Recovering a frequency amplitude from iterated regularity

An oscillation can conceal the amplitude that determines a distribution's order. In a frequency graph the concealed factor is explicit. Removing it turns suitable operators on the distribution into ordinary derivatives of the amplitude. Dyadic $L^2$ estimates then recover the exact symbol order.

The graph argument below is written in full. The selected operator, $L^2$ and dyadic analytic proofs are supplied in the linked companions, including the proper local representation needed to apply the endpoint estimate. This draft does not certify the whole intrinsic lesson, its general phase theorem, its global bundle theorem, or publication eligibility.

1. Conventions and exact earlier proofs

We use $\widehat f(\xi)=\int e^{-ix\cdot\xi}f(x)\,dx$, inverse coefficient $(2\pi)^{-n}$, and $D_j=-i\partial_{x_j}$. All orders are real unless specified. An ordinary symbol $b\in S^r(\mathbb R^n)$ satisfies

For a symbol depending on a base variable, every base derivative has the same order. Matrix symbols use any fixed finite-dimensional norm.

The earlier programme proofs used here are exact, bounded selections:

These are programme proof dependencies, not replacements by human-source citations. Section 1 of AN03-U008 proves its endpoint estimate directly by Fourier kernel bounds and Schur's inequality. Only that selection is needed here; the transmission and complex-analysis parts of that lesson are not. The selected companion fills the endpoint summations and extension steps, without asserting Schwartz density in the endpoint norm. Its supplied local representation hypothesis keeps the operator-calculus dependency visible. Earlier whole-lesson approvals are not adopted as current source decisions.

Write $B^s=B^s_{2,\infty}$. If $A_j$ is the low-frequency ball for $j=0$ and a dyadic annulus for $j\ge1$, then

up to the fixed Plancherel constant. Membership includes local square integrability of the Fourier transform. The local space requires (G2) after every compactly supported smooth multiplication. Summing the weighted annular squares gives $H^s\subset B^s\subset H^{s-\epsilon}$ for each $\epsilon>0$: the first inclusion bounds a supremum by an $\ell^2$ norm; the second uses $\sum_j2^{-2j\epsilon}<\infty$.

Let $H$ be real, smooth on $\mathbb R^n\setminus0$, and homogeneous of degree one. Fix a real smooth extension $H_e\in S^1$ equal to $H$ for $|\xi|\ge2$. The graph is

It is Lagrangian: the pullback of $\sum_jd\xi_j\wedge dx_j$ vanishes because $H''$ is symmetric; its frequency projection is the identity, so its dimension is $n$ and the parametrization is an embedding.

An admissible first-order operator is a properly supported ordinary $\Psi^1$ operator whose principal symbol vanishes on this graph. For symbols not assumed classical this means that the restriction of a representative to $x=H'(\xi)$ belongs to $S^0$, locally in the base variables; changing the representative by $S^0$ preserves the condition. All compact frequency regions can be changed freely. Define $\mathcal I^m(\Gamma_H)$ by

This definition retains the exact dyadic endpoint.

2. A real frequency phase with a properly supported kernel

G5. There is a real $h\in S^1$ with $h-H_e\in S^{-\infty}$ such that the inverse Fourier transforms of $h$ and all $h_j=\partial_jh$ have compact support.

Proof. Put $K=\mathcal F^{-1}H_e$. Take a smooth dyadic partition in frequency and write $K=\sum_{j\ge0}K_j$ as a tempered-distribution sum. For the large-frequency pieces, changing variables $\xi=2^j\eta$ and integrating by parts on a fixed annulus gives

Indeed every derivative of the rescaled symbol is bounded by $C2^j$; each $x$-derivative contributes at most $2^j$, and the volume is $2^{jn}$. Moving $(1-\Delta_\eta)^M$ off the exponential supplies arbitrarily high even powers of the final denominator, which imply the displayed estimate for every integer $N$. The low-frequency piece is Schwartz by the same integration by parts.

On $|x|\ge\epsilon>0$, choose $N>n+1+|\beta|$ and sum the geometric series. Increasing $N$ proves arbitrary decay as $|x|\to\infty$, for every derivative. Thus $K$ is smooth off zero and $(1-\kappa)K$ is Schwartz whenever $\kappa\in C_c^\infty$ is one near zero. Choose $\kappa$ real and even. Set

Schwartz Fourier invariance, proved in AN04-U001's quadratic companion, shows that the difference is Schwartz. Since $H_e$ is real, $K(-x)=\overline{K(x)}$ distributionally. The same identity holds for $\kappa K$, hence its Fourier transform is real. Finally $\mathcal F^{-1}h_j=-ix_j\kappa K$, which has the same compact support. Convolution with each of these compact kernels is properly supported: if one variable is in a compact set, the other is in its sum with a fixed compact set. This proves every assertion. ∎

The use of a real $h$ matters: multiplication by $e^{ih}$ preserves Fourier absolute values. The compact kernel matters independently: words in the operators below preserve compact support, with a larger compact set allowed for each fixed word.

3. Removing the frequency phase after a base localization

For a base-compact ordinary symbol $a(x,\theta)$ of order $r$, set

This is a distribution: in its pairing with a compact test function move $(1-\Delta_x)^N$ off $e^{ix\cdot\theta}$, divide by $\langle\theta\rangle^{2N}$, and take $2N>r+n$. All resulting integrals are absolutely convergent and controlled by finitely many test seminorms. A frequency cutoff and the same estimate justify its removal and every integration by parts used below.

G8 (exact-order reduction). The smooth function

belongs to $S^r$, with finite-seminorm bounds. Its first coefficient is

There is a full expansion with a remainder in $S^{r-N}$ after $N$ terms. No positive definiteness of $H''$ is required.

Proof. For $\xi=R\omega$, $R\ge2$, $|\omega|=1$, the defining double integral has phase $x\cdot(\theta-R\omega)-H_e(\theta)+H(R\omega)$. First split it into a region $|\theta/R-\omega|<c$, with fixed $0<c<1/4$, and its complement, using a smooth cutoff equal to one on a smaller such region.

The complement is rapidly decreasing in $R$, with all radial and angular derivatives. Here are the bounds for its unbounded frequency domain. On $|\theta|\le4R$, away from that smaller region, $|\theta-R\omega|\ge c'R$. Repeated integration by parts in $x$ gives any power $R^{-M}$ times a polynomially growing integral in $R$. A possible logarithm at a borderline order is bounded by an extra factor $R$. On $|\theta|>4R$, the same denominator is comparable to $|\theta|$; integration of $\langle\theta\rangle^{r-M}$ gives any negative power of $R$ once $M$ is large. Differentiation by any prescribed number of $R\partial_R$ and angular derivatives only adds polynomial factors in $R,\theta$; increasing $M$ absorbs them. Cutoff derivatives obey the same bounds. This proves the assertion about the complement without an unsupported compact-frequency truncation.

In the remaining region write $\theta=R\eta$. The phase becomes $R\Phi$, where

There is exactly one critical point in $(x,\eta)$: $(H'(\omega),\omega)$. Euler's identity $H'(\omega)\cdot\omega=H(\omega)$, obtained by differentiating $H(t\omega)=tH(\omega)$, also shows that its critical value is zero. The Hessian there is

The quadratic form $2X\cdot E-E^TH''(\omega)E$ becomes $2Y\cdot E$ under $Y=X-\tfrac12H''(\omega)E$. This change has determinant one. A further orthogonal change to $(Y+E)/\sqrt2$ and $(Y-E)/\sqrt2$ gives $n$ positive and $n$ negative squares. Consequently $|\det Q_\omega|=1$ and its signature is zero.

The rescaled amplitude $a(x,R\eta)$ is an ordinary symbol of order $r$ in $R$, uniformly with all compact $(x,\eta,\omega)$ derivatives. A fixed base cutoff can include the compact set $\{H'(\omega):|\omega|=1\}$; outside the support of $a$ this just extends a zero amplitude. Compact parameter patches and the finite cutoffs of AN04-U001, Appendix A.4, now allow its Sections 5–6 to apply uniformly. The stationary-phase coefficient $(2\pi/R)^n$ cancels the Jacobian and normalization $(2\pi)^{-n}R^n$. The determinant and signature factors are both one. This gives (G9), and each further term loses one power of $R$.

The same theorem gives the differentiated remainders, not merely undifferentiated $O$-bounds. On the zero critical value in (G10), it bounds every $(R\partial_R)^k\partial_\omega^\beta$ of the remainder by $C R^{r-N}$. In polar coordinates a Cartesian derivative is $R^{-1}$ times a smooth angular combination of $R\partial_R$ and angular derivatives. Induction therefore gives exactly $C_\alpha R^{r-N-|\alpha|}$. Low frequencies are smooth because $w$ is compactly supported. This proves (G8)–(G9), the full remainder orders and finite-seminorm continuity. ∎

In particular multiplication by a compact smooth function preserves the class \$\mathcal F^{-1}(e^{-iH_e}S^r)$. This conclusion follows by applying G8 to $a(x,\theta)=\chi(x)b(\theta)$; it is not an unproved assertion that multiplying an oscillatory distribution simply multiplies its frequency amplitude.

4. The graph criterion, in both directions

G12. If $u\in\mathcal E'(\mathbb R^n;\mathbb C^q)$, then

Conversely, without a compact-support requirement, every $\mathcal F^{-1}(e^{-iH_e}b)$, $b\in S^{m-n/4}$, belongs locally to the class on the left. Low-frequency changes add a smooth function and do not alter the assertion.

Forward proof. Choose $h$ from G5 and put

These are proper operators. Direct Fourier differentiation gives

The first identity follows from $i\partial_j(e^{-ih}v)-h_j e^{-ih}v=e^{-ih}i\partial_jv$. For the second, $[x_j,h_k(D)]=i(\partial_jh_k)(D)$ and mixed derivatives of $h$ commute. Every $Q_jD_k$ is an admissible first-order operator: its principal symbol is $(x_j-H_j(\xi))\xi_k$.

For $|\alpha|=|\beta|=a$, the operator $D^\beta Q^\alpha$ is a finite linear combination of words of length at most $a$ in the $Q_jD_k$, and the identity. To see this without assuming commutativity, pair the $a$ labels from $Q^\alpha$ with those of $D^\beta$ and form a product of these pairs. Moving all $D$'s to the left using $Q_jD_k=D_kQ_j+i\delta_{jk}$ gives $D^\beta Q^\alpha$ plus terms with one fewer $D$ and one fewer $Q$. Induction proves the claim.

Each resulting distribution has compact support because the word is proper and $u$ is compact. Its local bound (G4) is therefore a global bound: take a compact cutoff equal to one on its support. Using P3 and $|e^{-ih}|=1$, its dyadic estimate is

A fixed annulus meets only finitely many dyadic blocks, so powers of two are not required. On that annulus $\sum_{|\beta|=a}|\xi^\beta|^2\ge c_aR^{2a}$, as follows by expanding $(\sum_j\xi_j^2)^a$. Consequently

Put $v_R(\eta)=R^{-r}v(R\eta)$. Changing variables gives

because the power is $-2r+2|\alpha|-n+2r+n-2|\alpha|=0$. Cover the unit sphere by finitely many balls whose larger concentric balls lie in this annulus. P1's local frequency Sobolev estimate applied to each derivative of $v_R$ makes that derivative bounded on the smaller balls. Scaling back gives (G1).

The Fourier transform of a compact distribution is smooth with polynomially bounded derivatives. Here is the needed proof, without a dependency on the later tangent lesson. Choose a compact smooth cutoff $\chi$ equal to one near its support. A fixed finite-order distribution bound controls its pairing with any test supported in $\operatorname{supp}\chi$ by the supremum of that test's derivatives through some order $M$. Apply it to $\chi(x)(-ix)^\alpha e^{-ix\cdot\xi}$. On this fixed compact set the bound is $C_\alpha\langle\xi\rangle^M$. Taylor's formula in the parameter $\xi$, applied also to every test derivative through order $M$, justifies differentiating the pairing, successively for every $\alpha$. Thus the Fourier transform has all the asserted derivatives and bounds. Low frequencies cause no difficulty. Finally $H_e-h$ is Schwartz. Repeated chain and product rules give $e^{i(H_e-h)}-1\in S^{-\infty}$, since each nonzero derivative contains a derivative of that Schwartz difference, while the zeroth order uses $e^{it}-1=it\int_0^1e^{ist}\,ds$. Multiplication by this factor preserves $S^r$. Therefore $e^{iH_e}\widehat u\in S^r$.

Reverse proof. Let $u=\mathcal F^{-1}(e^{-iH_e}b)$, with $b\in S^r$. G8 shows that every compact localization has Fourier transform equal to $e^{-iH_e}$ times an $S^r$ symbol. Its squared Fourier mass in an annulus of radius $R$ is bounded by $CR^{2r+n}$. Thus it lies in $B^{-r-n/2}=B^{-m-n/4}$, including when the order is negative.

We must preserve that amplitude order under every admissible operator, not just the displayed generators. Work on a fixed output compact set. P2 gives a left symbol $p(x,\theta)$ of order one after output localization; a smoothing remainder contributes a smooth function. For large $\theta$, put

The restriction assumption gives $p_0\in S^0$. All $b_j$ belong locally to $S^1$: $H'$ is degree zero and each of its frequency derivatives lowers degree by one; repeated differentiation under the compact $t$-integral gives the required estimates. The fundamental theorem of calculus yields the exact identity

For the phase $\phi=x\cdot\theta-H(\theta)$, $(x_j-H_j)e^{i\phi}=(1/i)\partial_{\theta_j}e^{i\phi}$. Integration by parts therefore changes the amplitude $pb$ into

Every term is of order $r$, with the written matrix factor order unchanged. Insert a compact output cutoff and use G8 to remove its base dependence. Compact-frequency differences are smooth. This proves that the localized $Pu$ again has the form $\mathcal F^{-1}(e^{-iH_e}S^r)$.

For completeness, proper support makes this local argument iterable. For a chosen output compact set and a fixed finite word, successively choose input cutoffs equal to one on the compact kernel projections needed for the next factor. Terms outside these cutoffs vanish on the working output set; off-diagonal smoothing terms have the same smooth conclusion by P2. At each step G8 supplies the localized graph amplitude of the same order. Induction on word length gives (G4). This proves both the reverse implication and the noncompact local assertion. ∎

For $n=1$, take $H(\xi)=a\xi$ and $b(\xi)=\chi(\xi)\xi$, where $\chi=0$ for $\xi\le1$ and $\chi=1$ for $\xi\ge2$. On every positive annulus $R\le\xi\le2R$, $R\ge2$, the amplitude is exactly $\xi$. The three curves in the right panel are all $R^{-1}b(R\eta)=\eta$, and their squared integral is $7/3$. The corresponding unscaled squared integral is $7R^3/3$. This is the exact cancellation of powers in (G16)–(G17), illustrated for $r=1$, not a numerical substitute for the proof.

5. What localization does and does not establish

An ordinary proper $A\in\Psi^0$, including a matrix-valued one, preserves the class (G4). If $L$ is admissible, then $C=[L,A]\in\Psi^1$ is admissible as well. Its order-one principal symbol is the matrix commutator of the order-one symbol of $L$ and the order-zero symbol of $A$, so it vanishes on the graph. All frequency-derivative terms are lower order. This does not assert that the matrix commutator has order zero.

Here is the word induction. P1 supplies the assertion for a word of length zero. For a word $W=L_1\cdots L_{N-1}$, write

The first term is controlled by the induction hypothesis, since $L_Nu$ itself satisfies (G4); the second is already an admissible word of length $N$. No commutator of two matrix first-order operators has been inserted. Smooth multiplication is the special case needed for compactly supported receivers.

G12 is a theorem for a specified frequency graph. The complete geometric coordinate proof, C0–C5 now supplies such a graph near any nonzero point of a conic Lagrangian, using a base diffeomorphism. The complete coordinate and directional-localization proofs, T0–W5 transport both the intrinsic word condition and its wavefront set, and construct proper compact microlocal cutoffs. They also prove that the word condition forces wavefront into the Lagrangian. The complete conic inverse and localization proof, K0–K7 now supplies the converse for arbitrary elliptic tests, finite conic reconstruction, and independence of sufficiently small cutoffs and Lagrangian extensions. Its matrix proof retains the original factor order and every symbol remainder on one fixed cone.

The prescribed-phase proof, F0–F7 uses this graph criterion to prove both directions for a given ordinary nondegenerate or clean phase. It supplies the excess correction, the amplitude construction to every order, and the lower-order criteria. Thus the intrinsic definition is retained throughout. The full original lesson's example and exercise coverage, the global principal-symbol and Maslov interpretation, and the remaining course retain their separate outstanding obligations.

6. Three exercises with complete solutions

1. Check the shift. Why is the symbol order in G12 $m-n/4$?

Solution. An amplitude of order $r$ has squared annular energy bounded by $R^{2r+n}$. Multiplication of the block norm by $R^{-m-n/4}$ is bounded at the critical exponent exactly when $r+n/2=m+n/4$, or $r=m-n/4$. For a point mass the Fourier amplitude is constant, so $r=0$, $m=n/4$ and the endpoint is $B^{-n/2}$. Its $H^{-n/2}$ norm diverges because every large dyadic block contributes a fixed positive amount to the squared norm.

2. Verify a balanced word. In one dimension set $T=QD$ and assume $[Q,D]=i$. Express $D^2Q^2$ in terms of $T$.

Solution. $DQ=T-i$. Also $DQ^2=Q^2D-2iQ$, so $D^2Q^2=Q^2D^2-4iQD-2$. On the other hand $T^2=QDQD=Q^2D^2-iQD$. Substituting gives $D^2Q^2=T^2-3iT-2$. This checks both the shorter identity term and the sign in the reordering used for (G15).

3. Explain why an arbitrarily small Sobolev loss is insufficient. For $n=1$, take disjoint dyadic bumps $b(\xi)=\sum_{j\ge2}j\,\psi(2^{-j}\xi)$, where $\psi\in C_c^\infty((1,3/2))$ is nonzero. Show that this smooth polynomially bounded amplitude is not $S^0$, although its inverse Fourier transform and all words in $xD$ lie in $H^{-1/2-\epsilon}$ for every $\epsilon>0$.

Solution. On the $j$-th support, differentiation $k$ times is bounded by $C_kj2^{-jk}$. Its peak grows like $j$, so the zeroth symbol estimate of order zero fails. Applying $xD$ replaces the Fourier amplitude by $i\partial_\xi(\xi b)$. Every fixed word therefore replaces $\psi$ by a fixed finite sum of its derivatives times powers of its argument, retaining the factor $j$ and the support scale $2^j$. Its weighted squared Fourier integral is at most $C\sum_j j^2 2^{-2j\epsilon}<\infty$. To verify convergence, the ratio of successive terms tends to $2^{-2\epsilon}<1$, so a geometric series bounds the tail. In contrast, the endpoint annular sequence for the empty word is a nonzero constant times $j$, hence unbounded. The lost endpoint cannot recover the missing $S^0$ bound. This example uses $H=0$, so no frequency phase conceals the failure.

Free human sources and derivation

Lars Hörmander, Fourier integral operators. I, Acta Mathematica 127 (1971), pp. 150–154, supplies the free primary-source stationary-testing route. G8 gives the exact graph specialization using the already proved AN04-U001 parameter estimates, including the noncompact-frequency tail. The dyadic derivative argument in G12 is developed from the selected earlier programme conormal proof and the explicit identities (G14)–(G17). The free article is distinct from any paid book.

Jared Wunsch's version 3 lecture notes, Section 8.2, Proposition 8.6, provide a freely accessible comparison for iterated regularity. They expressly leave the precise order relationship aside. They are not the proof of the endpoint theorem here. Their classical-amplitude definition is not silently substituted for the ordinary-symbol class in G12.

The exposition and new figure in this receiving component are dedicated to CC0 to the extent rights exist. No human-source PDF or prose is reproduced. Any later incorporated programme selections must keep their own component terms, attribution and version history.