossings occur. At such a point $y$, each $v_i(y)$ is exactly the sum accumulated by group $i$ from its start through the first round at which its coefficient norm exceeds $\tau$.

Indeed, rows whose supports miss $y$ vanish at $y$. Rows whose supports contain $y$ are assigned to group $i$ exactly while that group is active along the path. Group numbers only increase, so no later nonzero contribution enters a completed group.

The coefficient norm of each completed group exceeds $\tau$. Its labels remain available on the final observation interval, so the lower bound in (P2) implies

Integrating and applying (Energy), we obtain

Substitution of the value of $\tau$ gives

This is D (4.7). The fifth power comes from the two coefficient comparisons: the unfinished part uses the upper bound $D^{3/2}$, while the completed part uses the lower bound $D^{-1}$. Squaring the combined factor yields $D^5$.

5.6 The unfinished part has norm at most $\lambda/4$

Outside $\Omega_T$, every partial sum over rounds is a prefix sum of completed groups plus the current unfinished coefficient sum. The latter has coefficient norm at most $\tau$, so its Hilbert-space norm is bounded, by (P2), by

Immediately after a crossing, include the newly completed group in the prefix and take the unfinished part to be zero; the same bound holds. Outside $\Omega_T$, no row containing the point has been omitted because the state reached $T+1$. The comparison therefore still concerns partial sums of all original rows.

Set

Then D (4.8) reads

5.7 Dyadic prefix decomposition costs only $\log T$

Since $T$ is a power of two, for each $0\le k\le\log_2T$, let $\mathcal D_k$ be the partition of $\{1,\ldots,T\}$ into consecutive dyadic intervals of length $2^k$. For $J\in\mathcal D_k$, set $v_J=\sum_{i\in J}v_i$.

Every prefix $\{1,\ldots,j\}$ is a disjoint union of dyadic intervals, with at most one interval of each length. Pointwise, therefore,

For fixed $k$, the intervals of $\mathcal D_k$ are disjoint. By (Adj),

The triangle inequality in $L^2(R)$ now implies

This is D (4.9). The factor $\log T$ comes from the dyadic decomposition of finite prefixes. Since $T$ depends only on $D$, no factor $\log N$ appears.

5.8 The weak-type bound and the $L^1$ maximal estimate

By (Rem),

up to a fixed null set. Combining (Hit), (Prefix), and Chebyshev's inequality yields

The second line uses $T\ge D^5$. Together with the elementary bound $|\{M>\lambda\}|\le|R|$, this proves the weak estimate in (Max).

If $BE_c^{1/2}=0$, the estimate directly implies $M=0$ almost everywhere. Otherwise set $H=(C_D/2)BE_c^{1/2}$. The distribution-function formula for a nonnegative function gives

Split the last integral at $\lambda=H/|R|^{1/2}$. Both parts equal $H|R|^{1/2}$, proving the $L^1$ estimate in (Max).

To check the simplified constant, set $x=\log_2D\ge1$. Then $L_D\le5x+2$, and

Thus $C_D\le(40/3)\log_2(2D)$. This completes the proof of Lemma 4.3.

6. Lemma 4.4: From maximal adjoint partial sums to varying inputs

Source: D p.22, Lemma 4.4, equation (4.10).

6.1 The full statement

Retain all assumptions of Lemma 4.3. Let $(F_s)_{s\in\mathcal S}$ satisfy (V), and choose nondecreasing sampling indices

All rows in round $n$ act on the same $F_{s_n}$. The sampling index need not equal the spatial depth of the row's support interval. The paper asserts that

The constant is independent of the number of rows, depths, or function blocks $W$.

6.2 Partitioning the rounds by input blocks

Fix an arbitrary test vector $c=(c_\nu)$. Since the sampling indices are nondecreasing, the rounds whose indices lie in one function block $\mathcal S_b$ form a consecutive interval $p\le n\le q$, possibly empty.

On this interval, define as in Lemma 4.3

The adjoint identity expresses the output pairing on this block as

All sums are finite, and each pairing is integrable by the $L^2$ Cauchy–Schwarz inequality.

6.3 Pointwise Abel summation

Using $g_n=S_n-S_{n-1}$, at every nonexceptional point we have

The inner product is linear in its first variable, so the differences occur in that variable. No complex conjugate is missing.

Taking absolute values and applying Cauchy–Schwarz bounds the right side by

The last step uses §2.3: nondecreasing sampling within one original block traverses each edge at most once, and repeated indices contribute zero. The amplitude plus variation is therefore still controlled by (V).

6.4 Cancellation of the block count

Write the test energy of this block as

For an empty block it is zero. Integrate the preceding bound and apply the $L^1$ maximal estimate of Lemma 4.3. The absolute value of the block pairing is at most

Sum over all $W$ blocks and apply Cauchy–Schwarz to the finite sequence:

Therefore

Taking the dual norm in the finite Hilbert direct sum $\bigoplus_{\nu\in\mathcal R}Y_\nu$ proves (Var). The factor $W^{-1/2}$ cancels exactly against the factor $\sqrt W$ from Cauchy–Schwarz across blocks.

7. Applying the lemma: what still has to be checked

The paper permits the constructed finite rows, partitions, and column family to depend on the given function sequence $(F_s)$. Once constructed, however, these objects must be fixed, and (B) must be verified for every test input $z$. An estimate only for the original $F_s$ would not support the test-dependent grouping in Lemma 4.3 or the duality argument in Lemma 4.4.

Each application must check at least the following: whole-row supports are preserved; the adjoints have constant-coefficient representations on observation cells using the same persistent column family; the next support partition refines the current observation partition; the joint Bessel bound holds for arbitrary inputs; and every row in a round samples the same index, with those indices nondecreasing across rounds.

No mixing is used in this unit, and the local quadrilinear Theorem 3.2 has not been deduced. What has been proved is one varying-input row estimate used in the paper. Applications to projection increments, smooth mean-zero rows, and subsequent local modes still require the stated structure to be verified in each case.

Guide to the original text

• D p.17, equation (4.1): amplitude and total variation within fixed input blocks.
• D pp.17–18, Lemma 4.1, equation (4.2): square sums of differences over overlapping index intervals and nondecreasing sampling.
• D pp.18–19, equation (4.3), Lemma 4.2, and equation (4.4): persistent private columns and two-sided coefficient bounds.
• D p.19, the full assumptions of Lemma 4.3 and equation (4.5): the maximal estimate for adjoint partial sums.
• D p.20, the whole-row grouping and equation (4.6): applying the original joint operator bound to test-dependent subfamilies.
• D p.21, equations (4.7)–(4.9): the set of full threshold crossings, the remainder, and dyadic prefix estimates.
• D p.22, the final part of Lemma 4.3: integration of the distribution function and simplification of the constant.
• D p.22, Lemma 4.4 and equation (4.10): pointwise Abel summation, Cauchy–Schwarz across blocks, and direct-sum duality.