Elliptic boundary regularity from rough data

A solution may start far below the Sobolev order at which its normal traces are available by an ordinary trace theorem. The first part of the proof raises its normal order while preserving the sum of its two regularity indices. After reaching the trace range, the actual volume and boundary inverse raises regularity at a fixed tangential index. We keep one final tester and one final collar throughout this argument.

This independent exposition is CC0. The mathematical source is Lars Hörmander, The Analysis of Linear Partial Differential Operators III, Springer 2007, ISBN 978-3-540-49938-1, particularly the elliptic boundary calculus of Section 20.1 and its use in Section 24.2. The existing AN-03 Boundary wave fronts for elliptic systems, Sections 1–9, supplied the source comparison. The proofs below use the exact earlier programme components recorded in the proof map, including the full volume parametrix VB. Earlier linked components retain their own licences.

1. The domain and the two conclusions

Use the fixed inward coordinate $r\ge0$, $D_r=-i\partial_r$, and the intrinsic supported representatives of the dual conormal class $\mathcal A'$. The earlier CNF and BW components prove the actual pairing, injectivity of interior restriction and the compressed wavefront set. Write

First take the generalized elliptic cylinder and complementing boundary system of VB:P1:

Second, for an original differential $P$ with noncharacteristic boundary, only local ellipticity at one nonzero tangential covector is required. Its boundary measurements may have redundant output coordinates. We will prove the local inclusion using injectivity, not require surjectivity of those original measurements. This second assertion supplies the elliptic part of the boundary-reflection lesson.

2. Intrinsic normal derivatives and their traces

We first prove the boundary-supported correction lemma. If $h$ is a boundary distribution, let $Th=h\otimes\delta_0$, with the density and bundle factors in CNF:J2. Then

At a boundary covector outside the equator, choose the full symbol of $A$ supported in a compressed cone disjoint from the equator, and elliptic at that covector. Its boundary symbol is residual, so $A_\partial h$ is smooth. At an equatorial covector outside $\operatorname{WF}(h)$, choose the full symbol supported over a smaller regular tangential cone. The full ordinary pseudolocal theorem and its residual terms make $A_\partial h$ smooth there; use an output cutoff inside the regular base region. In either case $T A_\partial h$ is conormal, with its exact delta order from CNF:S2. Thus $A$ is a regularizing tester in the defining compressed wavefront intersection. Proper realization and conic cutoffs are GB:C6 and E2–E3. This proves EW4, including the pure compressed normal directions.

For $u\in\mathcal A'$, its intrinsic normal derivative is

The intrinsic trace agrees with the mixed Sobolev trace whenever the latter exists. Here is the needed identification. For a compact tangential test $\phi$ and $u\in\overline H_{(s,t)}$, $s>k+1/2$, choose a full extension. Fourier Cauchy–Schwarz in the normal frequency uses the finite integral of $|\kappa|^{2k}(1+|\eta|^2+\kappa^2)^{-s}$; the arbitrary tangential decay of $\widehat\phi$ absorbs the remaining fixed powers. Dominated convergence gives continuity at zero of $\langle D_r^ku(r),\phi\rangle$, independently of the extension, as in MH:TR1–TR3. Pairing the intrinsic representative with the positive approximate delta from CNF:J2 tends to this value: its tangential approximate identity converges on $\phi$, and its positive normal support shrinks to zero. The same approximants converge in the conormal topology that defines the intrinsic pairing. The two limits agree. Thus later Sobolev estimates concern the same $\gamma_k$ as EW2, not newly assigned boundary values.

3. One collar for all smoothing remainders

Fix compact input and output base sets for the local proper operators. After multiplying by a larger fixed cutoff, regard them inside one compact boundary manifold. For $u\in\mathcal N$, there is a collar $0\le r<\delta$ on these compact sets with no pure-normal interior wavefront covectors of $u$. Indeed such covectors at distances tending to zero, normalized in the compressed cosphere, would have a convergent subsequence with a pure compressed normal boundary limit. Closedness and EW1 exclude that limit. The compactness and coordinate law used here are BW:W1–W2 and GB:G6. By EW6 the same collar excludes pure-normal interior covectors of every intrinsic normal derivative of $u$.

Let $R(r)$ be any smooth tangentially smoothing family with the specified compact proper input and output supports. Then, on this one collar,

The output thus has no wavefront anywhere in this collar. Multiply it by a normal cutoff compactly supported in the collar. It remains in $\mathcal A'$; BW:S1–S2, using the full conormal intersection theorem, makes it smooth up to zero. This proves EW7. The collar was chosen from the wavefront set of the original input, not from $R$ or $k$. Constants may depend on these operators and derivatives. This distinction allows arbitrarily long finite arguments on one fixed final collar.

Tangential operators of order $d$ act on every real mixed restriction space by

4. Localizers with the correct bundle domains

Choose a fixed base-cone neighborhood of $q_0\in T^*Y\setminus0$ where $f$ and every $g_j$ are microlocally smooth. BW:S5 supplies a tangential tester $T_f$, elliptic on a smaller such neighborhood, for which $T_ff$ is smooth on a fixed collar. Its construction permits a symbol independent of $r$ near zero. The ordinary boundary tester gives corresponding operators for the finitely many $g_j$. Shrink the base-cone neighborhood into all their elliptic regions, and choose the common collar of EW7 for $u,f$. All later normal cutoffs have support strictly inside this fixed collar.

For any prescribed finite number of steps, choose nested base-cone regions and nested normal cutoffs between two fixed regions. The innermost operator $A_0^E$ is fixed, scalar in local input frames and elliptic at $q_0$. Each outer $A_{j+1}^E$ is elliptic on a neighborhood of the full microsupport of $A_j^E$, including its normal derivatives, and its normal scalar factor is one on their normal support. Construct each tangential kernel with one fixed compact proper-support bound. Smooth conic cutoffs and the full proper parametrices K:K3–K4 and P2:OP4–OP6 give these operators. At zero their positive normal derivatives vanish. Write

Every $A_j^Ff$ is smooth on the fixed working collar. To see uniformity of the collar, construct the full parameter-dependent tangential parametrix for the one fixed $T_f$ on the closure of the chosen base-cone region. Finite symbolic recursion followed by the complete asymptotic sum K:K1–K3 gives $A_j^F=Q_jT_f+R_j$, with $R_j$ tangentially smoothing and all its normal derivatives so. The construction includes normal derivatives among its compact parameter seminorms, so the same full symbol identities hold throughout the normal support. The first term acts on the fixed smooth output $T_ff$; EW7 makes the second smooth on the same collar for every $j$. Similarly $A_j^{G_l}g_l$ is smooth by the ordinary boundary parametrix and smooth-kernel action. No order-one principal-symbol comparison is substituted for these full remainders.

For $A=A_j^E$, $A^F=A_j^F$, the exact normal expansion is

We now justify insertion of the outer localizer in every coefficient of EW11–EW12. The complete symbolic product shows that these coefficients have full microsupport inside that of the inner localizer, modulo tangentially smoothing families. A full outer parametrix therefore factors each as $L_kA_{j+1}^E+R_k$, with $L_k$ of the same order and $R_k$ tangentially smoothing. Choose the factorization with normal output support inside the inner coefficient support. On that support the outer normal cutoff is identically one and its positive derivatives are zero. Hence $L_kA_{j+1}^ED_r^ku=L_kD_r^k(A_{j+1}^Eu)$ there, exactly. In the boundary formula its zero-order normal plateau similarly gives $A_{j+1}^E(0)\gamma_ku=\gamma_k(A_{j+1}^Eu)$. All residual terms on intrinsic jets are smooth on the fixed collar by EW7, and all boundary residuals are smooth on distributions. This proves the actual insertion statement, including the terms outside the principal symbol.

5. The first stage reaches the trace range

A fixed compact localization of the supported representative of $u$ has finite distribution order. Testing it against the compactly cut-off exponential bounds its Fourier transform by $C\langle(\eta,\kappa)\rangle^L$. Choose one real $s_0<-L-(\dim Y+1)/2$, and set $t_0=0$. The Fourier integral is then finite. Every outer order-zero tangential localizer belongs to $\overline H_{(s_0,t_0)}$ by EW8. The indices $s_0,t_0$ are fixed as the number of nested operators increases; only norm constants can change.

Assume $w=A_{j+1}u\in\overline H_{(s,t)}$, and set $v=A_ju$. Outer factorization gives $v\in\overline H_{(s,t)}$. From EW11, its insertion just proved and the smooth localized forcing,

For a lower original term, the same calculation gives

Repeat EW16 a fixed finite $J$ times until $s_1=s_0+J\ge m$, with $t_1=t_0-J$. The sum $s_1+t_1=s_0+t_0$ is unchanged. Both $J$ and $t_1$ are independent of how many later regularity gains are requested.

6. The second stage and the generalized equality

Now suppose the outer localized input is in $\overline H_{(s,t_1)}$, $s\ge m$. EW13 gives forcing in $\overline H_{(s+1-m,t_1)}$. The mixed traces of the outer input exist and, by Section 2, are its intrinsic traces. EW12 and the proved outer insertion give

For every integer $L\ge0$, insert enough intermediate cutoffs between the same fixed outer and inner supports to make $J$ first-stage steps and then $L$ second-stage steps. The initial order, $J$, $t_1$, final $A_0$, and working collar are fixed. The common smoothing argument in Section 3 and the one fixed forcing tester in Section 4 ensure that the same collar works for every finite family. Hence

For the forward inclusion, EW6 and BW:T6–T12 show that every $P_kD_r^ku$ lies in $\mathcal N$ and has boundary wavefront contained in that of $u$. The finite-sum theorem proves the same for $f$. Each intrinsic $\gamma_ku$ has the inclusion EW6, and ordinary tangential pseudolocality applies to every actual $B_{jk}$. Therefore the forcing and all boundary-data wavefronts are contained in $\operatorname{WF}_b(u)|_Y$. Together with EW20 this proves EW3, with separate input and output bundles. In tangential dimension zero the nonzero tangential cotangent bundle is empty; EW1 and the compact-cosphere/smoothness argument already give local smoothness, and the equality is vacuous as a statement about those covectors.

7. The locally elliptic differential problem

Let now $P$ be the original differential operator of order $m$, with invertible highest normal coefficient, on an arbitrary boundary chart. Its homogeneous principal normal polynomial is

Fix $q_0=(y_0,\eta_0)$ outside EW22 and the data wavefronts, and choose local frames. Let $d_+=\dim V^+_{q_0}$. The injective matrix $\beta$ has a nonzero square minor of size $d_+$, by finite Gaussian elimination QF:A0. Retain those actual output coordinates and their original row orders. The resulting $\beta^0:V^+_{q_0}\to\mathbb C^{d_+}$ is bijective. For $d_+=0$, retain no rows; the map of zero spaces is bijective. All discarded equations remain original data equations, but are unnecessary for regularity.

We construct the required global generalized auxiliary problem explicitly. Rescale the tangential chart to a small cube inside a flat torus $\mathbb T^d$, and take trivial bundles of the original input/output ranks there. Choose a smooth base and normal cutoff $\theta(y,r)$, equal to one on a smaller cube and collar about $(y_0,0)$, supported strictly inside the original coordinate region. Choose a degree-zero angular cutoff $\psi(\eta)$, equal to one in a narrow cone about $\eta_0$, supported in a slightly larger narrow cone, with a fixed smooth low-frequency completion. Put $e_0=\eta_0/|\eta_0|$. If $a_k(y,r,\eta)$ and $c_{jk}(y,\eta)$ are the complete original coefficient symbols, choose smooth classical symbols $a_k^0(\eta),c_{jk}^0(\eta)$ whose high-frequency terms are

Quantize these periodic-base symbols as actual tangential operators on the torus, with the following local construction. In the Euclidean lifted cube use their inverse Fourier kernels in the tangential difference, multiply by a fixed smooth difference cutoff equal to one near zero and supported at distance less than one third of the torus period, and sum the translates of that cut-off kernel. This is a well-defined distributional kernel on the torus: on each compact lifted coordinate pair only finitely many translates occur. Its near-diagonal left symbol is the specified symbol modulo a tangentially smoothing symbol, by the complete off-diagonal integrations and amplitude reduction P2:OP4–OP5. Those integrations include every normal-parameter derivative, since the coefficient bounds are uniform. Away from the diagonal its kernel is smooth tangentially. Thus its full local agreement with the original operator is modulo smooth tangential kernels, with every normal power retained. Quantize $\widetilde M$ as exact multiplication. This produces the complete normal polynomial $\widetilde P$ and the retained boundary rows $\widetilde B$ on a compact boundary manifold, of exactly the class used in GC and VB.

We prove their ellipticity uniformly. At $|\eta|=1$, let $p^0(\kappa)=p(y_0,0,e_0,\kappa)$. Choose $L\ge1$ so large that, for $|\kappa|\ge L$, the sum of lower terms in $M_0^{-1}p^0(\kappa)\kappa^{-m}$ has norm at most $1/4$. On the compact interval $[-L,L]$, the minimum singular value $c_0$ of $p^0$ is positive: otherwise a converging sequence of unit vectors and real parameters would give a nonzero null vector. Shrink the support of $\theta$ and the angular support of $\psi$ until all normalized original lower coefficients there, and the original $M$ on the base support, are close enough to their frozen values. On $[-L,L]$, the sum of their differences times $|\kappa|^k$ is then less than $c_0/2$; convex multiplication by $0\le\theta\psi\le1$ preserves that estimate. On the tails, make $\|M_0^{-1}(\widetilde M-M_0)\|<1/4$ and the lower normalized sum less than $1/2$, increasing $L$ once if needed. The convergent finite-matrix Neumann argument gives a bound

The stable measurement remains bijective. Normalize Cauchy component $k$ by $|\eta|^{-k}$ and measurement component $j$ by $|\eta|^{-m_j}$, only as pointwise homogeneous matrix weights. The normalized companion matrix and its fixed-contour projection vary continuously with the finitely many normalized coefficients, by NP:P1 and G1. The original spectrum has a positive distance from the real axis; compact contour resolvent bounds and the exact resolvent identity preserve that contour after the same small perturbations. If $q^0$ and $\widetilde q$ are the stable projections, shrink until $\|\widetilde q-q^0\|<1$. The map $\widetilde q:\operatorname{ran}q^0\to\operatorname{ran}\widetilde q$ is injective, since a vector in its kernel satisfies $\|v\|\le\|q^0-\widetilde q\|\|v\|$. The reverse restriction of $q^0$ is also injective, so the two finite dimensions agree and this map is an isomorphism. On the fixed space $\operatorname{ran}q^0$, the retained normalized map $\widetilde b\widetilde q$ is arbitrarily close to $\beta^0$. If its difference has norm less than half the positive minimum singular value of $\beta^0$, it is injective, and equality of dimensions makes it bijective. In rank zero this statement is immediate. This proves the stable complementing condition at every nonzero tangential covector of the auxiliary problem, with all original row weights retained.

Choose cutoffs equal to one on a smaller neighborhood of $q_0$ and extend the localized original $u$ by zero to the torus collar. It belongs to $\mathcal N$: smooth multiplication and the original coordinate/extension identifications preserve the intrinsic class and its local wavefront condition. Across the artificial tangential edge it is identically zero on a neighborhood. Its normal cutoff is smooth and equals one near the boundary. Let this extension be $\widetilde u$. The auxiliary forcing $\widetilde f=\widetilde P\widetilde u$ and retained measurements $\widetilde g=\widetilde B\widetilde u$ are then actual data of the globally elliptic generalized system. By EW6 and tangential action they belong to the required intrinsic classes.

Choose an order-zero tangential tester $A$ with full microsupport in the smaller region where $\theta=\psi=1$, and choose its actual proper kernel with both base variables inside the region where the leading normal coefficients agree. Complete symbol agreement and the full composition formula give

Apply EW3 to the constructed auxiliary system and $\widetilde u$. It gives $q_0\notin\operatorname{WF}_b(\widetilde u)$, hence the same for the original $u$, which agrees near that point. This proves EW23. Coordinate invariance follows from the exact compressed law GB:G6 and the ordinary matrix/frame transformations; the conclusion is local on the original manifold. Forward pseudolocality proved in Section 6 also gives equality between solution and data wavefronts on the complement of $\operatorname{Char}(P;B)$, including all discarded original measurements.

8. Checks, examples and the reflection interface

1. The delta correction for a smooth boundary value. Let $u(r)=e^{-r}$ on $r\ge0$, represented by $He^{-r}$. Then

2. Why the output localizer depends on the normal variable. Take two trivial rank-two bundles, $M(y,r)=\left(\begin{smallmatrix}1&r h(y)\\0&1\end{smallmatrix}\right)$, and any scalar tangential operator $A$. Direct matrix multiplication gives

3. The complete cubic commutator. Write $P=P_3D_r^3+P_2D_r^2+P_1D_r+P_0$, $P_3=M$, and $A^F=MAM^{-1}$. Then

4. A rough start and a fixed final tester. Take $m=2$, $s_0=-7/2$, $t_0=0$. Six applications of EW16 give $(s_1,t_1)=(5/2,-6)$; the total index remains $-7/2$. Seven further applications of EW19 give $(19/2,-6)$, which embeds into ordinary $H^{7/2}$. For any larger target, keep the same final tester and $t_1=-6$, and insert more intermediate cutoffs. Merely proving smoothness on collars whose widths tend to zero would not give this statement. The common collar in EW7 and fixed initial Fourier order are exactly what make the construction valid.

5. Redundant equations and the elliptic reflection region. For the scalar frozen polynomial $\kappa^2+a^2$, $a>0$, the bounded solutions are $v(r)=c e^{-ar}$, with Cauchy vector $(c,ia c)$. The two measurements $v(0)$ and $D_rv(0)$ give

6. A convex mixture needs the small-neighborhood proof. Scalar invertible symbols $1$ and $-1$ have singular midpoint zero. For matrices the same example times the identity suffices. Therefore ellipticity of two endpoint polynomials alone would not justify EW25. Our proof uses uniform proximity to one frozen polynomial on a compact real-frequency interval and a separate leading-coefficient bound on the tails. For a scalar example $p^0(\kappa)=\kappa^2+1$, perturb it to $\kappa^2+1+e$, with $|e|<1/2$. Every convex mixture has real part at least $\kappa^2+1/2$ for real $\kappa$, so it is uniformly elliptic. These are the quantitative mechanisms used in EW26, not a general convexity claim about invertible matrices.

9. The two gains on one neighbourhood

The first panel plots the exact index sequence in Exercise 4. Diagonal arrows keep $s+t$ fixed until $s\ge m$; horizontal arrows then increase $s$ at fixed $t$. The second panel distinguishes the original possibly redundant boundary map from the selected invertible minor and shows the fixed final collar inside the common input collar. It represents the support nesting, not a change of normal coordinate.

The complete proof covers intrinsic jets, every full commutator, the fixed-collar smoothing argument, all real initial indices, both stages of the bootstrap and the local differential theorem with redundant data. Its local auxiliary operator is explicitly constructed on a compact boundary; no global ellipticity of the original problem has been presumed. The companion establishes the elliptic input for U051. Reflection and hyperbolic or glancing propagation require their own hypotheses and proofs.