Symbols, finite defects, and the index on a closed manifold

An elliptic symbol describes the directions that an operator can recover at high frequency. The missing directions form a finite-dimensional space, but their number is not usually determined by the symbol at one point. This lesson connects three operations: removing compact errors, moving symbols through invertible maps, and taking products on different manifolds. The analytic estimates come first, so the topological index introduced later does not enter its own proof.

Throughout, $X$ is a compact Hausdorff second-countable smooth manifold without boundary, and bundles are smooth complex bundles of finite rank. Their ranks may differ in preliminary constructions; every two-sided elliptic symbol below has equal source and target ranks on each positive-dimensional component. Write $\mathcal E=E\otimes\Omega_X^{1/2}$ and $\mathcal F=F\otimes\Omega_X^{1/2}$, where $\Omega_X$ is the density line. Half-densities allow integration without orienting $X$. Orders $m,s$ are real. Unless explicitly called continuous, a symbol belongs to the standard estimate class $S^m=S^m_{1,0}$; it need not have a homogeneous expansion. The notation $\Psi^m_{\mathrm{cl}}$ additionally requires a polyhomogeneous expansion with leading degree $m$. The Fourier convention is $D=-i\partial$.

The analytic tools used here are in Symbols, operators and Sobolev scales, especially Sections 5–6 and 8–9 for matrix composition, adjoints, all-order Sobolev bounds and inversion, and in Detecting regularity without choosing coordinates, especially Sections 4–6, 11 and 13 for assembly, smoothing parametrices, bundle Sobolev spaces and anti-duals. Finite defects under perturbation, Sections 2–7, supplies Fredholm composition, compact perturbations, two-sided parametrices and strong families with two collectively compact error families. Section 9 of Traces that survive passage to cohomology proves a bounded tensor identity used for comparison; the smooth-domain calculation needed here is proved in Section 11.4.

The entry facts are smooth partitions, chart changes and bundle gluing; distributional pairing, restriction and finite order on compact sets; Fourier inversion and Plancherel with distributional Fourier extension; and elementary integration and calculus. For Hilbert spaces we use orthogonal projections, adjoints and Parseval, as developed in Traces that survive passage to cohomology, Section 1. The all-real-order interpolation step uses the annular Fourier proof in Symbols, operators and Sobolev scales, Section 8, with sequence exponent two; Section 10.2 spells out its use here. Finite-bundle compactness, smoothing approximation, order-changing and controlled-quantization adapters are proved below. No K-theoretic or characteristic-class index formula is assumed or concluded.

1. Compactness and duality in bundle Sobolev spaces

Choose a finite family of relatively compact bundle charts, smooth local frames, and cutoffs $\chi_j$ such that $\sum_j\chi_j^2=1$. One construction starts with nonnegative subordinate functions $\rho_j$ whose sum is positive and sets $\chi_j=\rho_j/(\sum_k\rho_k^2)^{1/2}$. A smooth Hermitian metric is obtained by adding the local metrics with a partition of unity. Define $H^s(X;\mathcal E)$ by the finite sum of local $H^s(\mathbb R^n)$ norms of $\chi_j u$, expressed as vector-valued half-density components and extended by zero. Detecting regularity without choosing coordinates proves equivalence of these norms under changes of frame, charts, and cutoffs. Completion can equally be taken inside distributions: the local limits agree on overlaps, because multiplication and coordinate changes are continuous. Thus the spaces are Hilbert, smooth sections are dense, and $$\bigcap_{s\in\mathbb R}H^s(X;\mathcal E)=C^\infty(X;\mathcal E), \qquad \bigcup_{s\in\mathbb R}H^s(X;\mathcal E)=\mathcal D'(X;\mathcal E). \tag{I1}$$ For the first equality, in each chart Cauchy–Schwarz gives absolute convergence of the differentiated inverse Fourier integral when $s>k+n/2$, hence $k$ continuous derivatives. Conversely a compact smooth section has rapidly decreasing local Fourier transforms. For the second, the finite-order distribution bound on each of the finitely many chart supports bounds the local Fourier transform by a polynomial; it is consequently square integrable after multiplication by a sufficiently negative Sobolev weight. A common negative order works for all charts. Density follows by local convolution and cutoffs, first for each of the finitely many local components and then by summing.

For every $\delta>0$, the inclusion $$H^{s+\delta}(X;\mathcal E)\longrightarrow H^s(X;\mathcal E) \quad\text{is compact.} \tag{I2}$$ Here is a local proof that includes negative $s$. For a distribution $v$ supported in a fixed compact chart set, let $L_R$ be the Fourier multiplier $\psi(\xi/R)$, with $\psi=1$ near zero and compactly supported. Then $\|(I-L_R)v\|_{H^s}\leq C R^{-\delta}\|v\|_{H^{s+\delta}}$. Choose compact smooth $\theta$ equal to one on that support. The same estimate after multiplication by $\theta$ bounds $v-\theta L_Rv$. On this support the latter operator has the smooth compactly supported kernel $\theta(x)\check\psi_R(x-y)\theta_1(y)$, where $\theta_1=1$ on the input support. Any such kernel gives a compact map between any two fixed Sobolev spaces. To verify this claim, enclose its support in two cubes, extend it smoothly and periodically by zero in larger cubes, and expand in a double Fourier series. Integration by parts gives coefficients decreasing faster than every power of both indices. Each finite sum is a finite-rank kernel with smooth input and output functions; Cauchy–Schwarz with the fixed Sobolev weights bounds the operator-norm tail by a convergent weighted square sum. The kernel is therefore an operator-norm limit of finite-rank maps. Letting $R\to\infty$ proves local compactness, and finitely many charts prove (I2). The same argument proves that a smooth bundle kernel is compact between all fixed Sobolev spaces and can be approximated there by smooth finite-rank kernels.

The anti-dual bundle $E^*$ consists of conjugate-linear functionals on $E$. As in Detecting regularity without choosing coordinates, use $\langle u,v\rangle=\overline{v(u)}$, linear in the first argument and conjugate-linear in the second. Integration identifies the continuous anti-dual of $H^s(X;\mathcal E)$ with $H^{-s}(X;E^*\otimes\Omega_X^{1/2})$. Locally, this is the Fourier identity between weighted $L^2$ spaces: Cauchy–Schwarz proves boundedness, and Hilbert representation gives every bounded functional by a Fourier function with the reciprocal weight. To pass to bundles, localize a functional with cutoffs, use that representation in each frame, and sum the resulting anti-dual sections. The pairing transition law from Detecting regularity without choosing coordinates makes this construction independent of charts. For a closed subspace of this Hilbert space, orthogonal projection separates every vector outside it by a bounded functional. Indeed its nonzero orthogonal component supplies such a functional by the inner product. Hilbert representation itself follows from the projection theorem: for a nonzero functional, the orthogonal complement of its kernel is one-dimensional, and its value on a unit vector in that line determines the representing vector. Thus a closed subspace is exactly the common nullspace of its annihilating functionals. These statements will determine the correct cokernel space, rather than an adjoint chosen from an unrelated Sobolev inner product.

2. The global elliptic alternative

Let $P\in\Psi^m(X;\mathcal E,\mathcal F)$. Its principal symbol is a class $$p\in S^m(T^*X;\operatorname{Hom}(\pi^*E,\pi^*F))/S^{m-1}. \tag{I3}$$ It is elliptic when a representative has an inverse at sufficiently large covectors with $\|p(x,\xi)^{-1}\|\leq C\langle\xi\rangle^{-m}$, in finitely many bundle charts. Equivalently there is $q\in S^{-m}$ such that $qp-I$ and $pq-I$ lie in $S^{-1}$. The equivalence follows from matrix inversion and differentiation as in Section 9 of Symbols, operators and Sobolev scales: a one-sided error of norm less than one is invertible, equal finite ranks give a genuine fiber inverse, and repeated differentiation of $p^{-1}p=I$ gives every inverse-symbol estimate. A cutoff across bounded covectors completes the smooth representative. Both remainders are required as typed bundle maps; the square-matrix algebra explains why either suffices for equal ranks.

Quantize $q$ using the local construction in Section 4 and the bundle assembly in Section 13 of Detecting regularity without choosing coordinates. Composition gives $QP=I-K_E$, $PQ=I-K_F$, with $K_E,K_F\in\Psi^{-1}$. In the realizations $$P_s:H^s(X;\mathcal E)\longrightarrow H^{s-m}(X;\mathcal F), \qquad Q_s:H^{s-m}(X;\mathcal F)\longrightarrow H^s(X;\mathcal E), \tag{I4}$$ the errors are compact by (I2). Section 5 of Finite defects under perturbation implies that $P_s,Q_s$ are Fredholm and have opposite indices. Iterating $u=K_Eu$ for a nullvector increases its Sobolev order by any positive integer. Thus $$N=\ker P_s=\{u\in C^\infty(X;\mathcal E):Pu=0\} \tag{I5}$$ is finite-dimensional and independent of $s$.

The distributional half-density adjoint is $$P^*:H^{m-s}(X;F^*\otimes\Omega_X^{1/2}) \longrightarrow H^{-s}(X;E^*\otimes\Omega_X^{1/2}). \tag{I6}$$ It is elliptic of order $m$, with the anti-dual symbol of $p$, by Section 13 of Detecting regularity without choosing coordinates. Applying the same regularity argument gives the finite-dimensional smooth space $N^*=\ker P^*\subset C^\infty(X;F^*\otimes\Omega_X^{1/2})$, independent of the realization. By the duality of Section 1 and the closed range in (I4), $$\operatorname{ran}P_s =\{f\in H^{s-m}(X;\mathcal F):\langle f,v\rangle=0\text{ for every }v\in N^*\}, \quad \operatorname{ind}P_s=\dim N-\dim N^*. \tag{I7}$$ The functionals furnished by a basis of $N^*$ are independent: otherwise their linear combination is the zero distribution, and all its coefficients are zero. Hence the codimension in (I7) is exactly $\dim N^*$, not merely bounded by it.

The smooth and distributional realizations have these same defects. A smooth $f$ satisfying the obstructions has a Sobolev solution by (I7); the full smoothing parametrix from Section 6 of Detecting regularity without choosing coordinates, or iteration of the order-one gain, makes that solution smooth. A distributional $f$ belongs to one Sobolev space by (I1), so the same argument gives a distributional solution when it annihilates $N^*$. Conversely every solution annihilates $N^*$ by the distributional adjoint identity. Thus the smooth and distributional ranges are precisely the respective finite intersections of kernels of these continuous functionals, and in particular are closed in their natural topologies. Their index means the difference of finite algebraic defects, as in the Sobolev realizations.

If $R\in\Psi^{m-1}$, then $R:H^s\to H^{s-m}$ factors through $H^{s-m+1}$ and is compact. Therefore $P+R$ has the same index. This proves dependence only on (I3). For composable elliptic $P_2:E\to F$ and $P_1:F\to G$, use the Sobolev spaces $H^s(E)$, $H^{s-m_2}(F)$, $H^{s-m_2-m_1}(G)$ in Section 4 of Finite defects under perturbation to obtain $$\operatorname{ind}(P_1P_2)=\operatorname{ind}P_1+\operatorname{ind}P_2, \qquad \operatorname{ind}P^*=-\operatorname{ind}P. \tag{I8}$$ The second formula follows equally from (I5)–(I7) and the double anti-dual identification. If $E=F^*$ and $P-P^*\in\Psi^{m-1}$, the operators have the same principal class and opposite indices, so their integer index is zero. Hermitian metrics turn this statement into the familiar assertion for a formally selfadjoint principal symbol; they are an explicit identification, not a deletion of the anti-dual bundles.

3. Exact order changes with index zero

On positive-dimensional components choose a smooth positive homogeneous function $h(x,\xi)$ of degree one away from the zero section, for example a cotangent norm from a Riemannian metric. On a zero-dimensional component use the identity as every order-changing map; its section spaces are finite-dimensional and every operator is smoothing. On a Hermitian bundle $E$, and for each real $r$, quantize $h^r I_E$ with a cutoff near zero and symmetrize the operator in the geometric $L^2$ pairing. The resulting $A_r\in\Psi^r_{\mathrm{cl}}$ is formally selfadjoint, elliptic, and has principal symbol $h^rI_E$. Its index is zero by Section 2. Let $\Pi$ be the $L^2$-orthogonal projection onto its smooth finite-dimensional nullspace, using the Hermitian metric. This is a smooth finite-rank kernel: for an orthonormal basis $e_1,\ldots,e_d$, it is $\sum_j e_j\langle\,\cdot\,,e_j\rangle$.

Then $$J_E^r=A_r+\Pi:H^s(X;\mathcal E)\longrightarrow H^{s-r}(X;\mathcal E) \tag{I9}$$ is bijective for every real $s$. Indeed, the range of $A_r$ is the annihilator of that same nullspace in every target Sobolev space; the projection supplies its missing finite-dimensional component. Its kernel is zero, since pairing $(A_r+\Pi)u=0$ with each $e_j$ gives $\Pi u=0$, and then $A_ru=0$. The inverse is bounded by the Banach inverse theorem. These inverses agree on overlaps of their domains by uniqueness of the distributional solution.

The inverse is itself in $\Psi^{-r}$. If $B$ is a smoothing-error parametrix, then $(J_E^r)^{-1}=B+(J_E^r)^{-1}(I-J_E^rB)$. The second term is a smooth kernel: its input remainder maps every Sobolev order to every higher order, and the coherent inverse is bounded with shift $r$ at all orders. Applying the same reasoning to the adjoint proves smoothness in the input variable as well; more explicitly, local Fourier coefficients of its kernel decrease faster than every polynomial in both indices by these two-sided bounds. This is the smooth-kernel criterion used in Section 1. Thus the claimed inverse belongs to the calculus. We use $J_E^r$ as a notation for this construction, not as an assertion that $J_E^rJ_E^t=J_E^{r+t}$.

In particular any operator norm problem $H^s(\mathcal E)\to H^{s-m}(\mathcal F)$ is converted, by the exact isomorphisms $$A=J_E^{-s}:L^2(\mathcal E)\to H^s(\mathcal E), \qquad B=J_F^{s-m}:H^{s-m}(\mathcal F)\to L^2(\mathcal F), \tag{I10}$$ into an $L^2$ problem for $BPA$. Its principal symbol is $h^{s-m}ph^{-s}=h^{-m}p$, when the same cotangent function is used. Both conjugating operators have index zero. This construction does not presuppose the norm-limit theorem proved next.

4. Measuring a principal symbol by operator norms

First let $P\in\Psi^m_{\mathrm{cl}}$ and take compactly supported vector functions $u,v$ in one chart, a nonzero real covector $\theta$, and $u_t(x)=e^{itx\cdot\theta}u(x)$. Fourier translation and dominated convergence give, for every real $a$, $$t^{-a}\|u_t\|_{H^a(\mathbb R^n)}\longrightarrow |\theta|^a\|u\|_2. \tag{I11}$$ For completeness, the squared integral has weight $(t^{-2}+|\theta+\eta/t|^2)^a$. On $|\eta|\leq t|\theta|/2$ it converges with a fixed polynomial majorant. On the complement, the rapid decay of $\widehat u(\eta)$ absorbs any fixed positive or negative power of $t$ and of $\langle\eta\rangle$; choosing a larger decay exponent makes its integral tend to zero. This also proves the assertion for negative $a$, where a uniform lower bound on the weight over all frequencies would be false.

The local exponential-test formula of Section 2 of Detecting regularity without choosing coordinates, or directly Fourier translation in the defining symbol integral, gives $$t^{-m}\langle Pu_t,v_t\rangle\longrightarrow \int\langle p_m(x,\theta)u(x),v(x)\rangle\,dx. \tag{I12}$$ Here $p_m$ is the homogeneous leading symbol. In the integral representation, $t^{-m}a(x,t\theta+\eta)\to p_m(x,\theta)$ uniformly on compact $x,\eta$ sets. Symbol bounds and the rapidly decreasing test transforms handle the remaining frequencies exactly as in (I11). A localized smoothing remainder contributes $o(t^m)$ by integration by parts in both compact input and output variables. Pairing through $H^{s-m}$ and its anti-dual $H^{m-s}$, equations (I11)–(I12) imply a local bound for $p_m$ by the norm of $P:H^s\to H^{s-m}$. Testing with $u=\phi w$, $v=\phi z$, shrinking $\phi$ to a point, and then taking unit fiber vectors $w,z$, detects the full matrix norm. Finitely many charts and homogeneity show that $$\sup_{(x,\theta)\in S_h^*X}\|p_m(x,\theta)\| \leq C_{s,m}\|P\|_{H^s\to H^{s-m}}, \qquad S_h^*X=\{h=1\}. \tag{I13}$$ The constant depends only on the fixed norms and charts, not on $P$. In geometric $L^2$ with $m=0$, chart cutoffs and local orthonormal frames can be chosen with multiplication norms at most one, so the constant is one. This is a statement about the homogeneous symbol; an arbitrary full symbol can be changed at bounded frequencies without being detected by this limit.

We need the converse estimate modulo a smooth finite-rank error. If $P\in\Psi^0_{\mathrm{cl}}(\mathcal E,\mathcal F)$, set $c=\sup_{S_h^*X}\|p_0\|$. For every $\eta>0$ there is a smooth finite-rank operator $R$ such that $$\|P-R\|_{L^2\to L^2}\leq c+\eta. \tag{I14}$$ To prove it, choose $M>c^2$. The positive definite bundle endomorphism $M I_E-p_0^*p_0$ has a smooth positive square root $b_0$. This finite-dimensional fact needs no operator spectral theorem: in a compact parameter set choose $L$ larger than its largest eigenvalue and use the binomial power series for $L^{1/2}(I-(I-L^{-1}(M I-p_0^*p_0)))^{1/2}$. The argument of the power series has norm strictly below one uniformly. Termwise differentiated series converge, proving smoothness in charts; uniqueness of the positive square root makes the local definitions agree. Radial extension gives an order-zero homogeneous symbol. Quantize it as $B$; the calculus gives the selfadjoint compact error $$P^*P+B^*B=M I+K,\qquad K\in\Psi^{-1}. \tag{I15}$$ Let $\Pi_N$ be increasing $L^2$-orthogonal projections onto finite-dimensional spaces of smooth sections with dense union. Such spaces are obtained by choosing a countable dense sequence of smooth sections and Gram–Schmidt, deleting zero vectors. Compactness of $K$ implies $\|K(I-\Pi_N)\|\to0$: finite-rank approximation reduces this to strong convergence of the projections on finitely many adjoint vectors. For $w=(I-\Pi_N)u$, (I15) yields $\|Pw\|^2\leq(M+\|K(I-\Pi_N)\|)\|w\|^2$. Choose $M$ and then $N$ so that the square root of this bound is at most $c+\eta$. The operator $R=P\Pi_N$ has smooth input and output vectors and proves (I14). This proof allows derivatives of $p_0$ to be large; only the finite-rank correction depends on them.

Consequently, if a sequence of smooth invertible degree-zero bundle symbols $b_j$ has uniformly bounded pointwise norms, it admits quantizations $Q_j$ with those principal symbols and uniformly bounded $L^2$ operator norms. Start with arbitrary quantizations and apply (I14), subtracting the corresponding smooth finite-rank errors. Uniform symbol-derivative estimates for $b_j$ are unnecessary. This is the exact norm-control statement used below.

5. Ellipticity survives an operator-norm limit

Suppose $P_j\in\Psi^m_{\mathrm{cl}}(X;\mathcal E,\mathcal F)$ converges to a bounded map $P:H^s\to H^{s-m}$ in operator norm at one real $s$. By (I13), the homogeneous symbols $p_j$ are uniformly Cauchy on the compact cosphere; their limit $p$ is continuous, homogeneous of degree $m$, and the convergence is uniform on every compact subset of $T^*X\setminus0$. If $p$ is invertible there, then $P$ is Fredholm and $$\operatorname{ind}P=\operatorname{ind}P_j\quad\text{for all sufficiently large }j. \tag{I16}$$ No membership of $P$ in a pseudodifferential class is asserted.

First take $s=m=0$. Compactness of the cosphere and uniform convergence imply that $p_j$ are invertible for large $j$, with a common bound for $p_j^{-1}$. Quantize those inverses as $Q_j$ with uniformly bounded norms using Section 4. Then $Q_jP_j-I$ and $P_jQ_j-I$ have order minus one and are compact. For large $j$, both $Q_j(P-P_j)$ and $(P-P_j)Q_j$ have norm less than one. Hence $$Q_jP=\bigl(I+Q_j(P-P_j)\bigr)+\bigl(Q_jP_j-I\bigr), \quad PQ_j=\bigl(I+(P-P_j)Q_j\bigr)+\bigl(P_jQ_j-I\bigr) \tag{I17}$$ are Fredholm of index zero, by the geometric series and compact perturbation invariance. Composing their inverses modulo compact errors with $Q_j$ gives a left and a right parametrix for $P$; Section 5 of Finite defects under perturbation makes $P$ Fredholm. Now that Fredholmness is established, operator-norm stability gives (I16). For arbitrary $s,m$, apply this argument to $BP_jA\to BPA$, with (I10). Its symbols are $h^{-m}p_j$. The exact inverse maps $A^{-1},B^{-1}$ transport Fredholmness and index back to $P$. Thus no ill-defined Sobolev Hilbert adjoint occurs in the reduction.

Assume more strongly that the same sequence converges compatibly at every real Sobolev order. Compatibility means that the resulting bounded maps agree on intersections, as they do automatically when their limits on smooth sections agree distributionally. Write $N_s=\ker(P:H^s\to H^{s-m})$ and $M_{m-s}=\ker(P^*:H^{m-s}(F^*\otimes\Omega^{1/2})\to H^{-s}(E^*\otimes\Omega^{1/2}))$. If $s_2>s_1$, then $N_{s_2}\subset N_{s_1}$, while $M_{m-s_2}\supset M_{m-s_1}$. The index in (I16) is the same at every $s$, since each $P_j$ has that property. Therefore $$\dim N_{s_1}-\dim N_{s_2} =\dim M_{m-s_1}-\dim M_{m-s_2}. \tag{I18}$$ The left side is nonnegative and the right side nonpositive, so both vanish. Finite dimension and inclusion give equality of these spaces for every pair of orders. By (I1) both nullspaces consist of smooth sections, and the range at every order is the annihilator of the same smooth adjoint nullspace. Single-order convergence alone has none of this smoothness consequence; an example below makes the distinction concrete.

6. Bounded symbol homotopies with strong continuity

Let $t\in[0,1]$. Suppose $a_t,b_t$ are continuous in the compact-open smooth topology on $T^*X$, uniformly bounded in $S^m$ and $S^{-m}$, respectively, and $$a_tb_t-I_F\text{ and }b_ta_t-I_E \quad\text{are uniformly bounded in }S^{-1}. \tag{I19}$$ The maps have the appropriate pulled-back bundle types. Then all quantizations of the endpoint principal classes have the same index. Continuity in every full symbol seminorm is not assumed.

Here are the operator-family details. Fix one finite system of charts, bundle frames, density identifications and cutoffs with $\sum_j\chi_j^2=1$. Quantize each local representative and place $\chi_j$ on both sides; extend by zero outside the chart and sum. Coordinate transport and the product formula in Detecting regularity without choosing coordinates show that the resulting $A_t,B_t$ have principal classes $a_t,b_t$. Every seminorm in their products and their order-minus-one errors is bounded by finitely many seminorms in (I19) and the original families. Thus $A_t:H^s\to H^{s-m}$, $B_t:H^{s-m}\to H^s$, and $$K_t=I-B_tA_t:H^s\to H^{s+1}, \qquad L_t=I-A_tB_t:H^{s-m}\to H^{s-m+1} \tag{I20}$$ are uniformly bounded.

For a fixed smooth input, a local oscillatory formula proves continuity of the output in every fixed Sobolev norm: restrict frequency to a large ball, use compact-open smooth continuity there, and control the tail by a uniform symbol bound times the rapidly decreasing Fourier transform of the input. Derivatives of the output are handled by increasing that uniform bound. On the compact manifold these estimates patch through finitely many charts. Smooth density in each input Sobolev space and uniform operator bounds extend continuity to every fixed input. Consequently both $A_t$ and $B_t$, and hence the errors in (I20), are strongly continuous. By (I2) the union of the images of the unit balls under each of $K_t,L_t$ has compact closure in its original Sobolev space. This proves collective compactness on both sides. Section 7 of Finite defects under perturbation now implies constant index. Altering either endpoint by a lower-order operator is compact by Section 2, so the assertion holds for arbitrary endpoint quantizations.

This argument also works on any compact parameter space with these continuity and boundedness properties: the index is locally constant there, and is constant on each connected component. Compactness of the parameter space is used through the stated uniform bounds and the abstract family theorem; no sequential compactness or metric structure on that parameter space is needed.

7. Moving all high frequencies to one sphere

An elliptic symbol with estimates need not approach a homogeneous limit. Nevertheless its index can be read on one sufficiently distant cotangent sphere. Let $p\in S^m$ represent an elliptic principal class and let $h>0$ be smooth and homogeneous of degree one away from zero. For every sufficiently large $R$, the degree-zero symbol $$p_R(x,\xi)=p\bigl(x,R\xi/h(x,\xi)\bigr),\qquad \xi\ne0, \tag{I21}$$ is invertible and has the same analytic index as $p$.

First let $m=0$. Choose $q\in S^0$ equal to the exact inverse of $p$ for $h\geq R_0$, by a cutoff and the inverse derivative estimates. Fix $R>2R_0$ and a smooth positive function $k$ on $[0,\infty)$, equal to one for $r\leq1$, equal to $r^{-1}$ for $r\geq2$, with $r k(r)\geq1$ for $r\geq1$. Such a function is obtained by a smooth convex combination of $1$ and $1/r$ in $[1,2]$. Define $$F_t(x,\xi)=\bigl(x,\xi k(th(x,\xi)/R)\bigr), \qquad p_t=p\circ F_t,\quad q_t=q\circ F_t, \quad 0\leq t\leq1. \tag{I22}$$ At zero covectors use the identity map: the cutoff is exactly one in a neighborhood of the zero section, so no nonsmooth norm is evaluated in a nonconstant expression there. For $th/R\leq1$, the map is the identity. In the transition region $1\leq th/R\leq2$, its radial scale is comparable to one and its frequency derivatives of order $\ell\geq1$ have size at most $C_\ell h^{1-\ell}$; base derivatives have size at most $C h$. For $th/R\geq2$, the image is $(x,R\xi/(t h))$, of size comparable to $R/t$. In that region every frequency derivative costs $h^{-1}$, while each derivative of an order-zero input symbol costs the reciprocal of the image size. Iterated chain rules therefore prove uniform $S^0$ bounds for $p_t,q_t$, for all $t\in[0,1]$. These bounds include mixed base and frequency derivatives. Equivalently, on the transition annulus one may set $\eta=t\xi/R$; its compact angular and radial bounds reduce all estimates to the uniform order-zero estimates for $p(x,(R/t)\eta)$ and $q$. This also explains why uniform symbol bounds remain valid as $t\downarrow0$.

The families are continuous in the compact-open smooth topology, including at $t=0$, since on each fixed compact set they eventually equal the original symbols. Above $h\geq R$, their image has $h\geq R$: if $th/R\leq1$ this is immediate, and otherwise it equals $(R/t)(th/R)k(th/R)\geq R$. Hence their two pointwise products are exactly the identity there. All derivatives of the product errors are uniformly bounded on the remaining compact set, so those errors form a bounded family in every negative symbol order. Apply Section 6. At $t=1$, the resulting symbol equals $p_R$ for $h\geq2R$; a cutoff of its homogeneous extension therefore has that same principal class. This proves (I21) in order zero.

For general $m$, right-compose the operator with $J_E^{-m}$ from Section 3. The product has order zero, index unchanged, and principal representative $p h^{-m}$ outside a compact set. Its radial restriction is $R^{-m}p_R$. Multiplication by the positive scalar $R^{-m}$ is an invertible order-zero operator of index zero. The order-zero argument consequently proves the assertion for $p$. There is no need to assume $p$ has a limit along rays, and no deformation through an uncontrolled family of orders has been used.

8. The index of a continuous symbol

Let $p:T^*X\to\operatorname{Hom}(\pi^*E,\pi^*F)$ be continuous and invertible outside a compact set $K$. It need not satisfy symbol derivative estimates. Choose $h$ as above and a large $R$ with $K\subset\{h<R\}$. The restriction (I21) is a continuous degree-zero isomorphism on $T^*X\setminus0$, determined by its values on the compact cosphere.

A continuous section of a finite-rank smooth bundle over a compact smooth manifold can be approximated uniformly by smooth sections. To prove the version needed here, choose finitely many local frames and subordinate compactly supported partition functions. Express the product of each partition function with the section in that frame, extend its components by zero inside a larger coordinate domain, and convolve with a compact smooth approximate identity. Uniform continuity gives uniform convergence; for sufficiently small convolution radii the supports remain in their charts. Sum the resulting local sections. Compactness and bounded transition matrices give convergence in the bundle norm. This construction applies to the Hom bundle over the cosphere, without separately smoothing source and target frames.

If $a,b$ are two continuous invertible degree-zero symbols and $$\sup_{S_h^*X}\|a^{-1}b-I\|<1, \tag{I23}$$ then $(1-t)a+tb=a[I+t(a^{-1}b-I)]$ is invertible for every $t$. Choose a smooth approximation $b$ to $p_R$ so close that (I23) holds, extend it homogeneously, and quantize with a cutoff near zero. Define $$\operatorname{sind}(p)=\operatorname{ind}\operatorname{Op}(b). \tag{I24}$$ This definition is independent of the choices. For two sufficiently close smooth approximations, their convex combination remains within the uniform relative-error ball of radius less than one about $p_R$; its inverse is smooth and uniformly bounded with all derivatives on the compact parameter-cosphere product. Its homogeneous extensions satisfy Section 6. Thus both quantizations have the same index. For arbitrary approximations satisfying (I23), the same convex-combination argument works because the two error bounds have a common maximum strictly below one. Changing the quantization changes only a lower-order operator. The same reasoning proves that (I23) preserves the index when both symbols are merely continuous: approximate both endpoints so closely that the entire straight-line family remains invertible, and compare the resulting smooth path.

Changing the radius introduces the continuous path $p(x,((1-t)R_1+tR_2)\xi/h)$, entirely outside $K$. Changing $h$ to another positive continuous homogeneous function $h_1$ introduces $p(x,R\xi/((1-t)h+t h_1))$, for a radius large enough to keep all its images outside $K$. On the compact parameter-cosphere product the smallest singular value has a positive minimum. Uniform continuity partitions the interval into finitely many subintervals on which adjacent symbols satisfy (I23). This proves independence of both choices. It proves at the same time:

Continuous homotopy theorem. If $p_t$ is jointly continuous in $(t,x,\xi)$, $0\leq t\leq1$, and is invertible outside one compact set $K\subset T^*X$ for every $t$, then $\operatorname{sind}(p_t)$ is constant. The condition concerns one common compact set, not a separate set allowed to escape with $t$.

The characterization by this homotopy property and agreement with analytic index for smooth degree-zero homogeneous principal symbols is unique. Indeed, first deform any continuous $p$ to a radial extension using (I22); the same formula is continuous for continuous input and its noninvertibility remains in $\{h\leq R\}$. Then approximate its cosphere map and join it to that smooth approximation by a straight line. Extend the latter line continuously across bounded covectors by multiplying its homogeneous part with a fixed radial cutoff and retaining any continuous compact filling. Invertibility during the deformation is required only above a common radius. The final smooth homogeneous principal symbol determines the value by analytic agreement. This reduces every value to the prescribed one, proving uniqueness without a K-theory classification theorem.

Every elliptic $P\in\Psi^m$ has $$\operatorname{ind}P=\operatorname{sind}(p), \tag{I25}$$ for any continuous principal representative, by Section 7 and (I24). For a continuous degree-$m$ symbol defined only off zero, choose any continuous filling on a bounded cotangent region after an exterior cutoff; fillings differ through a homotopy supported in a common compact set, so the index is well defined. Such a filling exists even when $m\leq0$: make it zero near the zero section and interpolate with a radial scalar cutoff, with no requirement of invertibility in that region.

Combining (I25) with Section 5 gives the precise index in the norm-limit theorem: $$P_j\longrightarrow P\text{ in }\mathcal L(H^s,H^{s-m}),\quad p_j\longrightarrow p\text{ invertible on }T^*X\setminus0 \quad\Longrightarrow\quad \operatorname{ind}P=\operatorname{sind}(p). \tag{I26}$$ The relative-error condition applies to $p_j$ and $p$ on a cosphere for large $j$, so the index on the right equals the stabilized value in (I16).

For a zero-dimensional component, $X$ is a finite set and all section spaces are finite-dimensional. There is no cosphere. Define its contribution as $\sum_{x\in X}(\operatorname{rank}E_x-\operatorname{rank}F_x)$, the index of every linear map between its section spaces. All preceding statements reduce there to rank-nullity, and products below include these finite-dimensional factors. A compact manifold has finitely many connected components, so these contributions add to the positive-dimensional ones. This convention also covers the empty manifold and the zero bundle.

9. A circle calculation that fixes the sign

Write $X=\mathbb R/(2\pi\mathbb Z)$, with Fourier vectors $e_k(x)=e^{ikx}$. Suppose a scalar continuous elliptic symbol has exterior branches $a_+(x)\ne0$ for positive frequency and $a_-(x)\ne0$ for negative frequency, after radial restriction. Multiplication by $a_-^{-1}$ has symbol equal on both branches and contributes index zero: approximate it smoothly if necessary, and use the actual invertible multiplication operator. The ratio $r=a_+/a_-$ is a loop in $\mathbb C\setminus0$.

Its winding number $w$ can be defined without assuming differentiability. Divide the circle into arcs on which $r(x)/r(x_0)$ lies in the disk of radius less than one about one. The power series logarithm on that disk gives local continuous logarithms. Starting at $x=0$, add multiples of $2\pi i$ to make neighboring logarithms agree at endpoints. This constructs a continuous logarithm $\ell$ of $r$ on $[0,2\pi]$, with $\ell(2\pi)-\ell(0)=2\pi i w$ for one integer $w$. Different initial logarithms change $\ell$ by a constant integral multiple of $2\pi i$, leaving $w$ fixed. Thus $r(x)=e^{iwx+g(x)}$ with periodic continuous $g$, and the path $e^{iwx+(1-t)g(x)}$ reduces the calculation to $r=e^{iwx}$.

Let $\Pi_+$ project onto Fourier modes $k\geq0$, and set $\Pi_-=I-\Pi_+$. This projection is a classical order-zero pseudodifferential operator with principal branches one and zero. Here is a direct kernel justification. Choose a smooth function $\beta(\xi)$, zero for $\xi\leq-3/4$, one for $\xi\geq-1/4$; it equals $1$ at all nonnegative integers and $0$ at negative integers. The derivative of $\beta$ is compactly supported, so its inverse Fourier distribution is smooth away from zero and rapidly decreasing there by integration by parts. Periodize that distribution over translations by $2\pi$. Off the diagonal the derivative series converges locally uniformly, while near the diagonal only the untranslated singularity remains. Its Fourier coefficients are $\beta(k)$, by testing the periodized distribution against $e^{-ikx}$ and the Fourier inversion convention. The periodic convolution operator is therefore $\Pi_+$, and locally it has the Euclidean symbol $\beta$ plus a smooth kernel. This proves the pseudodifferential claim without a theorem about holomorphic boundary values.

Consider $$T_w=e^{iwx}\Pi_++\Pi_-. \tag{I27}$$ It sends $e_k$ to $e_{k+w}$ for $k\geq0$, and to $e_k$ for $k<0$. If $w\geq0$, the images are the disjoint orthonormal sets with frequencies $k<0$ and $k\geq w$. The kernel is zero and the missing frequencies $0,\ldots,w-1$ give a cokernel of dimension $w$. If $w=-d<0$, all frequencies occur, and the two image families overlap precisely at $-d,\ldots,-1$. For $0\leq j<d$, $e_j-e_{j-d}$ spans one relation. These $d$ relations are independent and exhaust the kernel, since all other frequencies have a unique preimage. Thus in every case $$\operatorname{sind}(a)=-\operatorname{wind}(a_+/a_-). \tag{I28}$$ All defect vectors are smooth, so the calculation agrees with every Sobolev realization.

To check the orientation version, orient $T^*X$ by $dx\wedge d\xi$. On the boundary of $\{|\xi|\leq R\}$, contraction with the outward normal gives orientation $-dx$ on the upper circle and $+dx$ on the lower circle. The degree of the phase map is therefore $-\operatorname{wind}(a_+)+\operatorname{wind}(a_-)$, exactly (I28). Reversing the phase-space orientation reverses that degree; it does not change the analytic convention for the index.

10. Partial operators can be approximated on every Sobolev scale

10.1. The local approximation statement

Put $z=(x,y)\in\mathbb R^n\times\mathbb R^{n'}$, $\zeta=(\xi,\eta)$, and $\langle\zeta\rangle=(1+|\zeta|^2)^{1/2}$. Let $m>0$. Suppose $a(z,\xi)$ is a classical symbol of order $m$ in the $\xi$ variables, uniformly over the entire $z$-space. Thus, for every pair of multiindices,

$$\sup_{z,\xi}\langle\xi\rangle^{-m+|\beta|} \|\partial_z^\alpha\partial_\xi^\beta a(z,\xi)\|<\infty. \tag{I29}$$

Its classical expansion has smooth components $a_{m-j}(z,\xi)$, homogeneous of degree $m-j$ for $\xi\ne0$, with the corresponding uniform bounds on $|\xi|=1$; after a fixed low-frequency cutoff, each remainder has order $m-N$, uniformly in the same sense. The symbols may be fixed-size complex matrices. No compactness assumption on the base variables replaces the uniform estimates in (I29).

Choose a smooth scalar function $\chi(\xi,\eta)$ with

$$\chi=1\quad\hbox{if }|\eta|\leq\max(1,|\xi|),\qquad \chi=0\quad\hbox{if }|\eta|\geq2\max(1,|\xi|), \tag{I30}$$

and homogeneous of degree zero in the region $|\eta|>2$. These conditions make $\chi$ a classical symbol of order zero on the full frequency space. Indeed, away from the cone where $|\eta|$ and $|\xi|$ are comparable it is identically zero or one; in that cone, outside a bounded set, the stated homogeneity supplies the symbol estimates. Write $\chi_0$ for its leading homogeneous part. For $0<\varepsilon\leq1$ define

$$a_\varepsilon(z,\xi,\eta) =a(z,\xi)\chi(\xi,\varepsilon\eta),\qquad A=a(z,D_x),\qquad A_\varepsilon=\operatorname{Op}_{x,y}(a_\varepsilon). \tag{I31}$$

For each fixed positive $\varepsilon$, $a_\varepsilon$ is a full-variable classical symbol of order $m$. Its leading symbol is

$$a_{\varepsilon,m}(z,\xi,\eta) =a_m(z,\xi)\chi_0(\xi,\varepsilon\eta), \tag{I32}$$

extended by zero across $\xi=0,\eta\ne0$. Extend $a_m(z,\xi)$ itself by zero at $\xi=0$. This extension is continuous because $m>0$. There are constants independent of $\varepsilon$ such that

$$\sup_{z,\,|\xi|^2+|\eta|^2=1} \|a_{\varepsilon,m}(z,\xi,\eta)-a_m(z,\xi)\| \leq C\varepsilon^m, \tag{I33}$$

and, for every real $s$,

$$\|A_\varepsilon-A\|_{H^{s+m}(\mathbb R^{n+n'})\to H^s(\mathbb R^{n+n'})} \leq C_s\varepsilon^m. \tag{I34}$$

The constants in (I34) depend on finitely many of the uniform symbol seminorms of $a$, on $s,m,n,n'$, and on $\chi$; they do not require uniform full-variable symbol seminorms of $a_\varepsilon$ as $\varepsilon\downarrow0$. Those latter seminorms generally grow.

Full-variable symbol assertion. On the support of $\chi(\xi,\varepsilon\eta)$, or of any nonzero derivative of that cutoff,

$$\varepsilon|\eta|\leq2\max(1,|\xi|),\qquad \langle\xi,\eta\rangle\leq C_\varepsilon\langle\xi\rangle. \tag{I35}$$

The reverse inequality with constant one always holds. For fixed $\varepsilon>0$, the linear change $(\xi,\eta)\mapsto(\xi,\varepsilon\eta)$ preserves the order-zero symbol estimates for the cutoff. Apply the product rule to (I31). Each derivative of $a$ is estimated by (I29); (I35) converts its frequency weight to the full-variable weight. Each derivative falling on the cutoff contributes its full frequency-order loss. This proves the $S^m_{1,0}$ estimates with constants allowed to depend on $\varepsilon$.

For the classical expansion, multiply each $a_{m-j}$ by $\chi_0(\xi,\varepsilon\eta)$. On the punctured full frequency space the latter factor vanishes in a neighborhood of the set $\xi=0$, so the zero extension is smooth there even when $m-j\leq0$. Each product is homogeneous of degree $m-j$. The classical remainder estimate follows from (I35); changing from a cutoff in $\xi$ to a cutoff in $(\xi,\eta)$ changes only a bounded-frequency region for this fixed $\varepsilon$. This proves (I32) and the full classical assertion.

For (I33), the difference vanishes when $|\xi|\geq\varepsilon|\eta|$. On its remaining support, homogeneity and the uniform leading-symbol bound give

$$\|a_m(z,\xi)\|\leq C|\xi|^m \leq C\varepsilon^m|\eta|^m.$$

The uniform bound on $\chi_0-1$ completes the estimate on the total unit sphere. This also checks uniformity near the two frequency axes, where pointwise convergence alone would be insufficient.

10.2. The estimate at every real Sobolev order

Set $C_\varepsilon=\chi(D_x,\varepsilon D_y)$ and $T_\varepsilon(a)=A_\varepsilon-A$. Left quantization gives the exact identity

$$T_\varepsilon(a)=a(z,D_x)(C_\varepsilon-I). \tag{I36}$$

There is no composition remainder in (I36), since the right factor is a Fourier multiplier.

Output order zero. For fixed $y$, the symbol $a(x,y,\xi)\langle\xi\rangle^{-m}$ has uniformly bounded derivatives of the finite orders required in Section 6 of Symbols, operators and Sobolev scales. Therefore that section’s matrix-valued $L^2$ estimate yields

$$\|a(x,y,D_x)v(\cdot,y)\|_{L^2_x} \leq C\|v(\cdot,y)\|_{H^m_x}, \tag{I37}$$

with the same $C$ for every $y$. Integrate its square in $y$, then apply partial and full Plancherel to $v=(C_\varepsilon-I)u$. The multiplier can differ from one only where

$$\varepsilon|\eta|>\max(1,|\xi|),\qquad \langle\xi\rangle^2\leq2\varepsilon^2|\eta|^2 \leq2\varepsilon^2\langle\xi,\eta\rangle^2. \tag{I38}$$

Consequently

$$\begin{aligned} \|T_\varepsilon(a)u\|_2^2 &\leq C\int\langle\xi\rangle^{2m} |\chi(\xi,\varepsilon\eta)-1|^2 |\widehat u(\xi,\eta)|^2\,d\xi\,d\eta\\ &\leq C'\varepsilon^{2m}\|u\|_{H^m}^2. \end{aligned} \tag{I39}$$

The full Fourier factors in the coefficient $C$ of (I39) are calculated below in (FC1)–(FC3). The same proof applies to every base derivative $D_z^\gamma a$, because it still satisfies the uniform order-$m$ estimates.

Nonnegative integer output orders. The cutoff multiplier commutes with every $D_{z_j}$, and differentiation of the left-quantized integral gives

$$[D_{z_j},T_\varepsilon(a)]=T_\varepsilon(D_{z_j}a). \tag{I40}$$

Thus, on Schwartz functions,

$$D_z^\alpha T_\varepsilon(a)u =\sum_{\gamma\leq\alpha}{\alpha\choose\gamma} T_\varepsilon(D_z^\gamma a)D_z^{\alpha-\gamma}u. \tag{I41}$$

For $|\alpha|\leq k$, apply (I39) to every summand, use $\|D^{\alpha-\gamma}u\|_{H^m}\leq C\|u\|_{H^{m+k}}$, and sum over the finitely many derivatives defining the equivalent integer $H^k$ norm. This gives (I34) for every integer $s=k\geq0$.

Negative integer output orders. Let $k\geq1$. Expand the polynomial

$$(1+|\zeta|^2)^k =\sum_{|\alpha|\leq k}c_\alpha\zeta^{2\alpha}, \qquad c_\alpha>0, \tag{I42}$$

and define

$$\widehat{u_\alpha}(\zeta) =c_\alpha\zeta^\alpha\langle\zeta\rangle^{-2k} \widehat u(\zeta). \tag{I43}$$

Then

$$u=\sum_{|\alpha|\leq k}D^\alpha u_\alpha, \qquad \sum_{|\alpha|\leq k}\|u_\alpha\|_{H^m}^2 \leq C_k\|u\|_{H^{m-k}}^2. \tag{I44}$$

The identity follows by multiplying (I42) by $\langle\zeta\rangle^{-2k}\widehat u$. For the estimate, the sum of squared multipliers in (I43) is bounded by $C_k\langle\zeta\rangle^{-2k}$, again by (I42). This constructs the negative-order decomposition instead of assuming it.

Move the derivatives in (I44) to the left using (I40) in reverse:

$$T_\varepsilon(a)D^\alpha =\sum_{\gamma\leq\alpha}(-1)^{|\gamma|} {\alpha\choose\gamma} D^{\alpha-\gamma}T_\varepsilon(D_z^\gamma a). \tag{I45}$$

Every derivative outside $T_\varepsilon$ has order at most $k$, hence is bounded from $L^2$ to $H^{-k}$. Use (I39) for its remaining factor and then the finite-sum Cauchy–Schwarz inequality in (I44). The result is

$$\|T_\varepsilon(a)u\|_{H^{-k}} \leq C_k\varepsilon^m\|u\|_{H^{m-k}}, \tag{I46}$$

which is exactly (I34) for $s=-k$. All identities were proved on Schwartz inputs; density in each $H^r$ and the estimates give their consistent extensions.

Noninteger orders and the exact interpolation interface. We use only the two-endpoint implication proved in Section 8 of Symbols, operators and Sobolev scales, equation (E39) and its annular convolution argument, with sequence exponent two. In explicit terms: if a consistent linear map is bounded $H^{m+r_j}\to H^{r_j}$ for $r_0<s<r_1$, then it is bounded $H^{m+s}\to H^s$, with a constant controlled by the two endpoint constants and the distances of $s$ from the endpoints. This is a statement on the full Euclidean frequency space, not separate interpolation in $x$ and $y$.

For clarity, choose sharp dyadic full-frequency projections $\Pi_j$. Apply the two endpoint estimates to $\Pi_j u$, then project the output to band $k$. After inserting the weights $2^{ks}$ and $2^{j(m+s)}$, the two estimates have the summable majorant

$$c_{k-j}=C\min\{2^{(k-j)(s-r_0)}, 2^{(k-j)(s-r_1)}\}. \tag{I47}$$

The $\ell^1*\ell^2\to\ell^2$ convolution bound proves the claimed $H^{m+s}\to H^s$ estimate. Pass from finite band sums by density; the Fourier norm is equivalent to the band-weighted $\ell^2$ norm. Apply this implication to $\varepsilon^{-m}T_\varepsilon(a)$ and the consecutive integers around $s$. Its endpoint constants are independent of $\varepsilon$ by (I41) and (I46), proving (I34) with the same power $\varepsilon^m$ for every real $s$. No analytic interpolation theorem with an unspecified domain is needed.

The same argument, with $C_\varepsilon-I$ omitted and $\langle\xi\rangle\leq\langle\xi,\eta\rangle$ in place of (I38), also proves that the partial operator $A$ itself maps $H^{s+m}$ continuously to $H^s$ for all real $s$.

Every Fourier factor and finite derivative coefficient

We make the constants in (I39)–(I46) explicit. Use exactly the Fourier transform and inverse from Fourier transforms, finite spectra and convex separation: the forward integral has no prefactor and the inverse in dimension $d=n+n'$ has $(2\pi)^{-d}$. Thus

$$\begin{aligned} \widehat u(\xi,\eta)&=\int_{\mathbb R^n\times\mathbb R^{n'}} e^{-i(x\cdot\xi+y\cdot\eta)}u(x,y)\,dx\,dy,\\ u(x,y)&=(2\pi)^{-d}\int e^{i(x\cdot\xi+y\cdot\eta)}\widehat u(\xi,\eta)\,d\xi\,d\eta,\\ \|u\|_{H^r}^2&=(2\pi)^{-d}\int \langle(\xi,\eta)\rangle^{2r}|\widehat u(\xi,\eta)|^2 \,d\xi\,d\eta. \end{aligned} \tag{FC1}$$

The component norm is the original Euclidean bundle-frame norm; it does not identify different fibers. Put $M_\chi=\sup_{\xi,\eta}|\chi(\xi,\eta)-1|$. This number is finite by the stated smooth low-frequency and homogeneous high-frequency conditions. No sign condition on the cutoff is needed. For each multi-index $\gamma$, let $A_\gamma$ be a uniform constant in the actual partial mapping estimate (I37) for $D_z^\gamma a$:

$$\| (D_z^\gamma a)(x,y,D_x)v\|_{L^2_x} \leq A_\gamma\|v\|_{H^m_x}, \qquad y\in\mathbb R^{n'}. \tag{FC2}$$

Its value depends on the retained symbol seminorms and matrix sizes. Let $w=(C_\varepsilon-I)u$. Apply (FC2) at each $y$, integrate in $y$, and use partial Plancherel first in $x$ and then in $y$. The two factors are respectively $(2\pi)^{-n}$ and $(2\pi)^{-n'}$. Using the full inequality (I38) gives

$$\begin{aligned} \|T_\varepsilon(D_z^\gamma a)u\|_2^2 &\leq A_\gamma^2(2\pi)^{-n} \int_{y,\xi}\langle\xi\rangle^{2m} |\mathcal F_x w(\xi,y)|^2\,d\xi\,dy\\ &=A_\gamma^2(2\pi)^{-n}(2\pi)^{-n'} \int_{\xi,\eta}\langle\xi\rangle^{2m} |\chi(\xi,\varepsilon\eta)-1|^2 |\widehat u(\xi,\eta)|^2\,d\xi\,d\eta\\ &\leq A_\gamma^2 2^m M_\chi^2\varepsilon^{2m} (2\pi)^{-d}\int_{\xi,\eta} \langle(\xi,\eta)\rangle^{2m}|\widehat u(\xi,\eta)|^2 \,d\xi\,d\eta\\ &=L_\gamma^2\varepsilon^{2m}\|u\|_{H^m}^2, \qquad L_\gamma=2^{m/2}M_\chi A_\gamma. \end{aligned} \tag{FC3}$$

Where the cutoff difference is zero both sides of the pointwise comparison are zero; on its support (I38) applies. In particular, no Fourier factor or differentiated coefficient is dropped.

For an integer $k\geq0$, the actual coefficients in (I42) are

$$c_\alpha=\frac{k!}{\alpha_1!\cdots\alpha_d!(k-|\alpha|)!}, \quad |\alpha|\leq k,\qquad \|v\|_{H^k}^2=\sum_{|\alpha|\leq k}c_\alpha\|D^\alpha v\|_2^2. \tag{FC4}$$

The multinomial theorem applied to the unchanged polynomial $(1+\zeta_1^2+\cdots+\zeta_d^2)^k$ proves the first formula; Plancherel with (FC1) proves the second. Every multiplier $\zeta^{\alpha-\gamma}$ in (I41) is bounded by $\langle\zeta\rangle^k$ when $\gamma\leq\alpha$ and $|\alpha|\leq k$. Apply (FC3) to every displayed summand in (I41), retaining its binomial coefficient. Then

$$\begin{aligned} \|T_\varepsilon(a)u\|_{H^k} &\leq \varepsilon^m C_k^+\|u\|_{H^{m+k}},\\ (C_k^+)^2 &=\sum_{|\alpha|\leq k}c_\alpha \left(\sum_{\gamma\leq\alpha} {\alpha\choose\gamma}L_\gamma\right)^2. \end{aligned} \tag{FC5}$$

For the negative-order decomposition in (I43), set $c_{\max,k}=\max_{|\alpha|\leq k}c_\alpha$. Its exact squared norm sum is

$$\begin{aligned} \sum_{|\alpha|\leq k}\|u_\alpha\|_{H^m}^2 &=(2\pi)^{-d}\int\langle\zeta\rangle^{2m-4k} \left(\sum_{|\alpha|\leq k}c_\alpha^2\zeta^{2\alpha}\right) |\widehat u(\zeta)|^2\,d\zeta\\ &\leq c_{\max,k}(2\pi)^{-d}\int \langle\zeta\rangle^{2m-2k}|\widehat u(\zeta)|^2\,d\zeta\\ &=c_{\max,k}\|u\|_{H^{m-k}}^2. \end{aligned} \tag{FC6}$$