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

Nonnegative symbols and weighted brackets

A sum of squares provides an energy directly. For a scalar pseudodifferential operator, the corresponding information can be stated as an inequality between symbols. Quantization then introduces a bounded negative error. Once that error is controlled, the bracket argument applies even when no differential sum-of-squares representation is available.

We assume the scalar Fefferman–Phong theorem proved in When a nonnegative scalar symbol acquires a negative part, in the precise form stated in Section 1. We also use the ordinary composition, adjoint and Sobolev mapping theorems in Symbols, operators and Sobolev scales, changes of quantization in From Weyl symbols to operators and changes of coordinates, the refined symbol and conic calculus in Detecting regularity without choosing coordinates, and the full energy/bracket theorem in Brackets, drift, and general hypoellipticity. Every step that passes from those statements to the arbitrary-order theorem is proved below.

Lerner's openly available chapter [L] explains scalar positivity in phase space. Deng [D] discusses the dependence of the Weyl estimates on their structural constants. Hörmander's original paper [H] supplies the diffusion-and-drift antecedent. Our Fourier convention is \(D=-i\partial\); Weyl quantization uses the kernel \(e^{i(x-y)\cdot\xi}\). All symbol classes in this lesson are \(S^m_{1,0}\), with arbitrary real \(m\); no homogeneous expansion is assumed.

1. Positivity with a bounded error

Write \(w=\langle\xi\rangle\). The precise positivity prerequisite is \[ 0\leq a\in S^2 \quad\Longrightarrow\quad (a^wu,u)\geq-C_K\|u\|_0^2 \quad(u\in C_c^\infty(K)). \tag{1.1} \] The constant is controlled by finitely many symbol seminorms on a larger compact coordinate set. This is the scalar Fefferman–Phong inequality. In this application the ordinary phase-space metric is \[ g=|dx|^2+w^{-2}|d\xi|^2,\qquad h=w^{-1}; \] its inverse-square weight is \(h^{-2}=w^2\). Thus (1.1) is the classical specialization of the scalar theorem named in the prerequisites. It is stronger than the order-two sharp Gårding estimate, whose negative error can have order one.

We use proper Weyl quantizations. A smooth cutoff of their kernels near the diagonal changes the operators by a smooth kernel. On compact sets such errors are bounded on \(L^2\), and so are included in the constant in (1.1).

Proposition 1.1 (symbol-controlled energy). Let \(P\in\Psi^2\) have Weyl symbol \[ a+ib,\qquad a\in S^2,\quad b\in S^1, \quad a,b\text{ real}, \qquad a\geq-C_K. \tag{1.2} \] Suppose \(Q\in\Psi^1\) has real principal symbol \(q\), and \[ |q|^2\leq A_K a+B_K \tag{1.3} \] on the relevant compact coordinate set, for all sufficiently large frequencies. Then \[ \operatorname{Re}(Pu,u)\geq-C'_K\|u\|_0^2,\qquad \|Qu\|_0^2\leq A'_K\operatorname{Re}(Pu,u)+B'_K\|u\|_0^2. \tag{1.4} \]

Proof. Add a constant to \(a\) so it is nonnegative and apply (1.1). Since \(b^w\) is self-adjoint on test functions, \(i b^w\) has zero real quadratic form. The properization errors are bounded. This proves the first inequality.

Replace \(Q\) by its self-adjoint part; the difference has order zero. Choose a real full Weyl symbol for that part. Its difference from \(q\) has order zero, so its square still satisfies (1.3) after increasing both constants and adding a constant to \(a\). Denote this full symbol again by \(q\). The Weyl product gives \[ (q^w)^2=(q^2)^w+R_0,\qquad R_0\in\Psi^0, \tag{1.5} \] because the first correction is \((2i)^{-1}\{q,q\}=0\), and the second correction has order zero. Choose \(A'>0,B'\) such that \(A'a-q^2+B'\geq0\). Low frequencies are absorbed into \(B'\). Apply (1.1) to this nonnegative order-two symbol and use (1.5). The result is \[ \|q^wu\|_0^2\leq A'\operatorname{Re}(Pu,u)+C\|u\|_0^2. \] Adding back the order-zero part of \(Q\) proves (1.4). ∎

This gives an energy family from a pointwise inequality, rather than from an operator factorization.

2. The first derivatives of a nonnegative symbol

The bracket proof also requires energy control of the first derivatives of the order-two real principal symbol. Nonnegativity supplies exactly those bounds.

Lemma 2.1. Suppose \(a\in S^2\) is real and \(a\geq-C_K\). On each smaller compact coordinate set, \[ |\partial_{\xi_\nu}a|^2+ w^{-2}|\partial_{x_\nu}a|^2 \leq C'_K(a+C''_K). \tag{2.1} \]

Proof. We first recall a Taylor estimate with its domain restriction. Suppose a nonnegative smooth \(f\) is defined on a ball of fixed radius \(r\), with its gradient bounded by \(B\) and its Hessian norm bounded by \(M\). Work in the concentric ball of half the radius. Choose \(M_1\geq M+1\) and \(M_1\geq2B/r\), and take \[ h=-M_1^{-1}\nabla f(z). \] The segment from \(z\) to \(z+h\) stays in the original ball. Taylor's formula and nonnegativity give \[ 0\leq f(z+h) \leq f(z)-M_1^{-1}|\nabla f(z)|^2 +\frac M2 M_1^{-2}|\nabla f(z)|^2. \] Therefore \(|\nabla f(z)|^2\leq2M_1 f(z)\).

Add \(C_K\) to \(a\), enlarge the compact base set slightly, and consider \[ f_R(x,\eta)=R^{-2}(a(x,R\eta)+C_K), \quad R\geq1,\quad \tfrac12<|\eta|<2. \tag{2.2} \] The functions are nonnegative. Their derivatives of orders at most two are uniformly bounded, because \(a\in S^2\). Cover the unit sphere and the smaller base compact by finitely many balls of fixed radius contained in this domain. The Taylor estimate gives a uniform bound \[ |\partial_\eta f_R|^2+|\partial_x f_R|^2 \leq C f_R \quad\text{when }|\eta|=1. \] The chain rule reads \(\partial_\eta f_R=R^{-1}\partial_\xi a\) and \(\partial_x f_R=R^{-2}\partial_x a\). Multiply through by \(R^2\), use \(R=|\xi|\) and \(R\asymp w\) at high frequency, and absorb bounded frequencies into the constant. This proves (2.1). ∎

By Proposition 1.1 the operators with principal symbols \[ \partial_{\xi_\nu}a,\qquad w^{-1}\partial_{x_\nu}a \tag{2.3} \] are energy operators for \(P\). These are precisely the derivative operators needed for the energy-factor commutator in the preceding lesson.

3. Keeping two symbol orders

For a left symbol \(p\in S^m\), its Weyl symbol through the first correction is \[ p_{\mathrm w} =p+\frac i2\sum_\nu\partial_{x_\nu}\partial_{\xi_\nu}p \pmod {S^{m-2}}. \tag{3.1} \] This sign follows, for example, from the left symbol \(x\xi\): its operator is \(xD\), with Weyl symbol \(x\xi+i/2\).

Combining (3.1) with the ordinary left composition formula gives, for symbols of orders \(r,s\), \[ f\#g=fg+\frac1{2i}\{f,g\} \pmod {S^{r+s-2}}. \tag{3.2} \] Indeed convert each Weyl input to a left symbol through its first correction, compose there, and convert the output back using (3.1). The two derivatives of the product cancel the separate mixed derivatives of the inputs, leaving half the antisymmetric first composition term. All omitted terms have two net frequency derivatives and hence order \(r+s-2\). This proves the product formula used in (1.5) and Section 4 from the stated ordinary-calculus prerequisites.

The class in (3.1) is the refined principal symbol: it retains two orders, rather than just the highest one. On a manifold it is intrinsic for scalar half-density operators. We may equivalently work with scalar functions and a chosen smooth positive density, identifying functions with half-densities by multiplying by the square root of that density. The coordinate formula and its transformation law are proved in the geometric-calculus prerequisite.

Suppose the refined symbol has imaginary part in \(S^{m-1}\), and its real part becomes nonnegative after adding some real symbol in \(S^{m-2}\). In local Weyl quantization we can therefore write \[ p_{\mathrm w}=a+ib,\quad a\in S^m,\quad b\in S^{m-1},\quad a,b\text{ real}, \qquad a\geq-C_Kw^{m-2}. \tag{3.3} \] Conversely the last inequality makes \(a+C_Kw^{m-2}\) nonnegative, so it gives the same lower-order condition.

This condition can also be tested in left quantization. If \(\operatorname{Im}p\in S^{m-1}\), then the real part of the first correction in (3.1) has order \(m-2\), because it differentiates \(\operatorname{Im}p\). Thus the real parts in the two quantizations differ by \(S^{m-2}\). The imaginary part still has order \(m-1\). This proves the equivalence of (3.3) with the corresponding two conditions on the full left symbol.

For a real-principal \(Q\in\Psi^{m-1}\), the useful pointwise condition is \[ |q|^2\leq A_K a\,w^{m-2}+B_Kw^{2m-4}. \tag{3.4} \] Changing \(a\) by \(S^{m-2}\), or \(q\) by \(S^{m-2}\), preserves this condition after increasing constants: use \(|q+d|^2\leq2|q|^2+2|d|^2\). The additive weight in (3.4) is important. It becomes a constant when the operator is normalized to order two.

4. Normalizing the operator without losing its drift

Let \[ \alpha=w^{2-m}, \] and let \(A\in\Psi^{2-m}\) be a proper self-adjoint elliptic operator with Weyl symbol \(\alpha\) on the coordinate neighborhood under discussion, modulo smoothing. Put \[ \widetilde P=AP. \tag{4.1} \] The ordinary elliptic mapping theorem gives \[ Pu\in H^s\text{ at }\gamma \quad\Longleftrightarrow\quad \widetilde Pu\in H^{s+m-2}\text{ at }\gamma. \tag{4.2} \] Properization of the multiplier changes this only by a microlocally smoothing term.

Write \(P'=a^w\), \(T=b^w\), so \(P=P'+iT\), with harmless smoothing errors. The real and imaginary self-adjoint parts of \(\widetilde P\) are exactly \[ \widetilde P' =\frac{AP'+P'A}{2}+\frac i2[A,T], \qquad \widetilde T =\frac{AT+TA}{2}+\frac{[A,P']}{2i}. \tag{4.3} \] For the symmetric Weyl products the first Poisson terms cancel. The real part thus has Weyl symbol \[ \widetilde a=\alpha a+r_0,\qquad r_0\in S^0. \tag{4.4} \] Indeed the second product corrections have order \((2-m)+m-2=0\), and \([A,T]\) also has order zero. In particular \(\widetilde a\geq-C_K\).

The drift, however, has the additional first-order term \[ \widetilde b=\alpha b+e+r'_0,\qquad e=-\frac12\{\alpha,a\},\quad r'_0\in S^0. \tag{4.5} \] The sign uses \(\sigma([A,P'])=(1/i)\{\alpha,a\}\), divided by \(2i\). Omitting \(e\) would in general change the bracket condition.

Fortunately \(e\) is itself energy-controlled. Put \(f=\alpha a\), of order two and bounded below by a constant. Since \(\alpha\) is independent of \(x\), \[ e=-\frac12\sum_\nu \frac{\partial_{\xi_\nu}\alpha}{\alpha}\, \partial_{x_\nu}f, \qquad \frac{\partial_{\xi_\nu}\alpha}{\alpha}\in S^{-1}. \tag{4.6} \] Lemma 2.1 gives \[ |e|^2\leq C_K(f+C'_K) \leq C''_K(\widetilde a+C'''_K). \tag{4.7} \] Thus Proposition 1.1 makes any real-principal operator with symbol \(e\) an energy operator for \(\widetilde P\), when these bounds hold globally on the compact set in question.

Similarly, multiplication of (3.4) by \(\alpha^2\) gives \[ |\alpha q|^2\leq A_K f+B_K \leq C_K\widetilde a+C'_K. \tag{4.8} \] Hence the self-adjoint part of \(AQ\), which has principal symbol \(\alpha q\), is an energy operator. Its difference from \(AQ\) has order zero. Both the original admissible symbols and the normalization error therefore fit the energy family required by the order-two theorem.

5. A weighted bracket survives multiplication by a frequency weight

The condition on repeated brackets must also survive normalization. We prove this with order-zero coefficients, which supplies a uniform estimate even for symbols without homogeneous leading terms.

Consider finitely many real symbols \(f_j\in S^{m-1}\). A bracket word \(J\) has \(k\) letters and Poisson symbol \(G_J\), of order \[ d_k=k(m-2)+1. \tag{5.1} \] Normalize it by \[ F_J=\alpha^k G_J\in S^1. \tag{5.2} \] Give every letter a positive integer weight, and add those weights in a word.

Lemma 5.1. Each \(F_J\) is a finite sum, with coefficients in \(S^0\), of bracket words in the normalized letters \(\alpha f_j\) whose weights are at most the weight of \(J\). The identity holds modulo \(S^0\) if the symbols are specified only as principal classes.

Proof. For a one-letter word the assertion is equality. By Jacobi it suffices to treat words with a generator on the outside. Suppose \(G\) is the symbol for a \(k\)-letter word. The product rule gives the exact identity \[ \{\alpha f,\alpha^kG\} =\alpha^{k+1}\{f,G\} +k\{f,\alpha\}(\alpha^kG) +\bigl(\alpha^{k-1}\{\alpha,G\}\bigr)(\alpha f). \tag{5.3} \] Both displayed coefficients have order zero: \[ \operatorname{ord}\{f,\alpha\} =(m-1)+(2-m)-1=0, \] and \[ \operatorname{ord}\bigl(\alpha^{k-1}\{\alpha,G\}\bigr) =(k-1)(2-m)+(2-m)+d_k-1=0. \] Solve (5.3) for the first term on the right. By induction \(F_G\) is a finite sum of \(cH\), with \(c\in S^0\) and \(H\) an order-one normalized bracket of no larger weight. Then \[ \{\alpha f,cH\}=c\{\alpha f,H\}+\{\alpha f,c\}H. \] The new coefficient \(\{\alpha f,c\}\) has order zero, and the new bracket has weight at most the weight of the original word plus that of \(f\). The remaining terms of (5.3) have only the previous word or the generator. This proves the assertion by induction. The error from replacing any symbol by its one-order-lower representative becomes order zero after normalization, so the same conclusion holds for principal classes. ∎

If a word is noncharacteristic at its natural order, meaning \[ |G_J(x,\xi)|\geq c w^{d_k} \tag{5.4} \] in a conic neighborhood at high frequency, then \(|F_J|\geq c w\). The \(S^0\) coefficients in Lemma 5.1 are bounded there. Consequently a finite family of normalized bracket words of no larger weight satisfies \[ \sum_\ell|H_\ell|^2\geq c' w^2 \tag{5.5} \] on a smaller conic neighborhood. To justify the lower bound, move the \(S^0\) remainder to the other side, absorb its bounded size for sufficiently large \(w\), and use Cauchy–Schwarz on the finite sum. This is a uniform ellipticity assertion, not just nonvanishing at one frequency.

In our application the original energy letters are the \(q_j\), of weight one, and the original drift letter is \(b\), of weight two. The normalized energy letters are \(\alpha q_j\). Equation (4.5) writes \[ \alpha b=\widetilde b-e\pmod {S^0}. \tag{5.6} \] Here \(\widetilde b\) is the actual normalized drift, of weight two, while \(e\) is an energy letter, of weight one. Expand each normalized bracket using (5.6). Every resulting word has weight at most the original one. Thus (5.5) remains true for the actual energy family and drift of \(\widetilde P\).

6. Extending a conic hypothesis

The original conditions may hold only near one covector. We cannot apply a globally stated test-function estimate until its energy hypotheses hold throughout the support of the test function. We now construct such an extension.

Work in a compact coordinate neighborhood and a conic set on which (3.3)–(3.4) hold and the bracket under consideration is elliptic. Choose real order-zero cutoffs \(\chi_0,\chi_1\), homogeneous of degree zero at sufficiently large frequency, with \[ \chi_0^2+\chi_1^2=1. \tag{6.1} \] Let \(\chi_0=1\) on a smaller closed conic neighborhood of \(\gamma\), and let its high-frequency support lie in the set where all the hypotheses hold. Choose its plateau so all derivatives of \(\chi_0-1\) vanish on the closed set where \(\chi_1=0\). For example take the cosine and sine of a smooth cutoff angle which is zero on that plateau, positive off it, and \(\pi/2\) outside the larger conic neighborhood. Include a base cutoff and a low-frequency cutoff in that angle. The nonsmoothing terms with \(\chi_0\ne0\) then have compact base support and lie in the given cone. Any bounded low-frequency discrepancy is absorbed into a constant.

Let \(\widetilde a,\widetilde b\) be the real Weyl symbols in (4.4)–(4.5). Extend their cutoff products smoothly by zero where necessary, and properly quantize \[ a_{\mathrm{ext}}=\chi_0^2\widetilde a+\chi_1^2w^2, \qquad b_{\mathrm{ext}}=\chi_0^2\widetilde b,\qquad P_{\mathrm{ext}}=a_{\mathrm{ext}}^w+i b_{\mathrm{ext}}^w. \tag{6.2} \] The real symbol is order two and bounded below by a constant; the imaginary symbol has order one. The extension agrees with \(\widetilde P\) microlocally near \(\gamma\), modulo smoothing.

Proposition 1.1 and Lemma 2.1 give the derivative energy hypotheses everywhere for \(P_{\mathrm{ext}}\). The cutoffs of the symbols \(\alpha q_j\) and \(e\) by \(\chi_0\) are energy symbols as well: (4.7)–(4.8), multiplied by \(\chi_0^2\), are bounded by a constant times \(a_{\mathrm{ext}}\) plus a constant. The additional symbols \[ \chi_1\xi_\nu,\qquad 1\leq\nu\leq n, \tag{6.3} \] are energy symbols, since \(\chi_1^2|\xi_\nu|^2\leq a_{\mathrm{ext}}+C\).

Where \(\chi_1\ne0\), these last symbols have no common characteristic covector. Where \(\chi_1=0\), the extension and all its derivatives agree with the normalized operator and letters, so the bracket family in (5.5) has no common characteristic covector. Its lower bound is uniform on that closed plateau. Symbol seminorms give uniform continuity after rescaling \(\xi=R\eta\), \(|\eta|=1\), so this lower bound persists on a fixed neighborhood of the plateau. On the remainder of any compact cosphere, \(\chi_1\) has a positive minimum, and (6.3) supplies a uniform bound. This proves the full finite-family ellipticity condition for the order-two bracket theorem, with the same maximum weight \(N\). Outside the compact base support, \(\chi_1=1\) and the extension is elliptic.

All these constructions use ordinary proper operators and smoothing remainders. Applied to a compactly localized distribution, the latter are smooth near \(\gamma\). They therefore do not alter the microlocal regularity conclusion.

7. The arbitrary-order theorem

Theorem 7.1. Let \(P\in\Psi^m(X)\) be scalar and properly supported, where \(m\) is any real number. On a conic neighborhood of \(\gamma\in T^*X\setminus0\), suppose the refined principal symbol has imaginary part of order \(m-1\), and its real part becomes nonnegative after addition of a symbol of order \(m-2\). Equivalently, choose the real Weyl representatives \(a,b\) in (3.3).

Let the admissible energy letters be the real principal symbols \(q\) of operators in \(\Psi^{m-1}\) satisfying (3.4). Give these letters weight one and give \(b\) weight two. Suppose some repeated bracket with \(\nu_1\) energy letters and \(\nu_2\) drift letters is noncharacteristic at its natural order, with \[ \nu_1+2\nu_2\leq N. \tag{7.1} \] Then for every distribution \(u\) and every real \(s\), \[ Pu\in H^s\text{ at }\gamma \quad\Longrightarrow\quad u\in H^{s+m-2+2\varepsilon}\text{ at }\gamma, \qquad 0<\varepsilon\leq2^{-N}. \tag{7.2} \] In particular smoothness of \(Pu\) at \(\gamma\) implies smoothness of \(u\) there.

Proof. Normalize to \(\widetilde P=AP\) as in Section 4, so the data have order \(t=s+m-2\) by (4.2). Section 4 shows that the normalized admissible letters and the error \(e\) are energy symbols, and that the real part is bounded below by a constant. Section 5 transfers the original noncharacteristic bracket to a finite family of actual normalized energy/drift brackets of weight at most \(N\). Section 6 extends these local symbol conditions to an order-two operator with the global compact-test energy and bracket hypotheses.

Apply Theorem 6.1 of Brackets, drift, and general hypoellipticity to \(P_{\mathrm{ext}}\), at Sobolev order \(t\). The conic equality with \(\widetilde P\) makes \(P_{\mathrm{ext}}u\) belong to \(H^t\) at \(\gamma\). That theorem gives \[ u\in H^{t+2\varepsilon}=H^{s+m-2+2\varepsilon} \quad\text{at }\gamma. \] Compactly localize \(u\) first if necessary. The commutator with a base cutoff equal to one near the base point is smoothing microlocally there, so this localization does not change the hypothesis or conclusion.

On a manifold, choose a coordinate chart and identify scalar functions with half-densities using a smooth positive density. The refined-symbol transformation rule supplies (3.3) in that chart; the principal symbol bracket and its ellipticity are preserved by coordinate transport. Sobolev regularity is intrinsic by the same geometric-calculus prerequisite. Hence the local proof establishes the statement on \(X\). Finally take arbitrarily large \(s\) when \(Pu\) is smooth. ∎

The guaranteed gain is \(m-2+2\varepsilon\), which can be negative for low-order operators. Even then the implication for smooth data follows, because the hypothesis is available at every \(s\). A single finite-order estimate and smooth-data hypoellipticity are different conclusions.

8. Examples and exercises with complete solutions

A symbol not presented as differential squares. In \((v,z)\), let \[ p_{\mathrm w}(v,z,\nu,\zeta) =w^{m-2}\bigl(\nu^2+i v\zeta\bigr), \quad w=(1+\nu^2+\zeta^2)^{1/2}. \tag{8.1} \] Its real part is nonnegative of order \(m\), and its imaginary part has order \(m-1\). The real symbol \(q=w^{m-2}\nu\) satisfies (3.4) with \(A=1,B=0\). Where \(\nu\ne0\), this letter is noncharacteristic in its natural order. At \(\nu=0,\zeta\ne0\), the original bracket has value \[ \{w^{m-2}\nu,w^{m-2}v\zeta\} =w^{2m-4}\zeta \quad\text{when }\nu=0. \] It is therefore noncharacteristic of order \(2m-3\) on a conic neighborhood of those directions. The normalized letters are \(\nu\), of weight one, and \(v\zeta\), of weight two, with bracket \(\zeta\). Here the error \(e\) in (4.5) vanishes because the real symbol is independent of the base variables; the remaining normalization errors have order zero. Thus the theorem gives, in every direction after compact localization, \[ Pu\in H^s\quad\Longrightarrow\quad u\in H^{s+m-7/4}. \tag{8.2} \] Here \(2\varepsilon=1/4\). This is the general guaranteed exponent. The example uses arbitrary real \(m\), so the normalized mechanism does not depend on an integer differential order.

Exercise 1 — the additive symbol weight, 6 points. Normalize (3.4) for \(m=3\) and for \(m=-1\). Identify the energy symbols and the bounded additive terms.

Solution. At \(m=3\), \(\alpha=w^{-1}\) and the hypothesis is \[ |q|^2\leq A a w+B w^2. \] Multiplication by \(w^{-2}\) gives \(|w^{-1}q|^2\leq A w^{-1}a+B\). At \(m=-1\), \(\alpha=w^3\), and the hypothesis is \[ |q|^2\leq A a w^{-3}+B w^{-6}. \] Multiplication by \(w^6\) gives \(|w^3q|^2\leq A w^3a+B\). In both cases \(\alpha q\) is of order one, \(\alpha a\) is of order two, and the additive term becomes a constant. Replacing \(w^{2m-4}\) by \(w^{2m-2}\) in the hypothesis would instead produce an order-two additive term; that would not imply the energy bound used here.

Exercise 2 — a normalization error that is present, 8 points. In one dimension take \(a(x,\xi)=h(x)w^m\), with smooth real \(h\geq0\). Compute \(e\) in (4.5), and prove its energy bound on compact sets without assuming \(h\) is strictly positive.

Solution. We have \(\alpha=w^{2-m}\) and \[ \partial_\xi\alpha=(2-m)\xi w^{-m}. \] Since \(\partial_xa=h'(x)w^m\), the Poisson convention gives \[ e=-\frac12(2-m)\xi h'(x). \] The one-dimensional Taylor estimate in Lemma 2.1, on a slightly larger compact interval, gives \(|h'|^2\leq C h\), with a constant depending on finite derivative bounds and the interval. Therefore \[ |e|^2\leq C\xi^2h(x)\leq C h(x)w^2=C\alpha a. \] The estimate also holds at zeros of \(h\), where its first derivative vanishes. For \(m=2\) the error is identically zero; otherwise it need not vanish.

Exercise 3 — two normalized letters, 10 points. Let \(f,g\in S^{m-1}\). Express \(\alpha^2\{f,g\}\) in terms of \(\{\alpha f,\alpha g\}\), \(\alpha f\), and \(\alpha g\), with order-zero coefficients. Explain how ellipticity of the original bracket implies a finite normalized bracket condition.

Solution. The product rule gives \[ \alpha^2\{f,g\} =\{\alpha f,\alpha g\} -\{f,\alpha\}(\alpha g) -\{\alpha,g\}(\alpha f). \tag{8.3} \] Both coefficients \(\{f,\alpha\}\) and \(\{\alpha,g\}\) have order zero, since \((m-1)+(2-m)-1=0\). The original bracket has natural order \(2m-3\). If its absolute value is at least \(c w^{2m-3}\), multiplication by \(\alpha^2=w^{4-2m}\) makes the left side at least \(c w\). The coefficients on the right are bounded, so Cauchy–Schwarz implies that the three order-one symbols there have squared sum at least \(c'w^2\). The two one-letter terms have smaller weights than the two-letter bracket. This is exactly the finite-family condition required after normalization.

Exercise 4 — low-order regularity and smooth data, 8 points. Suppose \(m=0\), \(N=4\), and the hypotheses of Theorem 7.1 hold. State the conclusion from an \(H^s\) right-hand side at the largest permitted \(\varepsilon\). Explain why this still proves smooth-data hypoellipticity.

Solution. The largest value is \(\varepsilon=1/16\), so \[ Pu\in H^s\quad\Longrightarrow\quad u\in H^{s-15/8} \quad\text{at the covector.} \] This loses \(15/8\) derivatives relative to the data at that fixed order. If \(Pu\) is smooth at the covector, it belongs to \(H^s\) there for every \(s\). For any desired \(r\), choose \(s=r+15/8\); the implication gives \(u\in H^r\). Thus \(u\) is smooth microlocally, despite the negative relative gain at each fixed \(s\).

References

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