Iterated tests and frequency packet regularity

For ordinary symbols, the fixed dyadic $L^2$ endpoint and all iterated tangential tests recognize the precise Lagrangian order. For general symbols with $1/2<\rho<1$, a frequency packet can occupy a much smaller portion of an annulus. Its annular $L^2$ size can stay fixed while its height grows. That distinction matters in an intrinsic criterion.

This lesson proves three statements. General-symbol operators of order $2\rho-1$ whose principal symbols vanish on the Lagrangian preserve $I_\rho^m$, so all their finite words satisfy the usual fixed endpoint. The reverse implication, with precisely the fixed endpoint and test class specified below, is false: a compactly supported packet sum gives a counterexample for every $1/2<\rho<1$ and every real $m$. An exact criterion instead uses frequency cells of radius $\langle\xi\rangle^\rho$ and all weighted normal derivatives. We prove it in a frequency graph and then prove its intrinsic, microlocal meaning.

The primary antecedent is Lars Hörmander, The Analysis of Linear Partial Differential Operators IV, Springer 2009 eBook, ISBN 978-3-642-00136-9, §25.1, Definition 25.1.1 and Proposition 25.1.3 (printed 4–6), and the proposed general-symbol intrinsic extension on printed 16. Volume III, 2007 eBook, ISBN 978-3-540-49938-1, printed 98–100, explicitly identifies this endpoint as the Besov space and connects its annular bound to the conormal definition. The two scaled cutoff families on IV16 are checked explicitly below. The counterexample concerns the literal fixed-endpoint interpretation stated here; it is an independently proved source qualification, not a claim that a publisher has issued an erratum. The phase-based general-symbol class remains the class already constructed in the programme.

Exact earlier proofs are general stationary phase, full frequency reduction and invariant symbols, kernel maps and the general calculus, finite-seminorm operator bounds and graph inverses, the ordinary intrinsic theorem and proper phase regularization, dyadic and local Sobolev estimates, Fourier theory, and homogeneous symbol operations. The proof map records exact sources and locators. The 92 transitive human foundation providers remain external.

1. State the endpoint and the vanishing condition exactly

H0. Keep the three orders separate. Fix

For compactly supported distributions, the fixed endpoint $B^{-M}_{2,\infty}$ means that the low-frequency block is in $L^2$ and

Let $\mathcal J_\rho(\Lambda)$ consist of properly supported $L\in\Psi_{\rho,\delta}^{\epsilon}$ whose principal restriction vanishes. Precisely, on each working graph patch choose a full left symbol $l$ with

Call the fixed-endpoint condition

2. Every admissible operator preserves the phase class

A0. Prove preservation before iterating. Choose base coordinates making the Lagrangian a frequency graph,

For $L\in\mathcal J_\rho(\Lambda)$, the full symbol $l$ has the Taylor division

Act first in full left quantization on the inverse Fourier representation of $e^{-iH}v$. Fourier inversion in the intermediate variable, or regularized Schwartz approximation followed by the proved distribution action, gives its amplitude $l(x,\xi)v(\xi)$ with phase $x\cdot\xi-H(\xi)$. Since

Thus $L:I_\rho^m\to I_\rho^m$ locally. In particular order-zero multiplication and pseudodifferential localization preserve the class. The proof works with matrices in their original order, because $H$ and the phase are scalar.

A1. Deduce the actual endpoint and wavefront consequences. Iterating A0 gives the same class for every finite word. On a compact output the reduced symbol bound gives

Conversely, (IP5) alone does imply $\operatorname{WF}(u)\subset\Lambda$. At a covector outside the closed $\Lambda$, choose a scalar homogeneous angular/base cutoff $a$, supported away from $\Lambda$, equal to one on a smaller cone. A proper operator $L$ with principal symbol $a\langle\xi\rangle^\epsilon$ is admissible and elliptic there. Its $N$-th power is elliptic of order $N\epsilon$. From (IP5), $L^Nu\in H^{-M-\eta}_{\rm loc}$. The general-symbol elliptic parametrix and all real Sobolev bounds GRC:E0 and GRC:S0, applied to the identity graph, recover microlocal regularity $H^{N\epsilon-M-\eta}$ for $u$. Arbitrarily large $N$ prove the claim. The same argument works componentwise for finite bundles. Wavefront containment does not determine the precise symbol order.

3. Check both printed scaled cutoff families

T0. Both families really are admissible. Work first with the point conormal $\Lambda=T_0^*\mathbb R^n\setminus0$, so $H=0$. Choose $\chi\in C_c^\infty(\mathbb R^n)$ equal to one near zero, put $r=\langle\xi\rangle$, and consider

For the graph (IP6), replace $x$ by $x-H'(\xi)$. A frequency derivative of this extra term costs $r^{-1}$. Inside the scaled cutoff it costs $r^{\delta-1}=r^{-\rho}$; all further derivatives obey the same homogeneous chain rule. The support still gives $|x-H'|\leq Cr^{-\delta}$, and all estimates just proved persist. Thus the graph versions of both source families belong to $\mathcal J_\rho(\Lambda)$. Their proper quantizations differ from their full left forms only by smooth kernels locally.

The first family measures $r^\rho x$ inside a tube of base width $r^{-\delta}$. The second is elliptic of order $\epsilon$ beyond that tube. A packet of physical width $r^{-\rho}$ is much narrower than the tube when $\rho>\delta$. Its normal moments therefore remain bounded under the first family, and the second family meets only a rapidly decreasing tail. We now prove this for every admissible operator, not just these two examples.

4. A uniform scaled operator lemma

S0. Control an arbitrary test on the whole packet. Fix $1/2<\rho<1$, a compact base set containing zero, and a full left symbol

Write $l(x,\xi)=s(\xi)+\sum_jx_jc_j(x,\xi)$, where $s\in S_\rho^0$, $c_j\in S_{\rho,\delta}^\rho$, by Taylor division at zero. Put $\xi=Re_1+R^\rho\zeta$, $r_\xi=\langle\xi\rangle$. When $r_\xi\geq R/2$, a derivative of the $s$ term has bound

For $f\in\mathcal S$, differentiate $(2\pi)^{-n}\int e^{iz\cdot\zeta}l_R(z,\zeta)\widehat f(\zeta)\,d\zeta$ any fixed number of times in $z$. Transfer $(1-\Delta_\zeta)^K$ from the exponential to the remaining factors and divide by $(1+|z|^2)^K$. Formula (IP15) and sufficiently many Schwartz bounds of $\widehat f$ make every resulting integral absolutely bounded, uniformly in $R$. Choosing $2K$ larger than the requested $z$ weight plus one proves the asserted uniform Schwartz bound. Fourier continuity on $\mathcal S$ is the earlier complete proof P3:L1–L3.

S1. Retain proper kernels and smooth errors. A fixed proper test, restricted between the compact input and output sets needed for a finite word, equals the full left form in S0 plus a smooth compactly supported kernel. That residual also maps the scaled family in (IP12) to a uniformly bounded Schwartz family after conjugation. In fact, for a smooth compact kernel $k$, its conjugated output is

Initial compact multiplication has scaled input $\chi_0(R^{-\rho}z)f(z)$, a uniformly bounded Schwartz family. Properness supplies a compact output set at each step of a fixed word. Apply S0–S1 successively on those sets. We obtain: for every fixed finite admissible word $W$, every uniformly Schwartz-bounded input family $f_R$, and any scalar $d_R$,

5. A compactly supported counterexample at every order

C0. Construct the distribution, not just a formal packet. Fix $1/2<\rho<1$, any real $m$, and $n\geq1$. Choose a nonzero $\Phi\in C_c^\infty(B(0,1))$ with $\Phi(0)>0$, and let $\psi=\mathcal F^{-1}\Phi$. Choose $\chi_0\in C_c^\infty$ equal to one near zero. Let

The series is a tempered distribution: pairing the compact Fourier packet (IP20) against a Schwartz test gives an arbitrarily negative power of $R_j$ after choosing enough test decay, since its support stays comparable to $R_j$. The sum converges absolutely in that pairing. Multiplication by $\chi_0$ makes its support compact. Proper pseudodifferential continuity on distributions permits every fixed admissible word to pass through the distributional sum.

Away from zero, the series and all its base derivatives converge smoothly. On a compact set with $|x|\geq c>0$, the Schwartz factor $\psi(R_j^\rho x)$ defeats the fixed powers from modulation, differentiation and the coefficient. Thus its possible singular support is only zero. Its Fourier transforms outside a small cone about $e_1$ decrease rapidly by the same packet tail estimates proved next. In particular it has the correct point-conormal wavefront containment.

6. Every finite admissible word has the fixed endpoint

C1. Prove both lacunary tails. By S1, for every fixed word $W$ the $j$-th output has Fourier transform

For $R_j<S/4$, $|\xi-R_je_1|\geq cS$ on that annulus. For $R_j>4S$, the distance is at least $cR_j$. By the uniform Schwartz bound and the volume of the annulus, respectively,

7. Its actual Lagrangian order is higher

C2. Check the height after compact localization. Put $\psi_j(z)=\chi_0(R_j^{-\rho}z)\psi(z)$. This family is uniformly Schwartz and converges to $\psi$ in every Schwartz seminorm faster than any fixed negative power of $R_j$. Indeed $\chi_0(R_j^{-\rho}z)-1$ is supported where $|z|\geq cR_j^\rho$, and every derivative and weighted tail of the Schwartz $\psi$ has arbitrary decay there.

At the center $\xi=R_je_1$, its own Fourier contribution is $R_j^a(\Phi(0)+o(1))$. For the other packets, the uniform Schwartz bound gives

In fact its order is sharp. Differentiate (IP21) for the empty word. For any $\alpha$ replace $a$ in the preceding tail bounds by $a-\rho|\alpha|$, and use the corresponding uniform Schwartz derivative bound. On frequencies comparable to a center there are only finitely many packets and each derivative is bounded by $Cr^{a-\rho|\alpha|}$; the two geometric tails obey the same bound after increasing $N$. Hence $\widehat u\in S_\rho^a$. Formula (IP25) excludes every smaller order. Therefore

8. An exact frequency-cell criterion

P0. Test the scale on which a symbol can vary. Return to $1/2<\rho\leq1$, a frequency graph (IP6), and compactly supported $u$. The earlier proper phase regularization IR:R2 constructs a real smooth $h\in S_{1,0}^1$, with compactly supported inverse Fourier transform, for which $h-H$ is rapidly decreasing at high frequency on the working cone. Put

The exact criterion is

P1. Prove both implications with no loss of order. If $v=e^{iH}\widehat u\in S_\rho^q$, then $v_h\in S_\rho^q$, since $e^{i(h-H)}-1$ is rapidly decreasing. On a cell the radii $\langle\xi\rangle$ and $R$ are comparable. The pointwise symbol bound, the cell volume $O(R^{\rho n})$ and (IP29) give (IP30).

Conversely, at each center set

9. Intrinsic and microlocal meaning of the replacement

P2. Prove independence of the choices by the established class. For a closed conic Lagrangian $\Lambda\subset T^*X\setminus0$, the following is an intrinsic criterion. Require $\operatorname{WF}(u)\subset\Lambda$. Near each point of $\Lambda$, choose adapted base coordinates in which it is a frequency graph, a bundle frame, and a properly supported ordinary order-zero microlocal cutoff $A$, elliptic at the point and with compact output support and microsupport inside that graph patch. Require (IP30) for $Au$ in the corresponding interior cone, with $q=m-n/4$. Existence of these choices at every point is sufficient.

If $u\in I_\rho^m$, wavefront containment follows from its general-symbol phase representation. A0 for order-zero operators gives $Au\in I_\rho^m$. P1 proves the cell estimates. Conversely the cell estimates give $Au\in I_\rho^m$ microlocally by P1. A proper ordinary elliptic parametrix $B$ for $A$ yields $u=BAu$ modulo a microlocally smooth distribution on a smaller cone. A0 preserves the phase class under $B$; hence $u$ has the required local phase class there. Away from $\Lambda$ it is smooth by the wavefront assumption. A finite partition over each compact cosphere, followed by the exact phase-class assembly RSP:R0–RSP:R1, proves $u\in I_\rho^m(X,\Lambda)$.

This equivalence also proves invariance under adapted charts, bundle frames, proper regularizations $h$, interior cone margins and cell radius $c$: membership gives the estimates for every such choice. More directly, two admissible $h$'s differ by a rapidly decreasing high-frequency symbol; their $Q$ words differ on compact distributions by properly localized smooth outputs, which satisfy every cell estimate. A smaller or larger fixed cell radius is handled by a finite cover and the same local Sobolev estimate. No claim of invariance of an unrestricted graph-dependent Fourier norm is needed.

The replacement is an exact weighted-normal, frequency-cell characterization. It does not silently identify (IP30) with the weaker annular condition (IP5). In the ordinary case, a cell of radius $cR$ covers a fixed part of an annulus, and the ordinary balanced-word argument recovers the earlier intrinsic criterion.

10. Three complete solutions

X1. Locate the missing height factor. Set $n=2,\rho=3/4,m=0$. Compute $q,M,a$, the physical and frequency scales, and the fixed-endpoint and cell $L^2$ powers for the counterexample. Decide its precise order.

Solution. Here $\delta=1/4,\epsilon=1/2$, $q=-1/2$, $M=1/2$, and $a=-1/4$. Each Fourier packet has radius $R^{3/4}$, height $R^{-1/4}$ and $L^2$ size $R^{-1/4+3/4}=R^{1/2}$, exactly the fixed endpoint. The physical scale is $R^{-3/4}$, whereas the cutoff tube has width $R^{-1/4}$. The exact cell criterion for order $m=0$, with $\alpha=0$, would allow only $R^{q+\rho n/2}=R^{1/4}$. The ratio of actual to allowed norm is $R^{1/4}$. Equations (IP25)–(IP27) give precise Lagrangian order $m+\delta n/2=1/4$, and exclude every smaller order. This is a pointwise-height failure even though the physical base support is compact.

X2. Evaluate a scaled test on one packet. Take one dimension, $H=0$, and ignore a fixed proper base localization for the exact full-left computation. Apply the full symbol $r^\rho x$, in left order, to $U_R\psi$, where $r=\langle\xi\rangle$. Compute the conjugated expression. Explain why this test is bounded on the packet scale, although $r^\rho x$ is not a global $S_\rho^\epsilon$ symbol, and how the cutoff family repairs admissibility.

Solution. Left quantization gives $x\langle D\rangle^\rho$. Its conjugation by $U_R$ has symbol $R^{-\rho}z\langle R+R^\rho\zeta\rangle^\rho$, so on the frequency support of $\Phi$

The full uncut symbol has size $r^\rho$ at a fixed nonzero $x$, and $\rho>\epsilon$ when $\rho<1$; it is not an admissible global test. The argument does not use it to infer (IP5). It is an exact scale calculation supplementing the all-admissible-operator proof.

X3. Recover a derivative from its frequency cells. Let $v\in C^\infty$, fix an integer $k\geq0$, and suppose (IP30), with $\widehat{Q^\alpha u}$ replaced by $\partial^\alpha v$, holds through $|\alpha|\leq k+s$, where the integer $s>n/2$. Prove the $k$-th symbol derivative bound and explain why the corresponding annular assumption loses a factor.

Solution. Use $F_\zeta$ from (IP31). The change of variables gives

11. The illustration shows the different scales

F0. Exact figure data. The first panel plots the actual smooth bump $\Phi(z)=\exp(1-1/(1-z^2))$ for $|z|<1$, zero otherwise, as a function of $z=(\xi-R)/R^\rho$. It marks the annular-distance scale $R^\delta=16$ for $\rho=3/4,R=65536$. This is the Fourier profile, normalized to unit height; its height multiplier $R^a$ and width $R^\rho$ are stated in the caption. The second panel plots the exact exponent gap $\delta n/2$ for $n=1,2$, so it displays an order difference, not a numerical proof. The third compares the exact exponents $M$ and $q+\rho n/2$ in X1. The marked difference is $1/4$. The original script retains every scale and label.

12. Source qualification and packet regularity

The phase-based $I_\rho^m$ class, its invariant symbol quotient and its full calculus remain valid with the already proved orders and hypotheses. Every vanishing-symbol test in $\mathcal J_\rho$ preserves that class, and the two printed cutoff families are legitimate. The compact packet construction proves that the literal fixed-endpoint converse (IP5) cannot serve as its exact definition when $\rho<1$. The weighted-normal frequency-cell condition (IP30), with wavefront containment and the local choices in P2, is a complete, exact replacement.

The source qualification is confined to that fixed-endpoint equivalence under the stated definitions. No official erratum is asserted, and the ordinary intrinsic proof is retained.

The mixed-boundary reading guide locates the separate manifold and coefficient-family assembly arguments with their precise hypotheses. The 92 human proof providers remain external to this CC0 export and have exact earlier C10/C20 programme routes. Original expression, proof, counterexample and illustration are CC0-1.0; earlier linked components retain their recorded terms.

Related proofs: editable source.