Subellipticity and unique continuation · Self-checked by the writing AI

Spectral gaps and one derivative lost

The bottom energy of an oscillator gives a useful sufficient test. It does not give the exact test for a complex operator. Every oscillator level matters, and directions with zero energy can contribute a continuous cost. When a characteristic set is irregular, even the limiting quadratic models can carry affine terms. This lesson combines those effects into a necessary spectral condition and a full microlocal sufficient condition.

We use the packet, closed-graph and affine homogenization results in Concentrated packets and necessary quadratic models, the fixed-form criterion in Quadratic coercivity when some directions have zero energy, the arbitrary-family theorem in Shrinking oscillators and uniform quadratic bounds, and the complete analytic estimate in From frozen quadratics to a local estimate. The classical half-density subprincipal symbol, proper Sobolev mapping, elliptic weights and coordinate invariance are those of Detecting regularity without choosing coordinates.

Basic references are Sjöstrand's original multiple-characteristic paper [S], Lerner's phase-space chapter [L], Nonnenmacher's semiclassical lectures [N] and Hörmander [H]. The exact spectral criterion is Hörmander's. We retain every small generalized Hamilton space, rather than only eigenvectors, throughout the proof.

1. Keep all affine limits in the spectral test

Let \(P\in\Psi^m_{\mathrm{cl}}(X;\Omega^{1/2})\) be properly supported, of arbitrary real order \(m\). Assume its principal symbol satisfies \[ \operatorname{Re}p_m\geq0,\qquad |\operatorname{Im}p_m|\leq\gamma\operatorname{Re}p_m \tag{1.1} \] for one finite \(\gamma\). Work in a coordinate patch, using its Euclidean unit cosphere. For sequences \(x_\nu\to x_0\), \(R_\nu\to\infty\), and unit \(\theta_\nu\to\theta_0\), retain every finite pointwise limit \[ Q(z)=\lim_{\nu\to\infty} R_\nu^{1-m} p_m(x_\nu+y/\sqrt{R_\nu}, R_\nu\theta_\nu+\sqrt{R_\nu}\eta), \qquad z=(y,\eta). \tag{1.2} \] The packet theorem gives \[ Q=Q_2+\ell+d,\qquad Q_2=\tfrac12d^2p_m(x_0,\theta_0),\qquad p_m(x_0,\theta_0)=dp_m(x_0,\theta_0)=0. \tag{1.3} \] Here \(d\) is a scalar constant and \(\ell\) is a linear form. Define a homogeneous quadratic on one additional pair by \[ \widehat Q(z,t,\tau)=Q_2(z)+t\ell(z)+t^2d, \tag{1.4} \] independent of \(\tau\), and set \(\kappa=p^{\mathrm{sub}}_{m-1}(x_0,\theta_0)\). The form \(\widehat Q\) is sector valued: for \(t\ne0\) it equals \(t^2Q(z/t)\), and at \(t=0\) the assertion follows by continuity. The form, its full Hamilton map and its scalar shift are the data of the test.

For a homogeneous quadratic \(G\), write \(F_G\) for the map satisfying \(\omega(u,F_Gv)=G(u,v)\), with \(\omega((x,\xi),(y,\eta))=\xi\cdot y-x\cdot\eta\). Let \(\mu_j(G)\) be the eigenvalues of \(F_G/i\) with positive real part, counted algebraically. They lie in the same sector. Define \[ V_e(G)=\bigoplus_{|\lambda|<e} \ker(F_G-\lambda I)^{2n+2}. \tag{1.5} \] The exponent refers to the enlarged phase space. It keeps all nilpotent and higher Jordan directions.

The required condition is the following:

For every compact base set \(K\) there are \(e_K,b_K>0\) such that every limit (1.2) with \(x_0\in K\), every \(v\in V_{e_K}(\widehat Q)\), and every nonnegative occupation multi-index satisfy \[ \left|\kappa+\widehat Q(\overline v,v) +\sum_j(2\alpha_j+1)\mu_j(\widehat Q)\right| \geq b_K. \tag{1.6} \] One positive number can serve as both cutoff and gap: replace them by their minimum. Empty frequency sums and zero-dimensional blocks are included.

Theorem 1.1. If, for one real \(s\), every distribution satisfies \[ Pu\in H^s(X)\quad\Longrightarrow\quad u\in H^{s+m-1}(X), \tag{1.7} \] then (1.6) holds. The same conclusion follows from the corresponding local Sobolev implication.

Conversely, (1.6) implies, for every real \(s\), every distribution and every nonzero covector, \[ Pu\in H^s\text{ at }\rho \quad\Longrightarrow\quad u\in H^{s+m-1}\text{ at }\rho. \tag{1.8} \] Thus smooth data imply microlocal smoothness. No smoothness, constant rank or transverse ellipticity of the principal zero set is assumed in this theorem.

Condition (1.6) uses every moving model, including the extra pair. The spectral test for the Hessian at a single characteristic point can be weaker.

2. Why the uniform quadratic theorem applies to this family

The forms in (1.4) can have unbounded affine coefficients as the limits vary. This does not violate the invariant hypothesis of the uniform quadratic theorem.

In the order of coordinates \((z,t,\tau)\), the \(t\) component of \(F_{\widehat Q}\) is zero, and the form has no \(\tau\) derivative. Its \(z\) block is \(F_{Q_2}\). Expansion of the characteristic determinant along the \(t\) row and the \(\tau\) column gives \[ \det(\lambda I-F_{\widehat Q}) =\lambda^2\det(\lambda I-F_{Q_2}). \tag{2.1} \] This calculation does not assert a direct decomposition of the zero generalized spaces; the cross coefficients can enlarge their nilpotent structure.

Equation (2.1) also holds for the real parts. On a compact coordinate cosphere the Hessians of \(\operatorname{Re}p_m\) have bounded coefficients. Hence the eigenvalues of \(F_{\operatorname{Re}\widehat Q}\) are uniformly bounded. The shifts \(\kappa\) are bounded there too. These are precisely the two boundedness assumptions of the arbitrary-family theorem. A bound on the entire matrix of \(\widehat Q\) is unnecessary.

Consequently (1.6) is equivalent, on each compact set, to a common lower norm estimate \[ \|\Psi\|_0\leq M_K \|(\widehat Q^w+\kappa)\Psi\|_0, \qquad \Psi\in\mathcal S(\mathbb R^{n+1}), \tag{2.2} \] for the entire enlarged family. To obtain the affine slice estimate, take \(\Psi(t,y)=f_\epsilon(t-1)\psi(y)\), with normalized compact profiles \(f_\epsilon\) concentrating at zero. No \(t\) derivative occurs. Coefficient continuity and Fubini pass (2.2) to \[ \|\psi\|_0\leq M_K\|(Q^w+\kappa)\psi\|_0. \tag{2.3} \] Conversely the necessity theorem in the packet lesson obtains the full enlarged estimate directly from the original operator estimate, with the same compact-set constant. It uses all real slices, including \(t=0\), before integration.

3. Necessity retains the entire enlarged model

Assume (1.7). The closed-graph proposition in the packet lesson gives compact estimates \[ \|u\|_{s+m-1}\leq C_K \bigl(\|Pu\|_s+\|u\|_{s+m-2}\bigr). \tag{3.1} \] Conjugate by proper elliptic Sobolev weights to put the data exponent at zero. The principal symbol is unchanged. The subprincipal conjugation term is a first Poisson product containing \(dp_m\), so it vanishes at every limiting center in (1.3).

The strong packet limit therefore gives (2.3), with a common constant for every model whose limiting base lies in the compact set. Choosing a slightly larger compact support neighborhood keeps the constant uniform at its boundary. The packet lesson's positive-frequency rescaling, parity for negative slices and continuity at zero give (2.2) for every enlarged model.

The family satisfies the real-Hamilton and shift bounds proved in Section 2. The full uniform quadratic theorem now gives (1.6), with full small generalized spaces and all occupation indices. This proves necessity. The local Sobolev assumption gives the same compact closed-graph estimate, so the argument applies to it as well.

4. Sufficiency, with arbitrary orders and rough distributions

First let \(m=1\). Condition (1.6), the uniform quadratic theorem and the slice argument give exactly the common affine model bound (2.3) required by the frozen-quadratic lesson. That lesson proves, for all real \(r\), \[ \|u\|_r+\sum_j \bigl(\|A_ju\|_{r+1/2}+\|B_ju\|_{r+1/2}\bigr) \leq C_{K,r}\bigl(\|Pu\|_r+\|u\|_{r-1}\bigr). \tag{4.1} \] Its regularization and shrinking-cone argument applies to arbitrary distributions. It gives \(u\in H^r\) microlocally from \(Pu\in H^r\), rather than assuming the conclusion while applying an a priori estimate. This proves (1.8) at order one.

For general real \(m\), choose a proper elliptic operator \(E\) of order \(1-m\), with positive principal symbol \(|\xi|^{1-m}\) in the coordinate patch, and set \[ \widetilde P=EP\in\Psi^1. \tag{4.2} \] Its principal symbol remains in the same sector. In (1.2) its moving normalized principal symbol is multiplied by \[ |\theta_\nu+\eta/\sqrt{R_\nu}|^{1-m}\longrightarrow1. \tag{4.3} \] Thus its finite moving limits are precisely the same polynomials \(Q\). At their double characteristic centers its subprincipal value is also \(\kappa\): the other product terms contain \(p_m\) or \(dp_m\), and the positive weight is one on this unit cosphere. Hence the order-one spectral condition is exactly (1.6).

If \(Pu\) is \(H^s\) at \(\rho\), mapping for \(E\) puts \(\widetilde Pu\) in \(H^{s+m-1}\) there. The order-one result gives \(u\in H^{s+m-1}\). This proves (1.8) for every real order and every real data index.

For a different positive homogeneous norm, the model and the shift at a unit center are multiplied by a positive factor bounded above and away from zero on compact cospheres. The Hamilton eigenvalues scale by that factor, and the small-space cutoff can be decreased uniformly. Thus the existence of compact-set cutoffs and gaps is unchanged. Coordinate Hessians at double characteristics and half-density subprincipal values are intrinsic; real symplectic covariance preserves the model norms and polarized costs. A finite coordinate cover proves the statement on \(X\).

Multiplying \(P\) by a fixed unit complex scalar also preserves its regularity implications. Thus a fixed closed convex sector of opening less than \(\pi\) can be rotated into (1.1). The spectral data and costs rotate by the same factor, and their moduli are unchanged.

5. Smooth transverse geometry removes the moving affine data

Let \[ \Sigma=\{p_m=0\}\subset T^*X\setminus0. \] Assume it is smooth and \(p_m\) is transversely elliptic there. In the sector setting this means that its Hessian quadratic \[ Q_\rho(v)=\tfrac12d^2p_m(\rho)[v,v] \tag{5.1} \] has radical \(T_\rho\Sigma\) and is elliptic on the real normal quotient. The sector kernel identity makes the real part positive definite on that quotient. No constant symplectic rank of \(T\Sigma\) is required.

For each \(\rho\in\Sigma\), let \(V_0(Q_\rho)\) be the full generalized zero space of its Hamilton map.

Theorem 5.1. Under these additional geometric assumptions, condition (1.6) is equivalent to the pointwise exclusions \[ p^{\mathrm{sub}}_{m-1}(\rho) +Q_\rho(\overline v,v) +\sum_j(2\alpha_j+1)\mu_j(Q_\rho)\ne0, \quad v\in V_0(Q_\rho),\quad\alpha\in\mathbb N^{k(\rho)}, \tag{5.2} \] at every unit characteristic covector. Consequently (5.2) gives the full conclusion (1.8).

Proof. Smooth normal coordinates and the positive definite normal Hessian give, on each compact cosphere neighborhood, \[ c\,\operatorname{dist}(\zeta,\Sigma)^2 \leq |p_m(\zeta)| \leq C\,\operatorname{dist}(\zeta,\Sigma)^2. \tag{5.3} \] For the lower bound use the Taylor integral formula for \(\operatorname{Re}p_m\) in the normal variables and shrink the neighborhood so its normal Hessian stays positive. The upper bound uses \(p_m=dp_m=0\) on \(\Sigma\). Finitely many neighborhoods give uniform constants near its compact portion.

The translation theorem in the packet lesson now proves that every moving limit is a real phase translation \[ Q(z)=Q_\rho(z+b). \tag{5.4} \] Its Weyl operator is unitarily conjugate to \(Q_\rho^w\), with the scalar subprincipal shift unchanged.

At \(\Sigma\), the complex rank of the quadratic matrix is its normal codimension. The sector identity gives this also for the rank of its real part. It is locally constant. Cover a compact characteristic portion by finitely many smaller neighborhoods on which that rank is fixed. Their closed characteristic portions give compact coefficient families of pairs \((Q_\rho,p^{\mathrm{sub}}_{m-1}(\rho))\). Condition (5.2), the fixed-form criterion and the compact constant-complex-rank corollary give a common lower norm bound on each such family. Take the largest of the finitely many constants. By (5.4) this is a common affine model bound for every moving limit with base in the compact set.

For its homogenization, \[ \widehat Q(z,t,\tau)=Q_\rho(z+tb), \tag{5.5} \] each real \(t\) slice is a unitary real phase translation of the same fixed form. Apply the common bound slice by slice and integrate in \(t\). This gives the common enlarged estimate (2.2). Its real Hamilton spectrum is bounded by (2.1), so the full uniform quadratic theorem gives (1.6).

Conversely at an exact zero take fixed centers and directions in (1.2). Then \(Q=Q_\rho\), and its homogenization is \(Q_\rho\) with a free extra pair. The full generalized zero space contains \(V_0(Q_\rho)\) with that pair set to zero, and its nonzero frequencies are exactly those of \(Q_\rho\). Condition (1.6) therefore implies (5.2). This proves the equivalence. ∎

The rank used here is the rank of the quadratic matrix. A Hamilton frequency may disappear even while that rank is constant. The generalized-zero cost in (5.2) is needed precisely in such cases.

If the characteristic set is empty, the model condition is vacuous and ordinary elliptic regularity gives the stronger gain of \(m\).

6. Examples and exercises with complete solutions

Exercise 1 — recover the missing oscillator levels, 10 points. On \(\mathbb R_x\times\mathbb T_y\), consider \[ P_c=D_x^2+x^2D_y^2+cD_y,\qquad c\in\mathbb C. \] Use Theorem 5.1 to find the exact exclusions for a one-derivative-loss estimate in both frequency cones. Compare \(c=-2\) with a test that uses only the bottom oscillator energy.

Solution. The characteristic set is \(x=\xi=0,\eta\ne0\), smooth and transversely elliptic. Its quadratic form is \(\xi^2+\eta^2x^2\), with one frequency \(|\eta|\); the remaining tangent pair has zero cost. The subprincipal value is \(c\eta\). At unit covectors the exclusions are \[ 2j+1+c\ne0\quad(\eta>0),\qquad 2j+1-c\ne0\quad(\eta<0),\qquad j\geq0. \] Thus \(c\) must avoid every positive and negative odd integer. For \(c=-2\), both cones satisfy the exact condition, with nearest normalized spectral modulus one. A sufficient test that requires \(1+c\) to avoid the nonpositive real axis fails in the positive cone, because \(1+c=-1\). That test was only sufficient; the full oscillator spectrum proves the estimate here. For every real \(s\), the conclusion is \(P_cu\in H^s_{\mathrm{loc}}\Rightarrow u\in H^{s+1}_{\mathrm{loc}}\).

Exercise 2 — a resonant distribution, 10 points. For \(c=-1\), let \(\psi_n(x)=(n/\pi)^{1/4}e^{-nx^2/2}\), \(n\geq1\). Show that \[ u(x,y)=\sum_{n\geq1}\psi_n(x)e^{iny} \] defines a distribution, solves \(P_{-1}u=0\), and is not locally \(L^2\) throughout a neighborhood of \(\{x=0\}\times\mathbb T\).

Solution. The profiles have \(L^2(\mathbb R)\) norm one. A smooth compactly supported test has Fourier coefficients in \(y\) decreasing faster than every power, also in their \(L^2_x\) norms. Cauchy–Schwarz in \(x\) therefore makes the distributional pairing series converge absolutely. The oscillator identity \[ (D_x^2+n^2x^2)\psi_n=n\psi_n \] gives \(P_{-1}(\psi_ne^{iny})=0\). Distributional continuity permits termwise application.

On any strip \(|x|<a\), the \(L^2_x\) norms of \(\psi_n\) tend to one. Orthogonality of the periodic modes shows that the sum of their squared norms diverges. If \(u\) were locally \(L^2\) at every point of the compact circle \(x=0\), a finite neighborhood cover would make it \(L^2\) on some such strip, a contradiction. The datum is smooth, while the solution fails local \(L^2\) somewhere on that circle. This is the ground-level resonance in the positive frequency cone.

Exercise 3 — an exact symplectic homogenization, 8 points. Let \(Q_2\) be a homogeneous real-phase quadratic and let \(b=(b_x,b_\xi)\) be real. Verify that \[ z'=z+tb,\quad t'=t,\quad \tau'=\tau+\omega(b,z) \] is symplectic, and identify the Hamilton spectrum of \(Q_2(z+tb)\).

Solution. In coordinates, \(x'=x+tb_x\), \(\xi'=\xi+tb_\xi\), and \(\tau'=\tau+b_\xi\cdot x-b_x\cdot\xi\). Expansion gives \[ d\xi'\wedge dx' =d\xi\wedge dx+(b_x\cdot d\xi-b_\xi\cdot dx)\wedge dt. \] The added term cancels the corresponding term in \(d\tau'\wedge dt\). The map is invertible and preserves \(d\xi\wedge dx+d\tau\wedge dt\). In the primed coordinates the form is \(Q_2(z')\), independent of the extra pair. Hamilton covariance therefore preserves every original eigenvalue and adds two zero eigenvalues. The zero space is still interpreted with all generalized directions; this coordinate identity provides more information than eigenvalues alone.

Exercise 4 — rank can stay fixed while a frequency vanishes, 8 points. For \(t\) near one, put \[ G_t=(x_1+t x_2)^2+(\xi_1-t\xi_2)^2. \] Find its quadratic rank and nonzero Hamilton frequencies. Explain why no constant frequency count is required in Theorem 5.1.

Solution. The two real linear forms are independent, so the quadratic rank is two for every \(t\). Their symplectic pairing is \(-(1-t^2)\). Hence the nonzero Hamilton eigenvalues, when \(t\ne1\), are \(\pm i|1-t^2|\), and the positive frequency is \(|1-t^2|\). At one all Hamilton eigenvalues are zero, while the quadratic rank remains two. For the shift \(\kappa=1+i/2\), every cost and frequency contribution is nonnegative real, so every spectral value has imaginary part \(1/2\). There is a common lower bound despite the disappearing frequency. The constant-rank corollary, and therefore the geometric proof above, retains this situation.

Exercise 5 — a fractional order and the data exponent, 6 points. Suppose the full spectral condition holds for an operator of order \(m=3/2\), and \(Pu\) is microlocally \(H^{-2}\). State the reducing order, the reduced data space and the conclusion. What follows for smooth data?

Solution. The reducing operator has order \(1-m=-1/2\). It raises the datum to \(H^{-3/2}\), equal to \(s+m-1\). The order-one result gives \(u\in H^{-3/2}\). For smooth data, apply the theorem with arbitrarily large \(s\); every target Sobolev order is reached, so the solution is microlocally smooth. The normalization works for all real orders, including negative ones.

References

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