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

Melin's lower bounds

At a double characteristic, the principal symbol vanishes together with its first derivatives. Its quadratic Taylor term still has an operator energy: the positive trace proved in Quadratic energy and the positive trace. A subprincipal term can reinforce or cancel that energy. The sign of their sum determines the lower bounds in this lesson.

We use the ordinary composition, adjoint, proper-support, elliptic parametrix and Sobolev mapping theorems in Symbols, operators and Sobolev scales and Detecting regularity without choosing coordinates. The latter supplies the invariant subprincipal symbol on half-densities. Two positivity results are used: scalar sharp Gårding from Section 6 of Positivity through a moving family of scalar probes, and scalar Fefferman–Phong from Section 9 of When a nonnegative scalar symbol acquires a negative part. Their exact specializations appear in Section 2. The geometric splitting and every passage to the lower bounds are proved below.

Melin's original paper [M] gives the historical starting point. Hérau's openly available paper [H] discusses related lower bounds with less symbol regularity. Lerner's open chapter [L] develops the scalar positivity estimates used here. Our operators are scalar, classical, properly supported pseudodifferential operators; their orders may be arbitrary real numbers. We work on smooth manifolds without boundary and on half-densities, so formal adjunction and the subprincipal symbol have intrinsic meanings.

1. The two statements

Let \(P\in\Psi^{2m}_{\mathrm{cl}}(X;\Omega^{1/2})\) be self-adjoint on compactly supported smooth half-densities. Write \(p_{2m}\) and \(p^{\mathrm{sub}}_{2m-1}\) for its principal and subprincipal symbols. Both are real. Assume \(p_{2m}\geq0\), and put \[ \Sigma=\{\rho\in T^*X\setminus0:p_{2m}(\rho)=0\}, \qquad Q_\rho(v)=\frac12d^2p_{2m}(\rho)[v,v]. \tag{1.1} \] Nonnegativity implies \(dp_{2m}=0\) on \(\Sigma\). The Hessian there is therefore an intrinsic quadratic form on the symplectic tangent space \(T_\rho(T^*X)\). Its positive trace is defined as in the preceding lesson. It has degree \(2m-1\), matching the subprincipal symbol.

Theorem 1.1 (a non-strict lower bound). Suppose \[ p^{\mathrm{sub}}_{2m-1}(\rho)+\operatorname{Tr}_+Q_\rho\geq0 \quad(\rho\in\Sigma). \tag{1.2} \] Assume also that \(\Sigma\) is a smooth manifold, that \(\ker Q_\rho=T_\rho\Sigma\), and that the symplectic form restricted to \(T\Sigma\) has constant rank. Then for every compact \(K\subset X\), \[ (Pu,u)\geq-C_K\|u\|_{H^{m-1}}^2, \qquad u\in C_c^\infty(K;\Omega^{1/2}). \tag{1.3} \]

Theorem 1.2 (the strict Melin bound). Suppose instead that \[ p^{\mathrm{sub}}_{2m-1}(\rho)+\operatorname{Tr}_+Q_\rho>0 \quad(\rho\in\Sigma). \tag{1.4} \] Then for every compact \(K\subset X\) there are \(c_K>0,C_K\) such that \[ (Pu,u)\geq c_K\|u\|_{H^{m-1/2}}^2 -C_K\|u\|_{H^{m-1}}^2, \qquad u\in C_c^\infty(K;\Omega^{1/2}). \tag{1.5} \] Here no smoothness, transverse nondegeneracy, or constant-rank assumption on \(\Sigma\) is required.

If \(\Sigma\) is empty, the symbol is elliptic on each compact cosphere and both conclusions follow from an elliptic square. Local conic versions hold when the corresponding hypotheses hold only in that cone; the proof below states the localized estimates explicitly.

2. The analytic input and the sign of the correction

The precise positivity inputs, with \(w=\langle\xi\rangle\), are \[ 0\leq a\in S^1_{1,0}\Longrightarrow a^w\geq-C, \qquad 0\leq r\in S^2_{1,0}\Longrightarrow r^w\geq-C. \tag{2.1} \] The first is scalar sharp Gårding: set the parameters \(\rho=1,\delta=0\) and the error Sobolev exponent to zero in the prerequisite. Conversion of an order-one real symbol to Weyl form changes its real part by an order-zero term, so the same bound holds in Weyl quantization. The second is the scalar Fefferman–Phong specialization already used in Nonnegative symbols and weighted brackets. Constants depend on finitely many symbol seminorms on a larger compact set. Properizing the kernels adds bounded smooth-kernel errors.

Initially take an order-two operator. Its Weyl symbol in a coordinate patch has the form \[ p_2+p_1+S^0, \qquad p_1=p^{\mathrm{sub}}_1, \tag{2.2} \] where both displayed terms are real and homogeneous at high frequency. If \(c=a+ib\), with \(a,b\) real of order one, and \(X=c^w\), then \[ X^*X=(a^2+b^2+\{a,b\})^w+\Psi^0. \tag{2.3} \] This follows from the first Weyl product term; every further term has order at most zero. A real \(c\) contributes just \((c^2)^w+\Psi^0\).

Suppose \(p_2\) is split into squares whose first derivatives at a characteristic covector agree with the real and imaginary parts of the linear functions in Theorem 3.1 of the quadratic lesson. Pair them into complex symbols and leave the multiplication directions real. Then (2.3) has subprincipal value \[ \sum_j\{\operatorname{Re}c_j,\operatorname{Im}c_j\} =-\operatorname{Tr}_+Q_\rho \tag{2.4} \] at that covector. Consequently the remaining first-order symbol has value \(p_1+\operatorname{Tr}_+Q\). The minus sign in (2.4) is fixed by \(\{x,\xi\}=-1\), not by a choice of notation for the trace.

3. Splitting a nonnegative function

We need two versions of a local splitting argument.

Lemma 3.1 (splitting at one minimum). Let \(f\geq0\) be smooth near zero in \(\mathbb R^d\), with \(f(0)=0\), and let the rank of \(d^2f(0)\) be \(r\). On a smaller neighborhood there are smooth functions \(b_1,\ldots,b_r\) and \(R\geq0\) such that \[ f=\sum_{\nu=1}^r b_\nu^2+R, \qquad \frac12d^2f(0)=\sum_{\nu=1}^r(db_\nu(0))^2, \qquad d^jR(0)=0\ (0\leq j\leq2). \tag{3.1} \]

Proof. Choose linear coordinates \((y,z)\) so the Hessian is positive definite in the \(r\) coordinates \(z\) and zero in the remaining coordinates \(y\). The mixed Hessian terms vanish as well, since the Hessian is nonnegative and its \(y\)-space is its kernel. The implicit function theorem applied to \(\partial_zf\) gives a unique smooth \(z=z(y)\) near zero with \(\partial_zf(y,z(y))=0\). Shrink the neighborhood so the \(z\)-Hessian is positive definite. Taylor's formula gives \[ f(y,z)=f(y,z(y))+(z-z(y))^TA(y,z)(z-z(y)), \] \[ A(y,z)=\int_0^1(1-t)\partial_z^2f(y,z(y)+t(z-z(y)))\,dt. \tag{3.2} \] The matrix \(A\) is smooth and positive definite. Its positive square root is smooth: the derivative of the square map at a positive matrix \(S\) is \(T\mapsto ST+TS\), invertible on symmetric matrices because its eigenvalues are sums of positive eigenvalues of \(S\); the implicit function theorem and uniqueness of the positive root give smoothness. Define the vector \(b=A^{1/2}(z-z(y))\), and \(R=f(y,z(y))\).

The latter is nonnegative. At zero, implicit differentiation gives \(z'(0)=0\), so \(R\) has zero value, gradient, and Hessian. Twice differentiating the splitting gives the quadratic identity in (3.1). If \(r=0\), take the empty sum and \(R=f\). ∎

Lemma 3.2 (splitting along the zero manifold). Suppose the zero set \(M\) of \(f\geq0\) is smooth and \(\ker d^2f=T M\) along it. Then locally there are independent defining functions \(b_1,\ldots,b_r\) for \(M\) such that \[ f=\sum_\nu b_\nu^2, \qquad \frac12d^2f|_M=\sum_\nu(db_\nu|_M)^2. \tag{3.3} \]

Proof. Choose coordinates with \(M=\{z=0\}\). We have \(f(y,0)=0\), \(df(y,0)=0\), and a positive definite \(z\)-Hessian. Formula (3.2), now with \(z(y)=0\), gives \(f=z^TA(y,z)z\). Put \(b=A^{1/2}z\). Its normal derivative is invertible, so it defines exactly \(M\) on a smaller neighborhood. Differentiation proves (3.3). ∎

Apply either lemma to \(p_2/|\xi|^2\) on a local cosphere \(|\xi|=1\), and extend the resulting functions homogeneously: multiply the \(b_\nu\) by \(|\xi|\), and the residual by \(|\xi|^2\). This gives order-one homogeneous functions and an order-two nonnegative residual in a cone. At a zero the radial vector is in the Hessian kernel by Euler's identity, so the quadratic identities on the cosphere give the full phase-space identities.

If two lists of \(r\) independent real covectors have the same sum of squares, one is an orthogonal transformation of the other. Indeed their common radical gives the same quotient space of dimension \(r\); each list identifies that quotient isometrically with \(\mathbb R^r\), and composition of the two identifications is orthogonal. This observation allows the first derivatives of the \(b_\nu\) to be matched to the quadratic functions used in (2.4).

4. Proof of the strict local estimate

Fix \(\rho_0\in\Sigma\) and suppose \(p_1(\rho_0)+\operatorname{Tr}_+Q_{\rho_0}>0\). Lemma 3.1 and a constant orthogonal transformation give, in a conic neighborhood, \[ p_2=R+\sum_{\nu=1}^{2k+\ell}c_\nu^2, \qquad R\geq0, \tag{4.1} \] where at \(\rho_0\) the \(dc_\nu\) are the real and imaginary parts of the \(L_j\) for \(Q_{\rho_0}\). Quantize \[ X_j=(c_{2j-1}+ic_{2j})^w\quad(1\leq j\leq k), \qquad X_{k+a}=c_{2k+a}^w\quad(1\leq a\leq\ell). \] By (2.2)–(2.4), microlocally in that cone, \[ P=R^w+\sum_jX_j^*X_j+G+\Psi^0, \tag{4.2} \] where \(G\) is self-adjoint of order one and its principal symbol is \[ g_1=p_1-\sum_{j=1}^k\{c_{2j-1},c_{2j}\}, \qquad g_1(\rho_0)=p_1(\rho_0)+\operatorname{Tr}_+Q_{\rho_0}>0. \tag{4.3} \] On a smaller cone, \(g_1\geq2c|\xi|\) for some \(c>0\).

Here and below, microlocal equalities have a precise use. Choose a properly supported \(\Psi\in\Psi^0\) whose symbol is supported in a still smaller cone. Extend \(R\) by multiplying it by a nonnegative cutoff equal to one there, cutting off low frequencies, and extending by zero. The result is a global nonnegative \(S^2\) symbol on the coordinate patch. Its quantization differs from the term in (4.2), after composition with \(\Psi\), by a smoothing operator. The second part of (2.1) bounds its form below by \(-C\|\Psi u\|_0^2\).

Extend \(g_1\) to a real \(S^1\) symbol \(\widetilde g\geq c' w\), with \(c'>0\), agreeing at high frequency on the smaller cone. One construction is a smooth convex combination of \(g_1\) and \(c'w\), with the first coefficient supported where \(g_1\geq c'w\) and vanishing at low frequency. The first part of (2.1), applied to \(\widetilde g-c'w\), gives \[ (\widetilde g^wv,v)\geq c'\|v\|_{1/2}^2-C\|v\|_0^2. \tag{4.4} \] The lower-order part of \(G\) is bounded. The square terms in (4.2) are nonnegative. A smoothing error composed with \(\Psi\), and then paired with \(\Psi u\), is bounded by \(C\|u\|_0^2\). Therefore \[ (P\Psi u,\Psi u)\geq c'\|\Psi u\|_{1/2}^2-C\|u\|_0^2. \tag{4.5} \] This proves the strict local estimate without any assumption on how nearby Hessian ranks or zero sets behave.

5. Proof of the non-strict local estimate

Now assume the geometric hypotheses of Theorem 1.1. For order two, \(Q_\rho\) has constant rank along the smooth \(\Sigma\). Its radical is \(T\Sigma\), whose restricted symplectic rank is constant. Section 5 of the quadratic lesson consequently supplies smooth local choices of the \(L_j\), with fixed \(k,\ell\). Their real and imaginary parts form a smooth list \(\Lambda_\nu\) with \[ Q_\rho=\sum_\nu\Lambda_\nu^2. \tag{5.1} \] Lemma 3.2 gives \(p_2=\sum b_\nu^2\). On \(\Sigma\) the two lists \(\Lambda_\nu\) and \(db_\nu\) differ by a smooth orthogonal matrix. Smoothness follows by taking the inverse of the isomorphism defined by the \(db_\nu\) on the normal quotient, and then composing with the \(\Lambda_\nu\). Extend that matrix off \(\Sigma\) using local normal coordinates, first on the cosphere and then homogeneously of degree zero. With \(c=Ob\), \[ p_2=\sum_\nu c_\nu^2, \qquad dc_\nu|_\Sigma=\Lambda_\nu. \tag{5.2} \] The homogeneous extension preserves the equality at all radii: both bracket corrections and the positive trace have degree one.

Form the operators \(X_j\) as in Section 4. Then \[ P=\sum_jX_j^*X_j+G_1+\Psi^0, \qquad g_1|_\Sigma=p_1+\operatorname{Tr}_+Q\geq0, \tag{5.3} \] where \(g_1\) is the real principal symbol of \(G_1\). In local cosphere coordinates \((y,z)\) with \(\Sigma=\{z=0\}\), extend \(g_1|_\Sigma\) as the nonnegative function constant in \(z\), and then extend homogeneously of degree one. Denote it by \(\widetilde g_1\). The difference \(g_1-\widetilde g_1\) vanishes on \(\Sigma\). The \(c_\nu\) are defining functions with independent normal derivatives, so the integral identity \[ h(y,c)-h(y,0)=\sum_\nu c_\nu\int_0^1 \partial_{c_\nu}h(y,tc)\,dt \tag{5.4} \] gives smooth real degree-zero functions \(r_\nu\) with \(g_1-\widetilde g_1=\sum c_\nu r_\nu\).

For a complex pair choose the degree-zero symbol \((r_{2j-1}+ir_{2j})/2\); for a real square choose \(r_{2k+a}/2\). Let their quantizations be \(R_j\). Their leading products give \[ P=\sum_jX_j^*X_j+\widetilde G +\sum_j(R_j^*X_j+X_j^*R_j)+\Psi^0 =\sum_j(X_j+R_j)^*(X_j+R_j)+\widetilde G+\Psi^0. \tag{5.5} \] In the second equality the terms \(-\sum R_j^*R_j\) are included in \(\Psi^0\). All first composition errors also have order zero. Extend the nonnegative \(\widetilde g_1\) by a nonnegative cutoff as in Section 4. Sharp Gårding bounds \(\widetilde G\) below by a constant. Testing (5.5) on \(\Psi u\), and absorbing the smoothing comparison errors, proves \[ (P\Psi u,\Psi u)\geq-C\|u\|_0^2. \tag{5.6} \] This is the non-strict local estimate. The smooth geometry is used to match the quadratic squares along the whole characteristic manifold, and to divide the remaining symbol by its defining functions.

6. Elliptic patches and assembly

If \(p_2>0\) at a covector, choose a real first-order symbol with homogeneous terms \[ b_1=\sqrt{p_2},\qquad b_0=\frac{p_1}{2\sqrt{p_2}}. \tag{6.1} \] Its self-adjoint quantization \(B\) satisfies \(P=B^*B+\Psi^0\) in a smaller cone. The first Weyl correction in a square is zero, and (6.1) matches both homogeneous terms of \(P\). A localized parametrix for the elliptic \(B\) gives \(\|\Psi u\|_1^2\leq C\|B\Psi u\|_0^2+C\|u\|_0^2\). Thus (4.5), and in particular (5.6), also hold in elliptic patches.

Cover the compact cosphere over \(K\) by finitely many such patches and the characteristic patches already treated. Choose real homogeneous order-zero cutoffs \(\chi_j\) with \(\sum_j\chi_j^2=1\) near that cosphere. Their quantizations can be chosen self-adjoint with zero subprincipal symbols: use real Weyl symbols in each chart and symmetrize the properized kernels. Half-density subprincipal invariance makes this condition consistent across charts. Let these operators be \(\Psi_j\). Then, near \(K\), \[ S=\sum_j\Psi_j^2=I+\Psi^{-2}. \tag{6.2} \] Both the order-zero principal discrepancy and its order-minus-one term vanish: the first is \(\sum\chi_j^2-1\), and the latter is the sum of zero subprincipal square terms. Low frequencies and errors away from a base neighborhood of \(K\) are smoothing on inputs supported in \(K\).

The exact double-commutator identity is \[ \Psi_jP\Psi_j =\frac12(\Psi_j^2P+P\Psi_j^2) -\frac12[\Psi_j,[\Psi_j,P]]. \tag{6.3} \] The double commutator has order zero. Equations (6.2)–(6.3), paired with a test function supported in \(K\), therefore give \[ \sum_j(P\Psi_ju,\Psi_ju)=(Pu,u)+O(\|u\|_0^2). \tag{6.4} \] This explicitly accounts for the first-order localization terms.

For the strict case, the Sobolev mapping calculus also gives \[ \sum_j\|\Psi_ju\|_{1/2}^2 \geq c\|u\|_{1/2}^2-C\|u\|_0^2. \tag{6.5} \] One can verify this in a chart by comparing \(\sum\Psi_j^*\langle D\rangle\Psi_j\) with \(\langle D\rangle\): their order-one principal symbols agree, so their difference has order zero. A finite coordinate partition gives the same norm equivalence on a manifold. Summing (5.6) proves (1.3) for \(m=1\). Summing (4.5), using the smallest of the finitely many positive constants, and applying (6.4)–(6.5) proves (1.5) for \(m=1\). The same reasoning restricted to one cone proves the stated local versions.

7. Arbitrary real order

Choose a positive elliptic homogeneous norm \(\varrho(x,\xi)\) of degree one. Let \(E\in\Psi^{1-m}\) be a properly supported self-adjoint elliptic operator with principal symbol \(e=\varrho^{1-m}\), and put \[ \widetilde P=E^*PE\in\Psi^2. \tag{7.1} \] Its principal symbol is \(e^2p_{2m}\). At a characteristic covector, \[ \widetilde p^{\mathrm{sub}}_1=e^2p^{\mathrm{sub}}_{2m-1}, \qquad \widetilde Q_\rho=e(\rho)^2Q_\rho, \qquad \operatorname{Tr}_+\widetilde Q_\rho =e(\rho)^2\operatorname{Tr}_+Q_\rho. \tag{7.2} \] To check the first identity, the first Weyl corrections in the product \(e\mathbin\#p\mathbin\#e\) cancel in pairs. Contributions from the lower symbol of \(E\) multiply \(p_{2m}\), which vanishes on \(\Sigma\). For the second identity differentiate \(e^2p_{2m}\) twice: all terms except \(e^2d^2p_{2m}\) contain \(p_{2m}\) or its first derivatives and hence vanish there. The last equality is positive homogeneity of the trace. The characteristic set, Hessian radical, and restricted symplectic ranks are unchanged. Thus the corresponding order-two theorem applies to \(\widetilde P\).

Here is the transfer back to \(P\), including the lower-order error. Let \(F\in\Psi^{m-1}\) be a proper parametrix of \(E\), so \(EF=I-R\) with \(R\) smoothing. For \(u\) supported in \(K\), put \(v=Fu\). Proper support places \(v\) in a fixed larger compact set. We have \[ u=Ev+Ru,\qquad \|v\|_0\leq C\|u\|_{m-1},\qquad \|u\|_{m-1/2}\leq C\|v\|_{1/2}+C\|u\|_{m-1}. \tag{7.3} \] The last inequality uses the mapping order \(1-m\) of \(E\) and smoothing of \(R\). Moreover \[ (Pu,u)=(\widetilde Pv,v)+O(\|u\|_{m-1}^2). \tag{7.4} \] For example, \(P:H^{m-1}\to H^{-m-1}\), whereas \(R:H^{m-1}\to H^{m+1}\); their dual pairing bounds each cross term by the error in (7.4). The term with two smoothing factors has the same bound. Applying the order-two non-strict estimate and (7.3) proves (1.3). Applying the strict estimate, and then the squared last inequality in (7.3), proves (1.5). This also covers negative and noninteger \(m\).

A choice of smooth positive density identifies half-densities unitarily with scalar functions. Under that identification the forms, Sobolev norms up to local equivalence, and hypotheses transfer unchanged.

8. Oscillator cancellation and irregular characteristic sets

On \(\mathbb R^2\), consider the formally self-adjoint differential operator \[ P_a=D_x^2+x^2D_y^2+aD_y,\qquad a\in\mathbb R. \tag{8.1} \] Its principal symbol is \(\xi^2+x^2\eta^2\) and its subprincipal symbol is \(a\eta\). At \(x=\xi=0\), \(\eta\ne0\), the quadratic form is \(Q=\dot\xi^2+\eta^2\dot x^2\), so its positive trace is \(|\eta|\). The correction is \[ a\eta+|\eta|. \tag{8.2} \] It is strictly positive in both frequency directions exactly when \(|a|<1\), and nonnegative in both directions exactly when \(|a|\leq1\). Here \(\Sigma\) is smooth, the Hessian radical is its tangent space, and the restricted symplectic form has constant rank. The strict theorem gives a positive \(H^{1/2}\) form term for \(|a|<1\). At \(a=-1\), the positive-frequency oscillator ground state cancels the drift exactly; Exercise 3 verifies why the strict term disappears.

The strict theorem also applies when the zero geometry is singular. In \(\mathbb R^4\), use the real homogeneous principal symbol \[ p_2=\xi_1^2+x_1^2\xi_2^2+(x_3^2-x_4^2)^2\xi_2^2 \tag{8.3} \] near a covector with \(\xi_2\ne0\), and choose a self-adjoint classical operator with this principal symbol and zero subprincipal symbol. On its zero set, \(x_1=\xi_1=0\) and \(x_3^2=x_4^2\). The first two terms give an oscillator frequency \(|\xi_2|\). The last term gives either one multiplication-square direction or the zero quadratic form, depending on whether \((x_3,x_4)\ne(0,0)\). It contributes no positive trace. Thus \(\operatorname{Tr}_+Q=|\xi_2|>0\) throughout this local zero set. The two branches cross, and the Hessian rank changes at their intersection. Section 4 nevertheless gives the strict localized \(H^{1/2}\) bound. No smooth zero manifold is needed for this example.

9. Exercises and complete solutions

Exercise 1 — an intrinsic scaling, 6 points. Let \(p_r\geq0\) be homogeneous of degree \(r\), vanish at \(\rho\), and let \(a>0\) be a smooth function near \(\rho\). Compute the Hessian quadratic form of \(a p_r\) at \(\rho\), and its positive trace.

Solution. Nonnegativity gives \(dp_r(\rho)=0\). The product rule yields \(d^2(ap_r)(\rho)=a(\rho)d^2p_r(\rho)\), because all other terms contain \(p_r(\rho)\) or \(dp_r(\rho)\). Hence \(Q_{ap}=a(\rho)Q_p\), and \(\operatorname{Tr}_+Q_{ap}=a(\rho)\operatorname{Tr}_+Q_p\). This is the exact Hessian calculation needed in (7.2); it would generally fail away from a zero.

Exercise 2 — a smooth zero set is insufficient, 8 points. For \(f(y,z)=z^4\), verify that the zero set is smooth but the hypothesis of Lemma 3.2 fails. Show that one cannot write \(f=b^2\) with \(db\ne0\) along its zero set.

Solution. The zero set is \(\{z=0\}\), but the Hessian is zero there, so its radical is all of \(\mathbb R^2\), rather than the tangent line to the zero set. If \(b^2=z^4\), then \(b(y,0)=0\). Differentiating twice at a zero gives \(d^2f=2(db)^2\). The left side is zero, forcing \(db=0\). Thus no independent defining square exists. Transverse quadratic nondegeneracy is a separate hypothesis in the non-strict theorem.

Exercise 3 — failure of a strict term at the endpoint, 12 points. For \(P_{-1}\) in (8.1), construct compactly supported normalized \(u_\lambda\), \(\lambda\to+\infty\), with \((P_{-1}u_\lambda,u_\lambda)=O(\lambda^{-1})\) and \(\|u_\lambda\|_{1/2}^2\geq c\lambda\). Deduce that no bound (1.5) with \(m=1,c_K>0\) holds on a fixed compact set containing their supports. Also consider \(a<-1\).

Solution. Let \(g\) be (4.2) in the quadratic lesson, set \(g_\lambda(x)=\lambda^{1/4}g(\sqrt\lambda x)\), and choose real normalized \(\chi\in C_c^\infty(\mathbb R_y)\). Initially put \(v_\lambda=g_\lambda\chi(y)e^{i\lambda y}\). Gaussian integration gives \[ \|D_xg_\lambda\|^2=\lambda/2,\quad \|xg_\lambda\|^2=1/(2\lambda),\quad \|D_y(\chi e^{i\lambda y})\|^2=\lambda^2+\|\chi'\|^2, \] and \((D_yv_\lambda,v_\lambda)=\lambda\). Hence \[ (P_av_\lambda,v_\lambda)=(1+a)\lambda+ \frac{\|\chi'\|^2}{2\lambda}. \tag{9.1} \] Multiply the \(x\)-factor by a fixed real compact cutoff equal to one near zero, and renormalize. Every altered energy integral and every derivative-of-cutoff term is bounded by a polynomial in \(\lambda\) times \(e^{-c\lambda}\), so (9.1) changes by that amount. The supports now lie in a fixed compact set.

The \(y\)-Fourier transform is \(\widehat\chi(\eta-\lambda)\). Choose \(R\) containing at least half its squared mass. For \(\lambda>2R\), that mass lies in \(|\eta|\geq\lambda/2\). The \(H^{1/2}\) norm squared has Fourier weight \(\langle(\xi,\eta)\rangle\geq|\eta|\), so it is at least \(\lambda/4\). For \(a=-1\), the left side of the proposed strict bound tends to zero, while its right side tends to infinity. For \(a<-1\), (9.1) tends to minus infinity; even a lower bound by \(-C\|u\|_0^2\) fails on this compact set. This establishes the endpoint distinction by explicit test functions.

Exercise 4 — a real order below zero, 8 points. Suppose \(P\) has order \(-3\), is self-adjoint, and satisfies the strict trace condition. State the lower bound, and identify the orders of the reducing operator and its parametrix.

Solution. Here \(2m=-3\), so \(m=-3/2\). The theorem gives \[ (Pu,u)\geq c_K\|u\|_{-2}^2-C_K\|u\|_{-5/2}^2. \] The reducing operator \(E\) has order \(1-m=5/2\), and its parametrix \(F\) has order \(-5/2\). Thus \(F:H^{-5/2}\to L^2\), while \(E:H^{1/2}\to H^{-2}\). These are exactly the two mapping statements in (7.3). A negative original order does not change the order-two positivity argument.

References

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