Finite-chart symbol calculus and equivariant Fredholm realizations

Put $⟨z⟩=(1+\mathrm{\mid }z{\mathrm{\mid }}^2{)}^{1\mathrm{/}2}$. The triangle inequality gives $⟨v+w{⟩}^r\le C_r⟨v{⟩}^r⟨w{⟩}^{\mathrm{\mid }r\mathrm{\mid }}$ for every real $r$, also when $r<0$ by interchanging $v$ and $v+w$. For an output-compact symbol $p$ of order $m$, integration by parts in $x$ gives $\mathrm{\mid }{\widehat{p}}_x(\zeta ,\xi )\mathrm{\mid }\le C_N⟨\zeta {⟩}^{-2N}⟨\xi {⟩}^m$. Fourier transformation turns its quantization into the kernel $(2\pi {)}^{-n}{\widehat{p}}_x(\eta -\xi ,\xi )$. After conjugation by Sobolev weights this is bounded by $C_{N,s}⟨\eta -\xi {⟩}^{-2N+\mathrm{\mid }s-m\mathrm{\mid }}$. Choose $2N>n+\mathrm{\mid }s-m\mathrm{\mid }$. Both integrals of this bound, in $\eta$ and in $\xi$, are finite. Weighted Cauchy–Schwarz, followed by integration in the other variable, bounds the squared operator norm by the product of these two integral bounds. Plancherel proves $H^s\to H^{s-m}$ boundedness and continuity in finitely many output-derivative symbol bounds. This also applies to differences of symbols. No frequency-derivative bound is needed for this particular estimate.

An amplitude $a(x,y,\xi )$, compactly supported in $(x,y)$ and satisfying the symbol bounds of order $m$ with derivatives in both variables, has a quantization with kernel $(2\pi {)}^{-n}\int e^{i(x-y)\cdot \xi }a(x,y,\xi )d\xi$, interpreted distributionally. It equals a left quantization with symbol $r(x,\xi )=(2\pi {)}^{-n}\int e^{ix\cdot \zeta }{\widehat{a}}_y(x,\zeta ,\xi +\zeta )d\zeta$. The latter integral and all its derivatives converge: integration by parts in $y$ gives arbitrary decay in $\zeta$, and the weight inequality in step 1.1 absorbs the frequency shift. Fourier inversion on Schwartz test inputs proves the identity by Fubini; hence it defines the distributional kernel without an unregularized oscillatory exchange. Taylor expansion in the last variable gives $r\sim {\sum }_{\alpha }{i}^{-\mathrm{\mid }\alpha \mathrm{\mid }}({\mathrm{\partial }}_{\xi }^{\alpha }{\mathrm{\partial }}_y^{\alpha }a)(x,x,\xi )\mathrm{/}\alpha !$. Indeed the integral remainder after degree $N-1$ contains ${\mathrm{\partial }}_{\xi }^{\alpha }{\widehat{a}}_y(x,\zeta ,\xi +t\zeta ){\zeta }^{\alpha }$, $\mathrm{\mid }\alpha \mathrm{\mid }=N$, integrated against $N(1-t{)}^{N-1}\mathrm{/}\alpha !$; integration by parts in $y$ and the same weight inequality bound every differentiated remainder by $C⟨\xi {⟩}^{m-N-\mathrm{\mid }\beta \mathrm{\mid }}$. Thus it is a symbol of order $m-N$, with continuous seminorm estimates. In particular the leading symbol is $a(x,x,\xi )$.

The kernel of a symbol is smooth off $x=y$. On a compact set separated from that diagonal, repeatedly use $e^{i(x-y)\cdot \xi }=\mathrm{\mid }x-y{\mathrm{\mid }}^{-2}(x-y)\cdot ({\mathrm{\partial }}_{\xi }\mathrm{/}i)e^{i(x-y)\cdot \xi }$. Each transfer to the symbol lowers its order; after more than $m+n$ transfers the integral is absolutely convergent. To differentiate the kernel a prescribed number of times, transfer that many additional derivatives first. The resulting integrals give continuous derivatives by integrable majorants, and coincide as distributions by testing against Schwartz functions and integration by parts. Thus cutting off a kernel near the diagonal changes it by a smooth kernel. This also justifies arbitrary nested compact chart localizations.

For a coordinate change $z=\kappa (x)$, near the diagonal write $\kappa (x)-\kappa (y)=H(x,y)(x-y)$ with $H(x,y)={\int }_0^{1}d{\kappa }_{y+t(x-y)}dt$. Shrink the coordinate neighborhood so that $H$ is invertible, and change frequency by $\eta =H(x,y{)}^t\xi$. The transformed kernel is an amplitude of order $m$: differentiating $p(\kappa (x),H^{-t}\eta )$ introduces a frequency derivative together with at most one factor of $\eta$, preserving its order. Smooth frame and density factors have bounded derivatives. At $x=y$, the Jacobian $\mathrm{\mid }\det d{\kappa }_y\mathrm{\mid }$ from the source density cancels $\mathrm{\mid }\det H{\mathrm{\mid }}^{-1}$. Step 2.1 therefore gives exactly the cotangent transformation of $p_m$; all other terms have order $m-1$. Cover the compact diagonal by finitely many such neighborhoods and use step 2.2 off it. This proves covariance, including continuous parameter-dependent coordinate changes on compact parameter subsets. The ordinary chain rule, change of variables and the weight inequality prove equivalence of the integer positive Sobolev norms; negative norms follow by duality, and the Fourier completion defines the corresponding distribution spaces.

For two output-compact left symbols $p,q$, Fourier inversion gives the composition symbol $r(x,\xi )=(2\pi {)}^{-n}\int e^{ix\cdot \zeta }p(x,\xi +\zeta ){\widehat{q}}_x(\zeta ,\xi )d\zeta$. Arbitrary decay of ${\widehat{q}}_x$ and step 1.1 justify the integral with every derivative. Taylor expansion of $p$ gives $r\sim {\sum }_{\alpha }{i}^{-\mathrm{\mid }\alpha \mathrm{\mid }}({\mathrm{\partial }}_{\xi }^{\alpha }p)({\mathrm{\partial }}_x^{\alpha }q)\mathrm{/}\alpha !$; the explicit integral remainder is bounded exactly as in step 2.1, now by $C⟨\xi {⟩}^{m+m^{\mathrm{\prime }}-N-\mathrm{\mid }\beta \mathrm{\mid }}$. The adjoint kernel has amplitude $p(y,\xi {)}^{\ast }$ with the smooth density and metric factors inserted; its diagonal value is $p_m(x,\xi {)}^{\ast }$. Steps 2.1–3.1 prove the adjoint assertion, and yield ${\mathrm{\Psi }}^m{\mathrm{\Psi }}^{m^{\mathrm{\prime }}}\subset {\mathrm{\Psi }}^{m+m^{\mathrm{\prime }}}$. Partitions realize any homogeneous symbol: multiply it by a high-frequency cutoff, quantize in charts, and sum with cutoffs whose squares sum to one. The diagonal symbol is the given bundle map. The same construction is linear and continuous on smooth symbol families.

The symbol expansions just obtained can be summed. For a sequence $b_j\in S^{m-j}$ choose radii $R_j\uparrow \mathrm{\infty }$ and a fixed cutoff $\chi$ zero on the unit ball and one outside the ball of radius two. For $j\ge 1$, make the first $j$ seminorms of $\chi (\xi \mathrm{/}{R}_j)b_j$ in $S^{m-j+1}$ smaller than $2^{-j}$. This is possible because on its support the extra order gives a factor $R_j^{-1}$, and cutoff derivatives give the compensating inverse powers. For each fixed symbol seminorm and each fixed $N$, the tail starting beyond $N$ converges in $S^{m-N}$; the finite remaining terms cause no difficulty. Thus $b={\sum }_j\chi (\xi \mathrm{/}{R}_j)b_j$ has the prescribed expansion. AC permits these countably many choices. If $P$ is elliptic, first quantize $\sigma (P{)}^{-1}$ with cutoff to obtain $Q_0$ and $I-P{Q}_0$ of order $-1$. Sum the formal series $Q_0{\sum }_{k\ge 0}(I-P{Q}_0{)}^k$ by this construction to obtain a right parametrix modulo all negative orders. A left parametrix is constructed likewise; their difference is smoothing because $Q_L-Q_R=Q_L(I-P{Q}_R)-(I-Q_{L}P)Q_R$. An operator of every negative order has a smooth kernel by the absolutely convergent derivative integrals from steps 2.1–2.2. This proves the asserted two-sided smooth remainders.

The Fourier definition implies ${\bigcap }_s{H}^s=C^{\mathrm{\infty }}$: if $s>n\mathrm{/}2+k$, Cauchy–Schwarz makes $\mathrm{\mid }\xi {\mathrm{\mid }}^k\widehat{u}$ integrable and Fourier inversion gives continuous derivatives through order $k$. Conversely smooth chart coefficients have arbitrary Fourier decay. Every distribution on a compact chart subset has finite order, and its Fourier transform grows at most polynomially, by testing it on a cutoff times $e^{-ix\cdot \xi }$; hence every distribution on compact $X$ belongs to some $H^{-s}$. The parametrix identity $u=QPu+Ru$ now proves elliptic regularity, including $Pu\in H^s\Rightarrow u\in H^{s+m}$ whenever $u$ is a distribution. Order $-1$ operators send $L^2$ to $H^1$ by step 1.1; [F2] in finitely many balls makes their $L^2$ realizations compact. Applying the same fact to integer Sobolev derivatives and, for negative orders, duality gives compactness of $H^{s+1}\hookrightarrow H^s$. Thus the parametrix remainders are compact on all Sobolev spaces; [F3] proves Fredholmness. Distributional adjoint kernels are smooth, so cokernels are their finite-dimensional smooth kernels. These arguments include dimension one.

Choose real cutoffs ${\varphi }_i$ with ${\sum }_i{\varphi }_i^2=1$ in finitely many bundle charts. Trivialize unitarily, incorporating the square root of the density into the chart map $T_i$ to make it an $L^2$ isometry on supported sections. Define ${\mathrm{\Lambda }}_E={\sum }_i{\varphi }_i{T}_i^{-1}⟨D⟩T_i{\varphi }_i$, with zero extension before the Fourier multiplier. It is formally symmetric on smooth sections and satisfies $({\mathrm{\Lambda }}_{E}u,u)\ge {\sum }_i\mathrm{\parallel }{\varphi }_{i}u{\mathrm{\parallel }}_2^2=\mathrm{\parallel }u{\mathrm{\parallel }}_2^2$. Steps 3.1–4.1 give positive scalar homogeneous symbol $\mathrm{\ell }={\sum }_i{\varphi }_i^2\mathrm{\mid }\xi {\mathrm{\mid }}_i$, the same for all bundles if their coordinate cover is shared. Elliptic regularity makes the adjoint kernel smooth; formal symmetry and the inequality make both kernel and cokernel zero. Hence ${\mathrm{\Lambda }}_E:H^1\to L^2$ is bijective with bounded inverse. Its unbounded $L^2$ realization on domain $H^1$ is closed: if $u_k\to u$ and ${\mathrm{\Lambda }}_E{u}_k\to v$ in $L^2$, then $u_k={\mathrm{\Lambda }}_E^{-1}{\mathrm{\Lambda }}_E{u}_k\to {\mathrm{\Lambda }}_E^{-1}v$ in $H^1$, so $u={\mathrm{\Lambda }}_E^{-1}v$ and ${\mathrm{\Lambda }}_{E}u=v$. The parametrix identities show that this inverse differs from its classical parametrix by a smooth-kernel operator: the difference is ${\mathrm{\Lambda }}_E^{-1}$ times a smooth remainder, which maps every negative Sobolev space to every positive one by elliptic regularity and the bounded inverse; this implies a smooth kernel as follows. After localization, the matrix coefficient between cutoff plane waves of frequencies $\xi ,\eta$ is bounded by $C_{a,b}⟨\xi {⟩}^{-a}⟨\eta {⟩}^{-b}$ for arbitrary $a,b$, using the $H^{-a}\to H^b$ bound and the Schwartz Fourier decay of the cutoffs. It is the double Fourier transform of the localized kernel. Its rapid decay with the same bounds after coordinate multiplication makes inverse Fourier differentiation absolutely convergent to every order, giving a smooth kernel. At any integer level $r$, elliptic Fredholmness and regularity identify the kernel and cokernel of ${\mathrm{\Lambda }}_E:H^r\to H^{r-1}$ with smooth nullspaces of ${\mathrm{\Lambda }}_E$ and its formal adjoint; the quadratic lower bound and symmetry make both zero. Thus each such map is a bounded bijection with bounded inverse by [F3]. Iterating these isomorphisms and their inverses gives the stated ${\mathrm{\Lambda }}_E^s:H^s\leftrightarrow L^2$ identifications for all integers; the iterates are classical because ${\mathrm{\Lambda }}_E$ and its inverse have classical symbols.

For $P\in {\mathrm{\Psi }}^0$, let $p_0={\sigma }_0(P)$ on $S^{\ast }X$. Its essential norm on $L^2$ is exactly $\mathrm{\parallel }{p}_0{\mathrm{\parallel }}_{\mathrm{\infty }}$. For the upper bound fix $c>\mathrm{\parallel }{p}_0{\mathrm{\parallel }}_{\mathrm{\infty }}$ and quantize $h_0=(c^{2}I-p_0^{\ast }{p}_0{)}^{1\mathrm{/}2}$. This is smooth: a uniformly convergent power series for the square root on a slightly larger spectral interval can be differentiated on the compact base, or one may use the convergent resolvent integral for positive matrices. Step 4.1 gives $c^{2}I-P^{\ast }P-H^{\ast }H$ compact. A compact operator is uniformly approximated by finite-rank operators: project its relatively compact unit-ball image onto the span of a finite small net. On the common orthogonal complement of the finite-dimensional ranges of an approximant and its adjoint the compact quadratic error is arbitrarily small; hence $\mathrm{\parallel }P(I-\mathrm{\Pi })\mathrm{\parallel }\le (c^2+\epsilon {)}^{1\mathrm{/}2}$ for a finite-rank projection $\mathrm{\Pi }$. Since $P\mathrm{\Pi }$ is compact, the essential norm is at most $c$, and then at most $\mathrm{\parallel }{p}_0{\mathrm{\parallel }}_{\mathrm{\infty }}$.

For the lower bound choose chart data $(x_0,{\xi }_0,v)$ with $\mathrm{\mid }{\xi }_0\mathrm{\mid }=\mathrm{\mid }v\mathrm{\mid }=1$, and a smooth bump ${\phi }_{\delta }$ supported within distance $\delta$ of $x_0$, normalized in $L^2$. Put $u_k=e^{ik{\xi }_0\cdot x}{\phi }_{\delta }v$. It converges weakly to zero: for smooth test sections integrate by parts in the ${\xi }_0$ direction, and extend to $L^2$ tests by density. Modulation replaces the local symbol by $p(x,k{\xi }_0+\eta )$. For every fixed $\eta$, its order-zero bounds and classical expansion give uniform convergence on the compact output support to $p_0(x,{\xi }_0)$; its norm is bounded uniformly in $k,x,\eta$. Thus the absolutely convergent Fourier integral against the Schwartz function ${\widehat{\phi }}_{\delta }(\eta )$ converges uniformly in $x$, by first truncating its integrable tail. The output support is fixed, so $\mathrm{\parallel }P{u}_k-e^{ik{\xi }_0\cdot x}{p}_0(x,{\xi }_0){\phi }_{\delta }v{\mathrm{\parallel }}_2\to 0$. Chart cutoffs contribute only smooth kernels by step 2.2, and those kernels send this weakly zero sequence to zero. Every compact operator does likewise, because any convergent subsequence of its images has weak limit zero. Therefore the essential norm is at least $\mathrm{\parallel }{p}_0(x,{\xi }_0){\phi }_{\delta }v{\mathrm{\parallel }}_2$. Send $\delta \downarrow 0$, then take the supremum over the data, to obtain the reverse inequality to step 7.2. For zero-dimensional $X$, the sphere is empty and every operator is compact.

Smooth kernels include finite-rank operators with smooth vectors, which are norm dense in all compact operators by smooth density and the finite-net argument in step 7.2. The essential-norm identity extends the principal symbol continuously to the norm closure, with kernel exactly the compact operators. It is onto the continuous symbols: choose smooth approximants $a_j$ whose successive uniform differences are less than $2^{-j}$. Quantize each difference. The essential-norm upper bound allows subtraction of a compact operator so that the adjusted lift has norm at most $2\mathrm{\parallel }{a}_{j+1}-a_j{\mathrm{\parallel }}_{\mathrm{\infty }}$; compact operators already lie in the closure. The adjusted lifts form a norm-convergent series realizing the continuous limit. Continuous sections admit the required smooth approximation by finite local convolutions and partitions, with ordinary matrix bundle transitions. Conjugate an order-$m$ realization by ${\mathrm{\Lambda }}_F^{s-m}$ on the left and ${\mathrm{\Lambda }}_E^{-s}$ on the right. Step 7.1 turns it into an order-zero operator with symbol $p_m{\mathrm{\ell }}^{-m}$. This transports the exact sequence to every pair $(m,s)$.

Average positive densities and bundle metrics using [F4]; their averages are smooth and positive by differentiation of their local smooth expressions over the compact parameter group. With these choices $U_g$ is unitary on $L^2$ and uniformly bounded on every Sobolev space by the chart chain-rule bounds. Step 3.1, including its amplitude remainder estimates, shows that conjugation of a fixed classical operator varies continuously in each classical seminorm as $g$ varies: use finitely many group neighborhoods and source-target charts, whose transition amplitudes and all derivatives depend smoothly on $g$. The off-diagonal kernels have the same property by step 2.2. Consequently the Haar integral exists in every symbol seminorm, and differentiation under its uniformly bounded local amplitudes makes it classical with averaged leading symbol. Averaging the operator ${\mathrm{\Lambda }}_E$ constructed in step 7.1 therefore preserves its positivity, ellipticity and scalar leading symbol, making a common invariant $\mathrm{\ell }$ available. Approximation extends operator averaging to each operator-norm closure and the simultaneous closure; uniform boundedness of the conjugations at each Sobolev level passes continuity from classical approximants to their limit, and invariance of the average follows by changing the group integration variable. In particular, a lift from step 9.1 of an invariant continuous inverse symbol can be averaged in the single-level closure and retains that inverse symbol. Smooth paths are norm continuous by step 1.1 and the finite chart construction. This construction uses no Fredholmness or regularity assertion for the closures.

In the closure at one Sobolev level, an invertible continuous symbol has a lift of its inverse by step 9.1. Compositions differ from the identity by compact operators, so [F3] gives Fredholmness. An operator in the simultaneous closure has this property at every level when its symbol is invertible. Its index is independent of the level: approximate it simultaneously at two chosen levels and uniformly in symbol by a classical operator with invertible symbol. Averaging a classical approximant as in step 10.1 preserves classicality and its averaged symbol; sufficiently close approximation keeps that symbol invertible. For an equivariant Fredholm operator $P:H\to H^{\mathrm{\prime }}$, use the invariant decompositions $H=\ker P\oplus (\ker P{)}^{\perp }$ and $H^{\mathrm{\prime }}=(\mathrm{ran}P{)}^{\perp }\oplus \mathrm{ran}P$. The restriction from $(\ker P{)}^{\perp }$ to $\mathrm{ran}P$ is a bounded equivariant isomorphism. For every sufficiently close equivariant $Q$, its infinite block between these two subspaces remains invertible by the Neumann series. Equivariant invertible row and column operations reduce $Q$ to that block and a map $S:\ker P\to (\mathrm{ran}P{)}^{\perp }$. Invariant orthogonal splitting of $\ker S$ and $\mathrm{ran}S$ gives $[\ker Q]-[\mathrm{coker}Q]=[\ker S]-[\mathrm{coker}S]=[\ker P]-[\mathrm{coker}P]$ in $R(G)$. Thus the virtual index is locally constant at either Sobolev level. For the classical approximant, elliptic regularity identifies its kernels and adjoint kernels at both levels, so their $R(G)$-indices agree; local constancy then gives the same index for the original closure operator.

Since $\ker P:H^{s+1}\to H^{s+1-m}$ is contained in the kernel at level $s$, its dimension can only decrease as $s$ increases. The cokernel is the kernel of the distributional formal adjoint in $H^{m-s}$; its dimension can only increase. Constancy of their difference from step 11.1 forces both dimensions, and therefore both kernel spaces, to be constant. Membership in every Sobolev space gives smoothness by step 6.1. If $Pu=f$ with $f$ smooth, then at every higher level $f$ is orthogonal to exactly the same smooth adjoint kernel, so closed range gives a preimage $u_s$ at that level. The difference $u-u_s$ lies in the already smooth distributional kernel. Hence $u$ belongs to every Sobolev space and is smooth. This also proves the asserted regularity for the simultaneous closure without assuming that its inverse is classical. On a compact zero-manifold all spaces are finite direct sums, the sphere symbols are empty, and all assertions reduce to finite-dimensional linear algebra; the empty manifold gives zero spaces. This proves every assertion.