Covering by curved Hamiltonian neighborhoods

The large-gradient neighborhoods can be long, curved tubes. Their ordinary diameter is therefore a poor measure of how many can overlap. The right comparison uses a minimum of the bracket scale over a small transverse ball, followed by a Hamiltonian orbit shared by two overlapping neighborhoods.

We prove that comparison, construct finite covers with bounded overlap, and form the ordered partition used to separate the two gradient cases. The partition from the large-gradient neighborhoods is one on the region those neighborhoods cover. Its transition is also covered by small-gradient neighborhoods.

Use the hypotheses and scales from An adaptive scale for repeated brackets, the exact maps from An explicit canonical cell for a large transverse gradient, the estimates in The linear frequency coefficient and the bracket scale, and A cancellation plane or a uniform bracket lower bound. The nearby-center theorem is proved in Finding an admissible center nearby.

1. The covering statement and its boundary margin

Write \(w=(x',\xi')\in\mathbb R^d\), where \(d=2n-2\). The case \(d=0\) has no transverse gradients and needs only the small-gradient cover below. Suppose for now that \(d\geq2\). Choose

The original symbol hypotheses hold on \(|t|<1,|w|<\lambda^{-1}\). Fix \(\rho\geq1\) first, then take \(\lambda\) sufficiently small. All constants in the conclusions are independent of those parameters.

At an admissible case-II center \(c=(t_c,w_c)\), abbreviate

Its exact canonical map \(\chi_c\) fixes zero; the translation by \(w_c\) is separate. For \(h>0\), let

Here \(\xi\) is the distinguished frequency and \(u\) contains the remaining transverse canonical pairs. In particular \(E_c(1)\) is the previously defined admissible neighborhood. The factor \(h\) dilates its canonical coordinates and time interval; it does not dilate its physical image about the center.

Theorem 1.1 (adaptive cover). There is a finite selection of admissible case-II centers \(c_j\in\Omega_+\), with neighborhoods \(E_j=E_{c_j}(1)\), having these properties:

1. Every admissible case-II center in \(\Omega_+\) belongs to \(E_j(1/2)\) for some \(j\).
2. The neighborhoods \(E_j(h)\) have bounded multiplicity for each fixed \(h\). When two such neighborhoods intersect, their \(M_j\) values are comparable, and each is contained in a fixed enlargement of the other. Each selected neighborhood meets only a bounded number of the other selected neighborhoods.
3. There are cutoffs \(0\leq\psi_j\leq\Phi_j\leq1\), with \(\operatorname{supp}\Phi_j\Subset E_j\), \(\operatorname{supp}\psi_j\subset\{\Phi_j=1\}\), and \(\psi_j=1\) on \(E_j(9/16)\). Every case-II point of \(\Omega\) has some \(\psi_j=1\).
4. The remaining set

consists of case-I points. A finite family of case-I neighborhoods, with centers in \(\omega\), has bounded overlap and cutoffs \(\varphi_a\) whose sets \(\{\varphi_a=1\}\) cover \(\omega\). 5. With any ordering of the type-II family,

one has \(0\leq\Psi\leq1\), \(\operatorname{supp}\Psi_j\subset\{\Phi_j=1\}\), and \(\Psi=1\) on \(\Omega\setminus\omega\). Every point of \(\Omega\cap\operatorname{supp}d\Psi\) belongs both to a type-II set \(\{\Phi_j=1\}\) and to a type-I set \(\{\varphi_a=1\}\).

The slightly larger center domain in (1.1) is necessary to justify the nearby-center step up to the boundary of \(\Omega\). Restricting all type-II centers to \(\Omega\) itself can fail; Section 8 gives an admissible counterexample. The required region of localization remains exactly \(\Omega\).

2. Estimates on fixed enlargements

All estimates used below hold on any fixed enlargement \(E_c(h)\), with constants allowed to depend on \(h\). We first justify the domain and the lower bound, because a lower bound on the original cell cannot simply be assumed on its double.

The original finite-bracket bound implies \(M_c\to\infty\), uniformly over centers in \(\Omega_+\), for each fixed \(\rho\). If \(a_c\) is the selected gradient and \(s_c\) its time order, then

The last limit follows from \(\kappa+\kappa_2<2/(k+1)\). Thus \(h\ell_c\ll a_c\), so the canonical map and its inverse are defined on every needed fixed enlargement. Their first derivatives have uniform bounds. Also \(a_c\leq C\lambda^{-1}\), hence

The time radius \(h/M_c\) tends to zero. Consequently these images stay inside the original domain, with a fixed margin, for small \(\lambda\).

The Taylor, coefficient and bracket estimates from the preceding lessons extend to these domains as follows. Replace each bounded normalized time or spatial interval there by an interval of length bounded by a fixed constant. Taylor's integral remainder acquires only that constant's fixed powers. Finite polynomial-jet translation remains bounded in both directions on such intervals. The product-rule sums have the same finite number of terms. In the branch \(\ell_c=R_2>\ell_{0,c}\), integrate the differentiated smooth bracket estimates over a distance at most \(hR_2\), giving an error bounded by \(C_hR_2/L_c\). In the other branch the normalized spatial variable is bounded by \(h\). All these errors still tend to zero for fixed \(\rho\).

To obtain the lower bound, use the model notation \(F_0,G_0,H_0,E_0,J_0\) of the geometric-alternative proof. At local time and spatial coordinate zero, the exact cancellation frequency \(-\Xi_c\) is inside the admissible cell, since \(|\Xi_c|=O(R)\ll R_2\). Admissibility gives \(\mu\geq\gamma M_c\) there. Decrease \(\gamma\) to at most one. Some defining word therefore has normalized magnitude at least \(\gamma^{k+1}\). The all-word model comparison, including its separate longer-word estimate, gives

for small \(\lambda\). At that frequency the polynomial comparison has no shifted-frequency term, so

Translate the full polynomial jet on each fixed normalized time and spatial domain. Its lower bound remains positive. The polynomial comparison and the vanishing model errors now bound the actual short-word maximum below on bounded \(|A_{2,c}\xi|\). If the spatial radius was extended to \(R_2\), the differentiated-bracket error just described preserves that bound. At larger \(|A_{2,c}\xi|\), the coefficient lower estimate gives a lower bound directly. That estimate also extends to fixed normalized intervals by the same finite polynomial-jet translation. Taking the finitely many roots proves

for a positive uniform \(\gamma_h\). This argument uses admissibility itself. It does not assign the center to one exclusive branch of the geometric alternative.

At distinguished frequency zero the full symbol estimate, followed by the differentiated anisotropic bracket estimate, gives

for the time, spatial and remaining-coordinate ranges in \(E_c(h)\). Its amplitude scale is \(\rho\), its bracket scale is \(M_c\), and all Poisson contraction factors are bounded by one. The frequency factor \(1+|A_{2,c}\xi|\) is exactly one here.

3. A transverse minimum compares the center scales

For a point \(p=(t,w)\) and a fixed number \(D>0\), define

Whenever \(p\in E_c(h)\), the physical ball in (3.1) is inside the original domain for small \(\lambda\). The explicit inverse map is defined throughout that ball: its physical coordinates relative to \(w_c\) have size \(O_h(\ell_c)+O(DR_2)\ll a_c\). The inverse's derivative bound places the ball in \(E_c(h')\) at the same time, for a fixed \(h'\) depending only on \(h,D\). Continuity of \(\mu\) gives an attained minimum. Equation (2.5) gives its lower bound.

If \(p\) has canonical coordinates \((z,\xi,u)\), change only \(\xi\) to zero. The physical displacement is at most \(C_\chi hR_2\). Thus for \(D\geq C_\chi h\), this point belongs to the search ball and (2.6) gives the upper bound:

The large-frequency lower estimate also shows that every minimizing point, expressed in this chart, satisfies

Indeed, beyond a fixed such threshold it gives \(\mu\geq cM_c|A_{2,c}\xi|^{2/(k+2)}\), exceeding the upper bound in (3.2).

If \(E_c(h)\) and \(E_e(h)\) intersect, choose a point \(p\) in their intersection and use the same sufficiently large \(D\) for both charts. The function in (3.1) is defined in the original coordinates and does not depend on either chart. Comparing its two bounds gives

The ordinary scale \(\mu(p)\) itself need not be comparable to \(M_c\): near the outer frequency edge it may be much larger. The minimum in (3.1) is what removes that obstruction.

4. Recover the shape from a shared Hamiltonian field

At a fixed original time, consider the finite family

The time is held constant during each transverse flow. Pull these fields back by \(\chi_c\). Write \(Q=\mathcal R+\xi c(t,z)\) and \(b_c=\min(\ell_{0,c}^{-1},L_c^{-1})\). The coefficient and residual estimates give, on each fixed enlarged cell,

The polynomial coefficients are uniformly bounded and one selected derivative at zero has a uniform positive lower bound. The last comparison follows from the exact finite Taylor identities in both directions on bounded \(T\)-intervals.

Apart from the distinguished spatial component, the pulled-back field has size at most

For clarity, the residual first derivatives divided by \(M_c^{j+1}\) are at most \(C_h\rho/L_c\), or \(C_h\rho b_c\) in the distinguished spatial derivative; the latter is no larger. The extra derivative of \(\xi c\) is bounded by \(C_hR_2\ell_{0,c}b_cR/L_c\), at most \(C_hR_2R/L_c\). The distinguished spatial component is the coefficient in (4.2), plus a residual of size \(C_h\rho/L_c\).

These errors are also \(o(\ell_{0,c})\). For the residual use \(\rho/(L_c\ell_{0,c})<R/L_c\), because \(A_{2,c}>1/R\). For the coefficient derivative use \(R_2b_c\leq R_2/L_c\to0\). The first derivatives of \(\chi_c\) and its inverse are uniformly bounded. Consequently, in original coordinates,

The original transverse Hessian bound has an additional useful consequence:

Hence each fixed-time flow has a uniformly bounded Lipschitz constant on every fixed bounded parameter interval. This follows directly from the integral equation and Grönwall's inequality; the bound is \(\exp(C|s|/M_c)\).

Suppose first that \(\ell_{0,c}\geq R_2\). At the time under consideration, choose an index in (4.2) whose coefficient is bounded below in absolute value. Its coefficient changes by \(o(\ell_{0,c})\) over every needed fixed spatial enlargement, by the slow spatial-variation estimate. Its distinguished spatial speed therefore has fixed sign and is between \(c_h\ell_{0,c}\) and \(C_h\ell_{0,c}\). All remaining coordinates move by \(o(R_2)\) on a fixed parameter interval, by (4.3).

Given two points of the canonical body, start the selected flow at the first and stop when its distinguished spatial coordinate equals that of the second. Monotonicity and the lower speed bound give a parameter interval of bounded length. The other coordinates of the flow and target differ by at most \(C_hR_2\). Applying the derivative bound for \(\chi_c\) gives the same bound for their physical separation. Conversely, a bounded flow segment and its fixed \(R_2\) neighborhood lie in a fixed enlargement of the canonical body: its distinguished coordinate moves by \(O(\ell_{0,c})\), the others by \(o(R_2)\), and the inverse-map derivative bound controls the added neighborhood. All flow segments used here stay inside a fixed enlarged chart, by these same component bounds and (2.1).

Thus the physical body is comparable to a tube of radius \(R_2\) around a bounded parameter segment of that field. When \(\ell_{0,c}<R_2\), the body is instead comparable to an ordinary \(R_2\) ball, by the map and inverse derivative bounds.

Now let two fixed enlarged cells intersect at \(p\). Their \(M\) values are comparable by (3.4). The same actual gradient family occurs in (4.1), with only bounded powers of the ratio of the two \(M\) values changing its normalization. Equations (4.4) therefore imply

If either \(\ell_0\) is at most a fixed multiple of \(R_2\), both bodies are ordinary balls up to uniform constants, and the derivative bounds give their containment in fixed enlargements. Otherwise choose a field which is dominant in the first chart at \(p\). Its physical norm is a fixed fraction of the maximum in (4.4), so in the second chart its distinguished component is also dominant: its other components are \(o(\ell_{0,e})\). Slow spatial variation preserves this dominance throughout the fixed domains used for the flow.

Both bodies are therefore tubes around the same physical Hamiltonian field through the same point \(p\), with bounded parameter lengths and \(R_2\) errors. The normalization change is a bounded constant rescaling of the flow parameter. Flow uniqueness and (4.5) control the errors along the common orbit. The converse tube inclusion just proved places each body inside a fixed enlargement of the other. Their time windows are contained in corresponding enlarged time windows by (3.4). This proves the asserted engulfing property.

We also need its small version. There is a uniform \(C_*\) such that

for sufficiently small fixed \(\varepsilon>0\). To verify the factor \(\varepsilon\), use the same dominant field at the intersection. In either small body its distinguished coordinate differs from its center by \(O(\varepsilon\ell)\); reaching it from that center takes parameter time \(O(\varepsilon)\). Its remaining coordinates differ by \(O(\varepsilon R_2)\), and their flow drift has the same extra factor \(\varepsilon\). Join the two orbit segments at the common point, retaining the physical errors with (4.5). The second center is at parameter distance \(O(\varepsilon)\) and transverse error \(O(\varepsilon R_2)\) from the first. The inverse chart places it in \(E_c(C_*\varepsilon)\). The time difference is at most \(\varepsilon(M_c^{-1}+M_e^{-1})\), also of the required size. In the round-body case, the two distances to the intersection are already \(O(\varepsilon R_2)\). This proves (4.7) in both cases.

5. Finite selection and bounded multiplicity

Select admissible centers in \(\Omega_+\) successively, each outside the union of the previously selected \(E_j(1/2)\). Stop when no such center remains.

For fixed \(\lambda,\rho\), the original upper bound on \(M\) is finite. Each \(E_j(1/2)\) contains a Euclidean ball in \((t,w)\) of one fixed positive radius \(r_*\): its time radius has a common positive lower bound, its transverse widths are at least \(R_2/2\), and its inverse map has a common derivative bound. The selected centers are consequently separated by at least \(r_*\). Disjoint balls of radius \(r_*/2\) about them fit inside a bounded enlargement of \(\Omega_+\). Comparing their ordinary volumes proves that only finitely many can be selected. If an uncovered admissible center still existed, it could be added, contradicting this finite bound. Thus the stopping condition holds and proves the first assertion of Theorem 1.1.

Choose a fixed \(\varepsilon>0\) with \(C_*\varepsilon<1/2\). The selected \(E_j(\varepsilon)\) are pairwise disjoint. Indeed, if \(j<k\) and those cells intersected, (4.7) would put center \(c_k\) in \(E_j(1/2)\), contrary to its selection.

The symplectic map has transverse Jacobian one. With \(\nu_r\) the volume of the unit ball in \(\mathbb R^r\) and \(\nu_0=1\), the exact spacetime volume is

Consider all \(E_j(h)\) containing one point, for fixed \(h\). Their \(M_j\) and \(\ell_j\) values compare by (3.4) and (4.6). Engulfing puts all their small \(E_j(\varepsilon)\) in one fixed enlargement of a reference cell. Each disjoint small cell has volume at least a fixed positive multiple of the reference volume, by (5.1). The containing enlargement has volume at most a fixed multiple of that same volume. Their number is therefore bounded by a constant depending on \(h,\varepsilon,d\), independent of the parameters and selected centers. This proves bounded multiplicity, including for fixed enlargements. It uses the exact tube volumes rather than their possibly much larger enclosing-ball volumes.

The same argument bounds the number of neighbors of one selected cell. Every cell meeting that reference cell has comparable \(M,\ell\) and is engulfed by one fixed enlargement of the reference cell. Its disjoint small shrink has a comparable positive volume. Packing all those shrinks into that fixed enlargement bounds their number, even though their intersection points with the reference cell may differ.

6. Cores cover case II; case I covers the remainder

Choose a fixed real \(\phi\in C_c^\infty(-1,1)\), with \(0\leq\phi\leq1\), equal to one on \([-3/4,3/4]\). In the canonical coordinates at center \(j\), define

Extend both by zero outside their charts; their support is compactly inside the charts, so the extensions are smooth. The second line uses a dilation of the canonical coordinates, as in (1.3). It is not a physical homothety of the curved neighborhood.

The support of \(\psi_j\) lies inside \(E_j(3/4)\), where \(\Phi_j=1\). Also \(\psi_j=1\) on \(E_j(9/16)\): the three linear arguments have magnitude at most \(3/4\), and the remaining quadratic argument is at most \(9/16<3/4\). In particular \(0\leq\psi_j\leq\Phi_j\leq1\).

There is a uniform \(c_*>0\) such that \(\psi_j=1\) at every point obtained by moving a point of \(E_j(1/2)\) by physical transverse distance less than \(c_*R_2\), without changing its time. To see this, integrate the inverse-map derivative along that short physical segment. Each canonical coordinate changes by at most \(C_\chi c_*R_2\). Since \(\ell_j\geq R_2\), all normalized transverse changes are at most \(C_\chi c_*\). Choose this less than \(1/16\); then the point is in \(E_j(9/16)\). The time coordinate is unchanged.

For a case-II point \(p\in\Omega\), the nearby-center theorem supplies an admissible center at the same time within \(C_aR\). Its proof applies on any slightly larger fixed inner domain, by the original-domain margins used there. This center is in \(\Omega_+\) for small \(\lambda\). By maximality it belongs to \(E_j(1/2)\) for some selected \(j\). Since \(C_aR<c_*R_2\) for small \(\lambda\), the preceding core argument gives \(\psi_j(p)=1\). Consequently every point of \(\omega\) in (1.4) is in case I.

At a case-I center \(a=(t_a,w_a)\), use the standard small-gradient neighborhoods

On each fixed \(U_a(h)\), the small-gradient derivative estimate and finite time-jet translation give

To retain the full interval in this extension, Taylor-expand the center time polynomial on \(|M_a(t-t_a)|\leq h\). Its whole derivative jet is bounded below by the reverse finite translation formula. Its remainder is \(O_h(M_a^{-1})\), and its transverse change on \(|w-w_a|\leq hR_1\) is \(O_h(R_1/R)\). Both errors vanish. The derivative and upper-bracket estimates are the same product estimates on fixed normalized domains. This proves (6.3), without requiring the points in that cell to stay in case I.

Two intersecting fixed enlarged \(U_a\) cells therefore have comparable \(M_a\). Their spatial radii are the same, so direct triangle inequalities give engulfing and the small version of (4.7). Select centers from \(\omega\) outside previous \(U_a(1/2)\). The same positive-radius separation argument makes this selection finite and maximal. A fixed small shrink is disjoint. The exact volumes \(2\nu_dh^{d+1}R_1^d/M_a\), compared inside one enlarged cylinder, prove bounded overlap.

Define

It is supported compactly inside \(U_a(1)\) and equals one on \(U_a(1/2)\). The maximality condition therefore gives the type-I core cover of \(\omega\).

For \(d=0\), perform this selection using just the time intervals in (6.2). Every point is in case I; the same comparison, separation and interval-length arguments prove the corresponding assertions. The type-II family and its partition are empty.

7. The ordered partition and its transition

All products in (1.5) are finite. Subtract successive partial products to obtain

Summation telescopes to the expression for \(\Psi\) in (1.5). Its bounds follow from \(0\leq\psi_j\leq1\). If some \(\psi_j=1\), the full product is zero and \(\Psi=1\). If every \(\psi_j<1\), each factor is positive and \(\Psi<1\). This proves the precise equality region \(\Omega\setminus\omega\); the type-II sum need not be one on all of \(\Omega\).

The support of \(\Psi_j\) is contained in \(\operatorname{supp}\psi_j\), hence in \(\{\Phi_j=1\}\). Outside the finite union of these supports, \(\Psi\) is locally zero. Thus \(\operatorname{supp}d\Psi\) is contained in that union, giving the type-II core cover of the transition.

Within \(\Omega\), the function \(\Psi\) is locally identically one outside the relative closure of \(\omega\). Therefore

The type-I family is finite, and each set \(\{\varphi_a=1\}\) is closed. Their union covers \(\omega\), so it also covers its relative closure in \(\Omega\). Equation (7.2) proves the type-I core assertion even at boundary points of the transition. Together with the preceding type-II assertion, this completes Theorem 1.1. ∎

The type-I cutoffs in (6.4) have transverse radius \(R_1\). The type-II cutoffs and ordered components in (6.1), (1.5) have transverse radius at least \(R_2\). Their symbol and motion estimates must respect this distinction. Those estimates are subsequent results.

8. Why centers need an interior margin

The restriction of all admissible type-II centers to the exact inner \(\Omega\) is false under the given hypotheses. Fix \(\rho\geq1\), take \(k=1\), \(0<\kappa<1/2\), and set

The transverse gradient is \(a\), the time bracket is the constant \(a^2=\lambda^{-2}\), and all higher derivatives vanish. The original value and derivative bounds hold on the original domain with uniform constants, since \(R/a\to0\). The positive time slope satisfies the sign orientation. The point \(p\) is strictly inside \(\Omega\). For small \(\lambda\),

so it is in case II.

At any case-II center \(c=(t_c,z_c,\xi_c)\) in \(\Omega\), the canonical map can be the identity and

In particular \(|at_c+\xi_c-b|<R\). The cancellation frequency at the center time is within transverse distance \(R\), hence inside its frequency interval of radius \(R_2\). At that point the bracket scale is exactly \(m\). Admissibility with its fixed lower constant \(\gamma\) therefore requires

If the prescribed neighborhood at \(c\) contained \(p\), its time interval would give \(|t_c|<M_c^{-1}\leq m^{-1}\), so \(|at_c|<\sqrt\rho\). But \(\xi_c<a/2\), because \(c\in\Omega\). Thus

For \(R/4>\sqrt\rho(1+1/\gamma)\), equations (8.3) and (8.5) imply \(M_c>m/\gamma\), contradicting (8.4). No admissible type-II neighborhood centered in the exact \(\Omega\) contains \(p\), although \(p\) is case II.

Allowing centers in \(\Omega_+\) repairs this boundary restriction. The nearby-center distance is \(O(R)=o(\lambda^{-1})\), so every required center is in that larger fixed interior domain. All cells still lie inside the original symbol domain by (2.1)–(2.2). No localization region or differential hypothesis is weakened.

9. Graded exercises with solutions

Exercise 1 — basic. In transverse dimension two, compute the volume in (5.1) directly. Explain why the canonical map does not introduce a Jacobian factor.

Solution. The time, distinguished spatial and frequency intervals have lengths \(2h/M\), \(2h\ell\) and \(2hR_2\). Their product is \(8h^3\ell R_2/M\), agreeing with (5.1) for \(d=2,\nu_0=1\). A symplectic map preserves the transverse symplectic volume and has determinant one. It is independent of time, so its extension with the unchanged time variable has the same determinant.

Exercise 2 — intermediate. Why would using \(\mu(p)\), rather than \(\mathfrak m_D(p)\), fail to prove (3.4) by the argument in Section 3?

Solution. A point in a type-II cell can have \(|\xi|\) of order \(R_2\), while \(A_2R_2\) tends to infinity. The coefficient estimate then permits \(\mu(p)\) much larger than the center scale. The transverse minimum can move to \(\xi=0\) within a fixed \(R_2\) physical displacement, giving the uniform upper bound in (3.2). The enlarged-cell lower bound prevents that minimum from being too small. Both bounds concern the same chart-independent function.

Exercise 3 — intermediate. Show why a large scalar value of one time-gradient is not needed at every time in the tube argument. What replaces that requirement?

Solution. The normalized coefficient is a polynomial of bounded degree with a nonzero selected derivative at zero. At any bounded normalized time, the maximum of its entire derivative jet remains bounded below by reverse finite Taylor translation. One may choose a different index \(j\) at that fixed time. Slow spatial variation then keeps that chosen field dominant across the tube at that time. The transverse flow itself holds time fixed, so the index need not stay dominant at other times.

Exercise 4 — intermediate. Suppose selected cells have disjoint \(\varepsilon\)-shrinks, comparable \(M_j,\ell_j\) at a common point, and all shrinks fit in a fixed enlargement of one reference cell. Derive a multiplicity bound from (5.1).

Solution. If the comparison constant is \(C\), every small cell has volume at least \(\varepsilon^{d+1}C^{-2}\) times the reference volume: its \(\ell_j\) is at least \(\ell_0/C\), and its \(M_j\) at most \(CM_0\). An enlargement by \(H\) has volume \(H^{d+1}\) times the reference volume. Disjointness therefore bounds the count by \(C^2(H/\varepsilon)^{d+1}\). All constants are fixed independently of the parameters.

Exercise 5 — advanced. Compute the ordered components for three numbers \(0\leq\psi_1,\psi_2,\psi_3\leq1\). Check their sum when all three are \(1/2\), and when \(\psi_2=1\).

Solution. The components are \(\psi_1\), \(\psi_2(1-\psi_1)\), and \(\psi_3(1-\psi_1)(1-\psi_2)\). Their sum is \(1-(1-\psi_1)(1-\psi_2)(1-\psi_3)\). With all three equal to one half it is \(7/8\), so this family does not always form a partition summing to one. With \(\psi_2=1\), the sum is one and the third component is zero. The equality region in Theorem 1.1 keeps precisely this distinction.

Exercise 6 — advanced. Why does covering \(\omega\) by type-I cores also cover the points of \(\operatorname{supp}d\Psi\) on its relative boundary? Would the same closure argument work for an arbitrary infinite family of closed cores?

Solution. A finite union of closed cores is closed and contains the relative closure of \(\omega\). Equation (7.2) places all transition-support points in that closure. An arbitrary infinite union of closed sets need not be closed: the singleton sets \(\{1/j\}\) cover a sequence converging to zero but do not contain zero. Finiteness, or an appropriate local finiteness argument, is needed.

Exercise 7 — advanced. In (8.1), keep \(\rho\) and \(\gamma>0\) fixed. Identify all three inequalities needed to rule out an admissible center in \(\Omega\) whose cell contains \(p\). Explain why enlarging the center domain does not enlarge the prescribed localization region.

Solution. Case II puts the center's cancellation frequency within \(R\), hence in its cell, giving \(M_c\leq m/\gamma\) by admissibility. Containment of \(p\) in the time interval gives \(|at_c|<\sqrt\rho\). The exact inner-domain center restriction gives \(\xi_c<a/2\), hence (8.5), incompatible with the first inequality once \(R/4>\sqrt\rho(1+1/\gamma)\). The repaired construction uses centers in \(\Omega_+\) only to supply nearby cells and boundary margins. Its coverage and transition assertions are still stated on \(\Omega\), and all cells remain inside the original larger domain.

References

The adaptive neighborhoods, cover and ordered partition are developed in Hörmander, The Analysis of Linear Partial Differential Operators IV, Chapter 27, Section 27.4, following Proposition 27.4.10 and preceding Lemma 27.4.11, especially formulas (27.4.50)–(27.4.53). The proof above supplies the transverse-minimum, shared-flow, shrink-disjointness and volume arguments. It repairs the exact inner-domain center restriction by an explicit boundary-margin construction and counterexample. The type-II partition and the separate type-I core cover are retained; their later symbol and motion estimates are not assumed here.

Written by GPT-6.1 Sol (OpenAI), at Ultra reasoning effort, September 2026. Self-checked by the writing AI. Public domain (CC0).