Inverting an operator with unequal frequency scales

An elliptic system measures every component at one common frequency order. Some systems need different orders for different components. Others have a scalar symbol whose size changes substantially with the frequency direction. Both can have an inverse modulo a smooth error. The useful question is whether derivatives of the symbol remain small relative to the symbol itself, measured between appropriate norms.

We assume the composition, adjoint and asymptotic summation theorems in Symbols, operators and Sobolev scales, together with proper quantization, conic localization and the classical wavefront tests in Detecting regularity without choosing coordinates. We state the exact calculus formulas used below. Fourier transformation has forward kernel $e^{-ix\cdot\xi}$, and $D=-i\partial$. The elementary prerequisites are finite-dimensional linear algebra, the product rule, smooth partitions of unity, and distributions. For the bundle extension we also assume smooth finite-rank bundles on Hausdorff, second countable smooth manifolds without boundary. Basic references are Grubb [G], Melrose [M] and Hörmander [H].

1. A system that needs two orders

Write $w=\langle\xi\rangle=(1+|\xi|^2)^{1/2}$. Let $b$ be a smooth scalar function on an open set $X\subset\mathbb R^n$. Consider the matrix symbol $p(x,\xi)= \begin{pmatrix}w^2&b(x)w\\0&w\end{pmatrix}. \tag{1.1}$ On each compact base set it is a symbol of order two. Its smallest singular value is comparable to $w$: factor it as $p=B(x)\begin{pmatrix}w^2&0\\0&w\end{pmatrix}, \qquad B(x)=\begin{pmatrix}1&b(x)\\0&1\end{pmatrix}.$ Both $B$ and $B^{-1}$ are bounded on compact sets. The upper and lower sizes therefore have different orders. No ordinary order $m$ makes this simultaneously an order-$m$ symbol and uniformly elliptic of order $m$. The first column requires $m\geq2$, while the second column makes a lower bound of order $m>1$ impossible.

Instead give the domain and range the norms $\|z\|_E=(w^4|z_1|^2+w^2|z_2|^2)^{1/2}, \qquad \|v\|_F=|v|. \tag{1.2}$ The factorization gives $\|z\|_E\leq C_K\|pz\|_F$. Every frequency derivative costs its usual inverse power of $w$, and every base derivative remains bounded between these two norms. The pointwise inverse is $p^{-1}= \begin{pmatrix}w^{-2}&-b(x)w^{-2}\\0&w^{-1}\end{pmatrix}. \tag{1.3}$ It is bounded from $F$ to $E$, although its two rows have different ordinary orders. Quantizing (1.3) gives the first approximation to an operator inverse. Derivatives of $b$ cause composition errors; their successive corrections are the subject of this lesson.

2. Pointwise norms and a relative derivative condition

Fix $0\leq\delta<\rho\leq1,\qquad \kappa=\rho-\delta>0. \tag{2.1}$ All norms below are on fixed finite-dimensional spaces. They depend on $(x,\xi)$, but need not depend continuously or smoothly on those parameters.

Definition 2.1. A family $\|\cdot\|_{E,x,\xi}$ is polynomially bounded in both directions if for each $K\Subset X$ there are $C_K\geq1$ and $M_K\geq0$ such that $C_K^{-1}w^{-M_K}|z| \leq\|z\|_{E,x,\xi} \leq C_Kw^{M_K}|z|. \tag{2.2}$ The same condition is imposed on $F$. Constants and polynomial exponents may depend on $K$. We abbreviate these norms by $\|\cdot\|_E,\|\cdot\|_F$, always at the displayed point.

For a smooth matrix $a:E\to F$, write $a\in\mathcal A^r(E,F)$ when, for all multi-indices and compact base sets, $\|\partial_\xi^\alpha D_x^\beta a(x,\xi)z\|_F \leq C_{K\alpha\beta} w^{r-\rho|\alpha|+\delta|\beta|}\|z\|_E. \tag{2.3}$ The derivatives act on $a$, not on either norm. Polynomial norm comparison implies an ordinary symbol estimate $\|\partial_\xi^\alpha D_x^\beta a\|_{\mathrm{Eucl}} \leq C_{K\alpha\beta} w^{r+M_E(K)+M_F(K)-\rho|\alpha|+\delta|\beta|}. \tag{2.4}$ Thus all these symbols have finite order on each compact base set. On a noncompact $X$, a single order valid on every compact set is not required. Operators of this kind are defined by localization; every compact localization has an ordinary finite symbol order. A symbol with every negative ordinary order will be called smoothing. It also belongs to every $\mathcal A^r$, by (2.2).

Suppose now $E,F$ have the same dimension and $p\in\mathcal A^0(E,F)$. The essential additional assumption is $\|z\|_E\leq C_K\|p(x,\xi)z\|_F \quad(x\in K,\ |\xi|\geq R_K). \tag{2.5}$ This is a pointwise lower bound; it does not assert an $L^2$ estimate for an operator measured with the parameter-dependent norms.

Lemma 2.2. Above the compact-dependent frequency radius in (2.5), the matrix $p$ is invertible and $\|\partial_\xi^\alpha D_x^\beta p^{-1}v\|_E \leq C_{K\alpha\beta} w^{-\rho|\alpha|+\delta|\beta|}\|v\|_F. \tag{2.6}$

Proof. The lower bound makes $p$ injective; equal finite dimensions make it invertible. Equation (2.5) proves (2.6) without derivatives. Differentiate $pp^{-1}=I$. For a nonzero combined derivative $\mu=(\alpha,\beta)$, $p\,\partial_\xi^\alpha D_x^\beta p^{-1} =-\sum_{0<\nu\leq\mu}c_{\mu\nu} (\partial^\nu p)(\partial^{\mu-\nu}p^{-1}). \tag{2.7}$ Here $\partial^\nu$ has the corresponding frequency and $D_x$ derivatives, and the coefficients are the product-rule coefficients. Apply the inductive inverse bound to the rightmost factor, then (2.3) to the left factor. The powers add to $-\rho|\alpha|+\delta|\beta|$. Finally apply (2.5) to the vector on the left of (2.7). This proves the next derivative order. Matrix factors have retained their order throughout. ∎

Proposition 2.3. For a smooth scalar symbol, the existence of norms satisfying (2.2), (2.3) with $r=0$, and (2.5) is equivalent to the following high-frequency conditions on each compact base set: $p$ is nonzero, both $p$ and $1/p$ have polynomial bounds, and $|\partial_\xi^\alpha D_x^\beta p| \leq C_{K\alpha\beta}|p| w^{-\rho|\alpha|+\delta|\beta|}. \tag{2.8}$

Proof. A norm on the one-dimensional complex space is a positive multiple of absolute value. Let $M$ be the ratio of its domain and range multipliers. The derivative estimate with no derivatives and the lower bound give $|p|\leq CM$ and $M\leq C'|p|$, respectively. Polynomial comparison for the two norms gives polynomial upper and lower bounds for $M$, hence the asserted bounds for $p$ and $1/p$. The other derivative estimates imply (2.8).

Conversely choose a positive scalar weight $M$ equal to $|p|$ at sufficiently high frequency on each compact base set. It can be filled in positively at low frequencies by a locally finite base partition and frequency cutoffs: each summand is a partition function times $h_l|p|+(1-h_l)$, with $h_l=1$ only where nonvanishing is guaranteed. The sum is positive everywhere and equals $|p|$ when all active cutoffs equal one. Polynomial upper and lower bounds for $|p|$ give (2.2) for $M|z|$; positivity and compactness handle each remaining bounded-frequency region. Take that domain norm and the Euclidean range norm. Equation (2.8) supplies the high-frequency derivative estimates, and compactness supplies the remaining estimates. The lower bound is equality at high frequency. ∎

For a graded system, choose real exponents $t_j,s_i$ and norms $\|z\|_E=|D_tz|$, $\|v\|_F=|D_sv|$, where $D_t=\operatorname{diag}(w^{t_j})$ and $D_s=\operatorname{diag}(w^{s_i})$. Equation (2.3) is then equivalent, up to constants depending only on the matrix dimension, to the entrywise bounds $|\partial_\xi^\alpha D_x^\beta p_{ij}| \leq C_{K\alpha\beta} w^{t_j-s_i-\rho|\alpha|+\delta|\beta|}. \tag{2.9}$ Indeed the measured derivative matrix is $D_s(\partial_\xi^\alpha D_x^\beta p)D_t^{-1}$; each entry is bounded by its operator norm, and its operator norm is bounded by the sum of the entry magnitudes. The lower bound (2.5) is exactly a uniform lower singular-value bound for $D_spD_t^{-1}$. Taking all $t_j=t$ and all $s_i=s$ recovers ordinary ellipticity of order $t-s$. Unequal exponents give the graded situation illustrated in Section 1.

3. Composition with rough measuring norms

For ordinary symbols, the prerequisite calculus gives the full symbol $a\#b$ of a composition, modulo smoothing terms from proper quantization. Its expansion is $a\#b\sim\sum_{\gamma} \frac{(\partial_\xi^\gamma a)(D_x^\gamma b)}{\gamma!}. \tag{3.1}$ For symbols of ordinary orders $m_a,m_b$, the remainder after $|\gamma|<N$ is in $S^{m_a+m_b-N\kappa}_{\rho,\delta}$, with all differentiated estimates. Products are associative modulo smoothing, and smoothing symbols form a two-sided ideal for locally finite-order operators. These statements require $\delta<\rho$, but do not require the coordinate-invariance inequality $\delta\geq1-\rho$.

Lemma 3.1. If $b\in\mathcal A^s(E,F)$ and $a\in\mathcal A^r(F,G)$, then $a\#b\in\mathcal A^{r+s}(E,G),$ and for every integer $L\geq0$, $a\#b-\sum_{|\gamma|<L} \frac{(\partial_\xi^\gamma a)(D_x^\gamma b)}{\gamma!} \in\mathcal A^{r+s-L\kappa}(E,G). \tag{3.2}$

Proof. Each displayed coefficient in (3.1) has class $\mathcal A^{r+s-|\gamma|\kappa}(E,G)$. For any further derivative, the product rule splits its frequency and base counts between the two factors. Their bounds are composed through the same norm on $F$, at the same point. This proves all derivative estimates for that coefficient without differentiating a norm.

Fix a compact base set. On a slightly larger compact set, (2.4) supplies finite ordinary orders $m_a,m_b$. Expand (3.1) to an integer $N\geq L$ so large that $m_a+m_b-N\kappa+M_E(K)+M_G(K) \leq r+s-L\kappa. \tag{3.3}$ The ordinary remainder, converted back to the $E,G$ norms by (2.2), then has the required bound in (3.2). The same $N$ works for every derivative: its frequency and base derivative costs already have the required form. The extra finitely many coefficients with $L\leq|\gamma|<N$ also have the required class. Sum these bounds. This proves (3.2) on the chosen compact set, and hence on every compact set. Taking $L=0$ proves the first assertion. ∎

Lemma 3.2. Given $a_j\in\mathcal A^{r-j\kappa}(E,F)$, there is a smooth symbol $a$ such that $a-\sum_{j<N}a_j\in\mathcal A^{r-N\kappa}(E,F) \quad(N\geq0). \tag{3.4}$ It has locally finite ordinary order. Two such sums differ by a smoothing symbol.

Proof. Use a compact exhaustion $K_l$ of $X$, and the seminorms in (2.3), with finitely many derivatives at each stage. Choose a smooth frequency cutoff $\chi$ equal to one near zero and compactly supported. For $j\geq1$, choose $R_j\to\infty$ such that $A_j=(1-\chi(\xi/R_j))a_j$ has each seminorm with $l\leq j$, derivative count at most $j$, and order $r-j\kappa+\kappa/2$, bounded by $2^{-j}$. This is possible because the extra factor $w^{-\kappa/2}$ tends to zero at high frequency. Derivatives hitting the cutoff contribute $R_j^{-|\gamma|}$ on $|\xi|\asymp R_j$, at most a constant times $w^{-\rho|\gamma|}$, since $\rho\leq1$. These statements involve only bounds on derivatives of $a_j$. No regularity of the norms is used. Take any fixed cutoff for $a_0$ as well.

The sum $a=\sum_jA_j$ is locally finite in frequency, hence smooth. Fix $N,l$ and a derivative count. For sufficiently large $J$, all $j\geq J$ satisfy the stipulated seminorm bounds and $r-j\kappa+\kappa/2\leq r-N\kappa$. Their tail is bounded by the convergent series $\sum_{j\geq J}2^{-j}$. The remaining terms in $a-\sum_{j<N}a_j$ are finitely many $A_j$ with $j\geq N$, already of the required class, and finitely many cutoff differences supported in bounded frequency sets. Those differences are smoothing on each compact base set, by (2.2) and smoothness of $a_j$. This proves (3.4). Equation (2.4) proves local finite order. For two sums the difference has every negative adapted order, hence every negative ordinary order by the fixed polynomial comparison on each compact set. ∎

These are the two places where polynomial boundedness matters. It converts the ordinary composition remainder into the desired norm estimate and converts an arbitrarily small adapted tail into an ordinary smoothing error. Bounds comparing norms at different parameter points are unnecessary.

4. The inverse modulo a smooth error

Theorem 4.1. Let $P$ be a properly supported pseudodifferential operator on $X$, acting between two copies of $\mathbb C^N$. Suppose its full symbol $p$ satisfies (2.3) with $r=0$ and the lower bound (2.5), for polynomially bounded domain and range norms. Then there is a properly supported operator $Q$, with symbol $q\in\mathcal A^0(F,E)$, such that $PQ=I+R_F,\qquad QP=I+R_E, \tag{4.1}$ where $R_E,R_F$ have smooth kernels. The symbol $q$ also has the lower bound with the domain and range norms interchanged: $\|v\|_F\leq C_K\|q(x,\xi)v\|_E \quad(x\in K,\ |\xi|\geq R'_K). \tag{4.2}$ The parametrix is unique modulo smoothing operators.

Proof. Cover $X$ by relatively compact open sets $U_l$ with frequency radii from (2.5), and choose a locally finite smooth partition $\theta_l$ with compact support inside $U_l$. For each $l$, let $h_l(\xi)$ be zero before invertibility on $U_l$ is guaranteed and one beyond a larger frequency radius. Define $q_0=\sum_l\theta_l(x)h_l(\xi)p(x,\xi)^{-1},$ extending each summand by zero where its cutoff vanishes. Each summand is smooth. On a fixed compact base set only finitely many occur, and beyond the largest of their radii their sum is exactly $p^{-1}$. Derivatives of the base cutoffs cost zero, which is at most $\delta$ per derivative, and frequency cutoff derivatives cost at most their required powers since $\rho\leq1$. Lemma 2.2 therefore gives $q_0\in\mathcal A^0(F,E)$, and $pq_0-I,\quad q_0p-I$ are smoothing symbols. Properly quantize $q_0$; the properness correction is smoothing and preserves these estimates.

Use the symbol product modulo smoothing and set $r=I-p\#q_0\in\mathcal A^{-\kappa}(F,F), \qquad \ell=I-q_0\#p\in\mathcal A^{-\kappa}(E,E). \tag{4.3}$ Lemma 3.1 shows that $r^{\#j}$ and $\ell^{\#j}$ have adapted order $-j\kappa$. Lemma 3.2 constructs symbols $q_R\sim\sum_{j\geq0}q_0\#r^{\#j}, \qquad q_L\sim\sum_{j\geq0}\ell^{\#j}\#q_0. \tag{4.4}$ For each $N$, the finite algebraic identities are $p\#\sum_{j<N}q_0\#r^{\#j}=I-r^{\#N}, \qquad \left(\sum_{j<N}\ell^{\#j}\#q_0\right)\#p=I-\ell^{\#N}. \tag{4.5}$ The error in replacing a finite sum by its asymptotic sum has adapted order $-N\kappa$. Composing it with $p$ preserves that order. Thus $p\#q_R-I$ and $q_L\#p-I$ have every negative adapted order, and are smoothing by (2.4). No convergence of an operator geometric series has been asserted.

The two inverses agree modulo smoothing, because $q_L-q_R =q_L\#(I-p\#q_R)+(q_L\#p-I)\#q_R. \tag{4.6}$ The smoothing ideal makes the right side smoothing. Properly quantize either symbol to obtain (4.1). The same identity proves uniqueness by comparison with any other two-sided parametrix.

Finally (4.4) gives $q-q_0\in\mathcal A^{-\kappa}(F,E)$. At high frequency $q_0=p^{-1}$, so the pointwise product satisfies $\|pq-I\|_{F\to F}\leq C_Kw^{-\kappa}.$ Increase the radius until this is at most $1/2$. Then $\|v\|_F\leq2\|pqv\|_F\leq C'_K\|qv\|_E,$ using $p\in\mathcal A^0(E,F)$. This proves (4.2). ∎

Corollary 4.2. If $e\in\mathcal A^{-\epsilon}(E,F)$ for some $\epsilon>0$, then $p+e$ satisfies the same hypotheses after increasing the frequency radius.

Proof. Lemma 2.2 gives $\|p^{-1}e\|_{E\to E}\leq C_Kw^{-\epsilon}$. Make this at most $1/2$. The finite-dimensional Neumann series bounds $(I+p^{-1}e)^{-1}$ in the $E$ norm by two, and hence bounds $(p+e)^{-1}:F\to E$. The derivative condition follows by adding the derivative bounds. ∎

5. Why the inverse recovers every singular direction

The wavefront set of a vector distribution is the union of the wavefront sets of its components. We use its Fourier definition, or equivalently the classical order-zero elliptic tests proved in Detecting regularity without choosing coordinates.

Lemma 5.1. A properly supported, locally finite-order operator of type $(\rho,\delta)$, with $0\leq\delta<\rho\leq1$, satisfies $\operatorname{WF}(Au)\subset\operatorname{WF}(u). \tag{5.1}$

Proof. Fix a covector absent from $\operatorname{WF}(u)$. The Fourier cutoff construction in the prerequisite gives a proper classical order-zero operator $B$ such that $Bu$ is smooth and the full symbol of $I-B$ is smoothing on a neighborhood of that covector. Choose a classical order-zero cutoff $C$, elliptic at the covector, with compact base support and angular support strictly inside that neighborhood. Both classical cutoffs belong to the type-$(\rho,\delta)$ calculus.

Every coefficient in the composition expansion of $CA(I-B)$ is rapidly decreasing: on the support of a derivative of the symbol of $C$, the symbol of $I-B$ and all its derivatives are rapidly decreasing. Off that support the coefficient is zero. Remainders become arbitrarily negative because $\kappa>0$. Conic localization, with nested cutoffs, gives the same assertion for changes of the cutoff extensions. Consequently $CA(I-B)$ is smoothing. Properness reduces all these assertions to finite ordinary orders on compact sets, even when their orders are not globally uniform. Since $CA(Bu)$ is smooth as well, $CAu$ is smooth. The classical elliptic test for $C$ shows that the chosen covector is absent from $\operatorname{WF}(Au)$. ∎

This proof applies in a fixed Euclidean chart throughout (2.1). It uses the positive gain in the symbol expansion. It does not infer coordinate invariance for the symbol class outside its invariant parameter range.

Theorem 5.2. Under Theorem 4.1, every $u\in\mathcal D'(X;\mathbb C^N)$ satisfies $\operatorname{WF}(Pu)=\operatorname{WF}(u). \tag{5.2}$ In particular, $Pu$ smooth on an open set implies $u$ smooth there.

Proof. Lemma 5.1 gives one inclusion. From (4.1), $u=QPu-R_Eu$. A proper smooth-kernel operator maps distributions to smooth functions: on each output compact set, properness restricts the input to a compact set, on which the distribution has finite order, and derivatives of the smooth kernel can be paired with it. Applying Lemma 5.1 to $Q$ gives the reverse inclusion. ∎

Theorem 5.3. Suppose the derivative and lower-bound assumptions hold only in an open conic set $\Gamma\subset T^*X\setminus0$, uniformly on every compact set of ray generators. Then $\operatorname{WF}(Pu)\cap\Gamma =\operatorname{WF}(u)\cap\Gamma. \tag{5.3}$

Proof. Fix a covector in $\Gamma$, and choose successively smaller base and angular neighborhoods with closures inside $\Gamma$. Construct a frequency-cutoff inverse $q_0$ on one of them, extend with a smooth cone cutoff, and restrict (4.3) to a smaller neighborhood. All pointwise derivative products there have the estimates of Lemma 3.1. The ordinary conic remainder theorem has arbitrarily negative remainders there; polynomial norm comparison proves the adapted remainders exactly as in (3.3). Summation by Lemma 3.2, with the cone cutoff kept in every term, yields left and right inverses whose errors are smoothing on a further smaller neighborhood. The comparison identity (4.6) applies there.

Thus $u=QPu+(I-QP)u$, where the second term is wavefront-regular at the chosen covector. Indeed, place a classical cutoff supported in the smoothing-error cone on its left; the product is smoothing by the composition expansion. Lemma 5.1 gives regularity of the first term whenever $Pu$ is regular at that covector. The forward inclusion is again (5.1). This proves (5.3) at each covector of $\Gamma$. ∎

6. Adjoints, coordinates, and bundles

For a norm on $E$, its dual norm is $\|\lambda\|_{E^*}=\sup_{\|z\|_E\leq1}|\lambda(z)|. \tag{6.1}$ It satisfies (2.2), with the same polynomial exponent. Identifying anti-duals by the Euclidean Hermitian pairing, a matrix $a:E\to F$ has adjoint $a^*:F^*\to E^*$, and $\|a^*\|_{F^*\to E^*}=\|a\|_{E\to F}. \tag{6.2}$

Proposition 6.1. The formal adjoint $P^*$ satisfies the hypotheses of Theorem 4.1, with domain norm $F^*$ and range norm $E^*$.

Proof. Equation (6.2), applied to each derivative, gives $p^*\in\mathcal A^0(F^*,E^*)$. The inverse $(p^{-1})^*$ is bounded from $E^*$ to $F^*$, so the pointwise matrix $p^*$ has the required lower bound. The full symbol of the operator adjoint is $p^\dagger\sim\sum_\alpha \frac{\partial_\xi^\alpha D_x^\alpha(p^*)}{\alpha!}. \tag{6.3}$ Every term with $|\alpha|\geq1$ has adapted order at most $-\kappa$. An ordinary remainder after sufficiently many terms has that same adapted order by polynomial norm comparison, as in (3.3). Hence $p^\dagger-p^*\in\mathcal A^{-\kappa}(F^*,E^*)$. Corollary 4.2 proves the assertion for the full adjoint symbol. The adjoint is properly supported because the two kernel projections are exchanged. ∎

For base-coordinate changes, impose the additional range $1-\rho\leq\delta<\rho\leq1. \tag{6.4}$ In particular $\rho>1/2$. In this paragraph the fiber spaces remain fixed, with the same component identifications in both charts.

Proposition 6.2. Under a smooth change of base coordinates $y=f(x)$, the hypotheses of Theorem 4.1 are preserved, with the norms transported by $\widetilde{\|z\|}_E(y,\eta) =\|z\|_E(x,f'(x)^T\eta), \tag{6.5}$ and the same rule for $F$.

Proof. On compact sets $\langle f'(x)^T\eta\rangle$ and $\langle\eta\rangle$ are comparable, so (2.2) survives. The leading transported symbol is $p_0(y,\eta)=p(x,f'(x)^T\eta)$, and its lower bound is unchanged. A frequency derivative of this pullback costs $\rho$. A base derivative either differentiates $p$ in its base variable, costing $\delta$, or differentiates the linear frequency argument, costing $1-\rho$. The latter cost is at most $\delta$ by (6.4). Repeated chain rules, all with scalar coefficients bounded on compact sets, prove $p_0\in\mathcal A^0(E,F)$ for the transported norms.

The coordinate expansion from the prerequisite has terms $\frac{\partial_\xi^\alpha p(x,f'(x)^T\eta)}{\alpha!} \left.D_z^\alpha e^{i r_x(z)\cdot\eta}\right|_{z=x}, \quad r_x(z)=f(z)-f(x)-f'(x)(z-x). \tag{6.6}$ The second factor is a scalar polynomial in $\eta$ of degree at most $\lfloor|\alpha|/2\rfloor$; its first-order term vanishes. These facts follow because $r_x$ vanishes to second order at $x$. The term has adapted order at most $-\rho|\alpha|+\lfloor|\alpha|/2\rfloor$. For $|\alpha|\geq2$, this is at most $-(2\rho-1)$. Additional frequency derivatives lower the polynomial degree, and additional base derivatives cost at most $\delta$, so these are full differentiated adapted estimates. The ordinary coordinate remainder after $|\alpha|<2N$ has order lowered by $N(2\rho-1)$. Since that gain is positive, choose $N$ large enough to absorb the polynomial norm-comparison cost. Consequently the full transformed symbol differs from $p_0$ by an adapted symbol of order at most $-(2\rho-1)$. Corollary 4.2 completes the proof. ∎

Theorem 6.3. Let $E,F\to X$ be smooth complex vector bundles of the same finite rank, and let $P:E\to F$ be a properly supported pseudodifferential operator of type $(\rho,\delta)$ in the range (6.4), with locally finite order. Suppose there is a covering atlas of coordinates and bundle frames in which the full symbols of $P$ satisfy (2.3) and (2.5). The measuring norms may be Hermitian norms on the fibers over $T^*X$, polynomially comparable on compact sets to smooth background norms; only their pointwise values are used. Then a proper global $Q:F\to E$ satisfies (4.1), with smooth global error kernels. Its symbols satisfy the reverse norm estimates and lower bounds in every chart of the stipulated atlas. Moreover, every distributional section $u$ of $E$ satisfies $\operatorname{WF}(Pu)=\operatorname{WF}(u) . \tag{6.7}$

Proof. Refine the stipulated atlas to a locally finite cover by relatively compact chart sets $U_i$. Restricting the hypotheses to a smaller chart preserves them. Local proper realizations differ from the original operator only by smooth kernels near the relevant diagonal, so Theorem 4.1 supplies local two-sided inverses $Q_i$. The ordinary coordinate calculus in (6.4), and composition with smooth frame-transition matrices, put all these operators in one intrinsic class of locally finite-order operators. This assertion is about their ordinary symbol classes; it does not transport their adapted norm bounds through a frame-transition matrix.

On an overlap, express two local inverses in any one set of coordinates and frames, and localize on a smaller relatively compact subset. Both invert the same $P$ modulo smooth kernels. The identity $Q_i-Q_j=Q_i(I-PQ_j)+(Q_iP-I)Q_j \tag{6.8}$ shows that their difference is smoothing there. Localizations away from the diagonal are already smooth by pseudodifferential kernel regularity. Thus the local inverse classes agree on every overlap.

Choose a smooth partition $\theta_i$ with compact supports inside $U_i$, and $\chi_i\in C_c^\infty(U_i)$ equal to one on a neighborhood of $\operatorname{supp}\theta_i$. These cutoff supports can be chosen locally finite because the cover is locally finite. Extend each compactly localized kernel by zero and set $Q=\sum_i\theta_i Q_i\chi_i. \tag{6.9}$ The sum is locally finite and defines a pseudodifferential operator. Its kernel support lies in the union of $\operatorname{supp}\theta_i\times\operatorname{supp}\chi_i$. Each projection is proper: a compact set meets only finitely many supports in either locally finite family, and the corresponding supports on the other side are compact.

To compare $Q$ with a local inverse $Q_j$, work on a smaller chart subset. Near its diagonal, every relevant $Q_i-Q_j$ is smoothing by (6.8). Also $\theta_iQ_j(\chi_i-I)$ is smoothing, since its two cutoff supports are separated from the diagonal. Summing these observations and using $\sum_i\theta_i=1$ shows $Q-Q_j$ is smoothing locally. It follows that both $PQ-I$ and $QP-I$ have smooth kernels near every diagonal point. They are smooth off the diagonal by the ordinary calculus, so they are globally smooth. In each stipulated chart, the full symbol of $Q$ differs from that of its local inverse by a smoothing symbol. Polynomial norm comparison preserves its reverse derivative estimates, and Corollary 4.2 preserves its reverse lower bound.

Finally use the local identity $u=Q_jPu+\text{smooth}$ and Theorem 5.2. Smooth bundle-frame changes preserve the componentwise wavefront set, as do coordinate changes by the classical tests in the prerequisite. The local wavefront equalities therefore yield (6.7) intrinsically. ∎

The hypothesis in Theorem 6.3 is stated on a framed atlas. Proposition 6.2 permits changes of base coordinates while retaining the same fiber component identification. The global inverse and the wavefront conclusion are independent of the atlas, by (6.8). No additional estimate on derivatives of the measuring norms is needed to glue the inverses.

7. A parabolic symbol and a curved time coordinate

The heat operator $P=\partial_t-\partial_x^2$ has symbol $p(\eta,\sigma)=\eta^2+i\sigma. \tag{7.1}$ Outside a fixed frequency ball, $c\langle(\eta,\sigma)\rangle\leq|p|\leq C\langle(\eta,\sigma)\rangle^2$. Take the scalar domain norm $\|z\|_E=\max(1,|p|)|z|$ and range norm $|z|$. At high frequency, $\frac{|\partial_\eta p|}{|p|} \leq2|p|^{-1/2}\leq Cw^{-1/2}, \qquad \frac{|\partial_\sigma p|}{|p|}\leq Cw^{-1}, \qquad \frac{|\partial_\eta^2p|}{|p|}\leq Cw^{-1}. \tag{7.2}$ All higher derivatives vanish. The bounded-frequency estimates follow from smoothness of $p$ and the lower bound one in the norm. Thus (2.3)–(2.5) hold with $\rho=1/2,\delta=0$. Theorems 4.1 and 5.2 prove hypoellipticity and equality of wavefront sets in these coordinates.

This parameter choice lies outside (6.4). A direct calculation explains the restriction. Set $y=x,\qquad s=t+x^3/3.$ Then $\partial_x=\partial_y+y^2\partial_s$, and the transformed operator is $\partial_s-\partial_y^2-2y^2\partial_y\partial_s -y^4\partial_s^2-2y\partial_s.$ Its full symbol is $\widetilde p(y,\eta,\sigma) =(\eta+y^2\sigma)^2+i(1-2y)\sigma. \tag{7.3}$ At $y=0,\eta=0$, its size is $|\sigma|$, while $\partial_y^4\widetilde p=24\sigma^2$. The scalar relative derivative bound with $\delta=0$ therefore fails. No alternative scalar measuring norms fix that bound: if $p$ satisfies (2.3) and (2.5), their ratio is comparable to $|p|$ at high frequency, which forces $|\partial_y^4p|\leq C|p|$ when $\delta=0$. The transformed operator remains hypoelliptic, since smooth coordinate pullback preserves both the equation and smoothness. A sufficient symbol test can depend on coordinates outside its invariant range.

7.1. A positive symbol whose local order changes

Let $c>0$, let $\nu\geq1$ be an integer, and let $\mu>0$ be real. Choose a smooth nonnegative frequency function $a_\mu(\xi)$ equal to $|\xi|^{2\mu}$ when $|\xi|\geq2$. Consider $p(x,\xi)=c+|x|^{2\nu}a_\mu(\xi). \tag{7.4}$ When $\mu$ is an integer we may take $a_\mu=|\xi|^{2\mu}$ everywhere. For a noninteger $\mu$, a smooth low-frequency extension defines an ordinary symbol without changing its large-frequency behavior.

At $x=0$ the symbol is the constant $c$, whereas at every $x\ne0$ its high-frequency order is $2\mu$. It has polynomial upper bounds on compact base sets, and $p^{-1}\leq c^{-1}$. Nevertheless a relative derivative estimate can hold uniformly through $x=0$.

Set $\delta=\mu/\nu$. If $|\xi|=\lambda\geq2$, homogeneity of the two factors gives, for $0\leq|\beta|\leq2\nu$ and any $\alpha$, $|\partial_\xi^\alpha\partial_x^\beta (|x|^{2\nu}|\xi|^{2\mu})| \leq C_{\alpha\beta}|x|^{2\nu-|\beta|} \lambda^{2\mu-|\alpha|}.$ For $|\beta|>2\nu$ the base derivative is zero. Write $r=|x|\lambda^{\mu/\nu}$. Then $\frac{|x|^{2\nu-|\beta|}\lambda^{2\mu-|\alpha|}} {c+|x|^{2\nu}\lambda^{2\mu}} =\lambda^{-|\alpha|+\delta|\beta|} \frac{r^{2\nu-|\beta|}}{c+r^{2\nu}}. \tag{7.5}$ The last quotient is bounded for $0\leq|\beta|\leq2\nu$, at both zero and infinity. The undifferentiated constant and the bounded-frequency region are handled directly using $p\geq c$. Thus $|\partial_\xi^\alpha\partial_x^\beta p| \leq C_{\alpha\beta,K}\,p\, w^{-|\alpha|+\delta|\beta|}. \tag{7.6}$ If $\mu<\nu$, then $\delta<1=\rho$. The scalar criterion in Section 2 applies, and Theorem 5.3 gives $\operatorname{WF}(Pu)=\operatorname{WF}(u)$ for a proper quantization of (7.4), for every distribution. This is a full inverse construction through the point where the ordinary frequency order changes.

For a positive real $\nu$ that is not an integer, the base power is smooth on the punctured set $x\ne0$; there the symbol is locally elliptic of order $2\mu$. A noninteger base power does not define a smooth symbol at $x=0$, so that point is outside the hypotheses. If either exponent is zero, the smooth cases reduce to an elliptic multiplier or multiplication by a smooth positive function.

Strict separation of the derivative costs cannot be discarded merely because the symbol is positive. On $\mathbb R^n$, the positive polynomial $p(x,\xi)=2n+|x|^2|\xi|^2$ has left quantization $P=2n+|x|^2|D|^2$. For the point mass at zero, $\langle |x|^2|D|^2\delta_0,\varphi\rangle =-\Delta(|x|^2\varphi)(0)=-2n\varphi(0). \tag{7.7}$ Hence $P\delta_0=0$, although $\delta_0$ is not smooth. This is the endpoint $\mu=\nu=1$, with $\delta=\rho=1$. The positive pointwise lower bound alone supplies no hypoellipticity theorem at that endpoint.

8. Exercises with complete solutions

Exercise 1 — foundational. For positive $w\geq1$, give $\mathbb C^3$ the norm $\|z\|_E=(w^6|z_1|^2+w^{-2}|z_2|^2+w^2|z_3|^2)^{1/2}.$ Find its dual norm and polynomial comparison exponents. State the direction of the adjoint of a matrix $E\to F$.

Solution. Cauchy–Schwarz after multiplying the components by $(w^3,w^{-1},w)$ gives $\|\lambda\|_{E^*} =(w^{-6}|\lambda_1|^2+w^2|\lambda_2|^2+w^{-2}|\lambda_3|^2)^{1/2}.$ Equality follows by choosing the weighted vector parallel to the weighted coefficients of $\lambda$. Both norms satisfy (2.2) with $M=3,C=1$; the common symmetric exponent three is necessary because one coefficient grows like $w^3$, and in the dual one decays like $w^{-3}$. An adjoint goes from $F^*$ to $E^*$.

Exercise 2 — intermediate. For (1.1), in one spatial dimension, compute the first composition error $p\#q_0-I$, modulo adapted order minus two, with $\rho=1,\delta=0$ and $q_0=p^{-1}$. Identify which matrix entry can be nonzero at first order.

Solution. The pointwise product is $I$. The first correction is $(\partial_\xi p)(D_xq_0)$. Only the upper-right entry of $D_xq_0$ is nonzero, and it equals $i b'(x)w^{-2}$. Thus the correction is $\begin{pmatrix}0&2i\xi b'(x)w^{-2}\\0&0\end{pmatrix}.$ It has order minus one between the Euclidean range norm and itself. All terms with two or more paired derivatives have adapted order at most minus two by Lemma 3.1. A frequency cutoff in $q_0$ changes this statement only by smoothing terms on compact base sets. This also displays why the pointwise inverse need not already be an operator inverse.

Exercise 3 — intermediate. Suppose $e\in\mathcal A^{-\epsilon}(E,F)$. Prove a derivative estimate for $(p+e)^{-1}-p^{-1}$, not just a bound for its values.

Solution. Corollary 4.2 and Lemma 2.2 give both inverses class zero, with every derivative estimate. The exact pointwise identity is $(p+e)^{-1}-p^{-1}=-(p+e)^{-1}e\,p^{-1}.$ Apply the product rule. Each differentiated inverse goes $F\to E$, the middle factor goes $E\to F$, and the norm types match in the displayed order. Their frequency and base derivative counts add, while the middle factor contributes $-\epsilon$. Hence the difference lies in $\mathcal A^{-\epsilon}(F,E)$ at high frequency. Smooth low-frequency extensions change it by a smoothing symbol.

Exercise 4 — advanced. Let the scalar symbol be $p(\xi)=w^{a}$, with $a\in\mathbb R$. Use domain norm $w^{a+t}|z|$ and range norm $w^t|z|$, where $t$ is arbitrary. Prove that the parametrix hypotheses hold for $\rho=1,\delta=0$. Explain why changing $t$ changes neither the ordinary order nor the inverse symbol.

Solution. The identity $\|pz\|_F=\|z\|_E$ proves the lower bound with constant one. The derivative formula for powers of $1+|\xi|^2$, or induction using the chain rule, gives $|\partial_\xi^\alpha w^a|\leq C_\alpha w^{a-|\alpha|}$. Dividing the output bound by the domain norm gives exactly $w^{-|\alpha|}$; base derivatives vanish. The norms are polynomially bounded, with exponents $|a+t|$ and $|t|$. The parameter $t$ cancels in the ratio of the norms. The symbol remains of ordinary order $a$, and its pointwise inverse is $w^{-a}$. This example separates a choice of measuring norms from an actual change of operator.

Exercise 5 — advanced. For the heat operator use $y=x,s=t+x^{d+1}/(d+1)$, where $d\geq2$ is an integer. Compute the transformed full symbol and a base derivative at $y=0,\eta=0$ that rules out the relative derivative condition with $\delta=0$.

Solution. Since $\partial_x=\partial_y+y^d\partial_s$, the transformed symbol is $\widetilde p=(\eta+y^d\sigma)^2+i(1-dy^{d-1})\sigma.$ At $y=0,\eta=0$, it is $i\sigma$. Its $2d$-th base derivative there is $(2d)!\sigma^2$, because the only surviving term of that degree is $y^{2d}\sigma^2$. The ratio of these magnitudes is $(2d)!|\sigma|$, unbounded at high frequency. The scalar consequence of (2.3) and (2.5) described after (7.3) makes the proposed bound impossible. Smooth coordinate equivalence still preserves hypoellipticity.

Exercise 6 — advanced, 8 points. In (7.4), on a base neighborhood of $0$, prove that every bound $|\partial_{x_1}p|\leq C p\,w^{\delta'}$ at large frequency requires $\delta'\geq\mu/\nu$. Deduce that when $\mu\geq\nu$ the scalar relative derivative criterion cannot hold for any $0\leq\delta'<\rho\leq1$. Does this prove failure of hypoellipticity for every such pair of exponents?

Solution. Take $\xi=\lambda e_1$ and $x=\lambda^{-\mu/\nu}e_1$, with $\lambda\to\infty$. These base points remain in the given neighborhood, $p=c+1$, and $\partial_{x_1}p=2\nu\lambda^{\mu/\nu}.$ The proposed estimate would give $2\nu\lambda^{\mu/\nu}\leq C(c+1)(1+\lambda^2)^{\delta'/2}$, which is impossible unless $\delta'\geq\mu/\nu$. If $\mu\geq\nu$, this forces $\delta'\geq1$, incompatible with the stated strict range. It proves failure of this symbol criterion for the whole family in that regime. It does not prove failure of hypoellipticity for every member. Equation (7.7) supplies an actual failure for one endpoint member; other operators would require their own argument.

References

[G] Gerd Grubb, Distributions and Operators, Springer, 2009. Author-hosted notes: Pseudodifferential operators on manifolds, index of elliptic operators. The coordinate amplitude method described there is due to Kuranishi.

[M] Richard B. Melrose, Introduction to Microlocal Analysis, author-hosted notes: Microlocalization and Pseudodifferential operators on manifolds.

[H] Lars Hörmander, Pseudo-differential operators, Communications on Pure and Applied Mathematics 18 (1965), 501–517.

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