Bergman and Calderón Projectors for Dirac Operators

Abstract

For a Dirac operator $D_{\bar{g}}$ over a spin compact Riemannian manifold with boundary $(\bar{X},\bar{g})$, we give a new construction of the Calderón projector on $\partial\bar{X}$ and of the associated Bergman projector on the space of L ² harmonic spinors on $\bar{X}$, and we analyze their Schwartz kernels. Our approach is based on the conformal covariance of $D_{\bar{g}}$ and the scattering theory for the Dirac operator associated with the complete conformal metric $g=\bar{g}/\rho^{2}$ where ρ is a smooth function on $\bar{X}$ which equals the distance to the boundary near $\partial\bar{X}$. We show that $\frac{1}{2}(\operatorname{Id}+\tilde{S}(0))$ is the orthogonal Calderón projector, where $\tilde{S}(\lambda)$ is the holomorphic family in {ℜ(λ)≥0} of normalized scattering operators constructed in Guillarmou et al. (Adv. Math., 225(5):2464–2516, 2010), which are classical pseudo-differential of order 2λ. Finally, we construct natural conformally covariant odd powers of the Dirac operator on any compact spin manifold.

Appendices

Appendix A: Polyhomogeneous Conormal Distributions, Densities, Blow-ups and Index Sets

On a compact manifold with corners $\bar{X}$, consider the set of boundary hypersurfaces $(H_{j})_{j=1}^{m}$ which are codimension 1 submanifolds with corners. Let ρ ₁,…,ρ _m be some boundary defining functions of these hypersurfaces. An index set $\mathcal{E}=(\mathcal{E}_{1},\dots, \mathcal{E}_{m})$ is a subset of (ℂ×ℕ₀)^m such that for each M∈ℝ the number of points $(\beta, j) \in\mathcal{E}_{j}$ with ℜ(β)≤M is finite; if $(\beta, k) \in\mathcal{E}_{j}$ then $(\beta+ 1, k)\in\mathcal {E}_{j}$, and if k>0 then also $(\beta, k-1) \in\mathcal{E}_{j}$. We define the set

$$\dot{C}^\infty(\bar{X}):=\bigl\{f\in C^\infty( \bar{X}); f\textrm{ vanishes to all orders on each } H_{j}\bigr\}. $$

Its dual $C^{-\infty}(\bar{X})$ is called the set of extendible distributions (the duality pairing is taken with respect to a fixed smooth 1-density on $\bar{X}$). Conormal distributions on manifolds with corners were defined and analyzed by Melrose [39, 40]; we refer the reader to these works for more details, but we give here some definitions. We say that an extendible distribution f on a manifold with corners X with boundary hypersurfaces (H ₁,…,H _m ) is polyhomogeneous conormal (phg for short) at the boundary, with index set $\mathcal{E}=(\mathcal {E}_{1},\dots,\mathcal{E}_{m})$, if it is smooth in the interior X, conormal (i.e., if it remains in a fixed weighted L ² space under repeated application of vector fields tangent to the boundary of $\bar{X}$), and if for each s∈ℝ we have

$$\Biggl( \prod_{j=1}^m \prod _{\substack{(z, p) \in\mathcal{E}_j\\ \mathrm{s.t.}\ \Re(z) \leq s}} (V_j - z) \Biggr) f = O \Biggl( \Biggl( \prod_{j=1}^m \rho_j \Biggr)^s \Biggr) $$

where V _j is a smooth vector field on $\bar{X}$ that takes the form $V_{j} = \rho_{j} \partial_{\rho_{j}} + O(\rho_{j}^{2})$ near H _j . This implies that f has an asymptotic expansion in powers and logarithms near each boundary hypersurface. In particular, near the interior of H _j , we have

$$f = \sum_{\substack{{(z,p)} \in\mathcal{E}_j\\ \mathrm{s.t.}\ \Re(z) \leq s}} a_{(z,p)} \rho_j^z (\log\rho_j)^p +O \bigl(\rho_j^s\bigr) $$

for every s∈ℝ, where a _(z,p) is smooth in the interior of H _j , and a _(z,p) is itself polyhomogeneous on H _j . The set of polyhomogeneous conormal distributions with index set $\mathcal{E}$ on $\bar{X}$ with values in a smooth bundle $F\to\bar{X}$ will be denoted by

$$\mathcal{A}^{\mathcal{E}}_\mathrm{phg}(\bar{X};F). $$

Recall the operations of addition and extended union of two index sets E ₁ and E ₂, denoted by E ₁+E ₂ and $E_{1} \,\overline{\cup}\, E_{2}$, respectively:

(23)

In what follows, we shall write q for the index set {(q+n,0)∣n=0,1,2,…} for any q∈ℝ. For any index set E and q∈ℝ, we write E≥q if ℜ(β)≥q for all (β,j)∈E and if (β,j)∈E and ℜ(β)=q implies j=0. Finally, we say that E is integral if (β,j)∈E implies that β∈ℤ.

On $\bar{X}$, the most natural densities are the b-densities introduced by Melrose [39, 40]. The bundle $\varOmega_{b}(\bar{X})$ of b-densities is defined to be $\rho^{-1}\varOmega(\bar{X})$ where ρ=∏ _j ρ _j is a total boundary defining function and $\varOmega(\bar{X})$ is simply the usual smooth bundle of densities on $\bar{X}$. In particular, a smooth section of the b-densities bundle restricts canonically on each H _j to a smooth b-density on H _j . The bundle of b-half-densities is simply $\rho^{-\frac{1}{2}}\varOmega^{\frac{1}{2}}(\bar{X})$.

A natural class of submanifolds, called p-submanifolds, of manifolds with corners is defined in Definition 1.7.4 in [41]. If Y is a closed p-submanifold of $\bar{X}$, one can define the blow-up $[\bar{X};Y]$ of $\bar{X}$ around Y; this is a smooth manifold with corners where Y is replaced by its inward pointing spherical normal bundle S ⁺ NY and a smooth structure is attached using polar coordinates around Y. The new boundary hypersurface is diffeomorphic to S ⁺ NY and is called the front face of $[\bar{X};Y]$; there is a canonical smooth blow-down map $\beta:[\bar{X};Y]\to\bar{X}$ which is the identity outside the front face and the projection S ⁺ NY→Y on the front face. See Sect. 5.3 of [41] for details. The pull-back β ^∗ maps continuously $\dot{C}^{\infty}(\bar{X})$ to $\dot{C}^{\infty}([\bar{X};Y])$ and it is a one-to-one correspondence, giving by duality the same statement for extendible distributions.

Appendix B: Compositions of Kernels Conormal to the Boundary Diagonal

In this section, we introduce a symbolic way to describe conormal distributions associated with the diagonal Δ _∂ inside the corner of $\bar{X}\times \bar{X}$, $\bar{X}\times\partial\bar{X}$, or $\partial\bar{X}\times\bar{X}$. In particular, we compare the class of operators introduced by Mazzeo–Melrose (the 0-calculus) to a natural class of pseudo-differential operators we define by using oscillatory integrals. We will prove composition results using both the push-forward theorem of Melrose [39] and some classical symbolic calculus. We shall use the notation from the previous sections.

2.1 B.1 Operators on $\bar{X}$

We say that an operator $K:\dot{C}^{\infty}(\bar{X})\to C^{-\infty }(\bar{X})$ is in the class $I^{s}(\bar{X}\times\bar{X},\Delta_{\partial})$ if its Schwartz kernel $K(m,m')\in C^{-\infty}(\bar{X}\times \bar{X})$ is the sum of a smooth function $K_{\infty}\in C^{\infty}(\bar{X}\times \bar{X})$ and a singular kernel K _s supported near Δ _∂ , which can be written in local coordinates (x,y,x′,y′) near a point (0,y ₀,0,y ₀)∈Δ _∂ under the form (here x is a boundary defining function on $\bar{X}$ and y some local coordinates on $\partial\bar{X}$ near y ₀, and prime denotes the right variable version of them)

(24)

where a is a smooth classical symbol of order s∈ℝ in the sense that it satisfies for all multi-indices α,α′,β

$$\big|\partial_{m}^\alpha\partial_{m'}^{\alpha'} \partial^\beta_{\zeta }a\bigl(m,m';\zeta \bigr)\big| \leq C_{\alpha,\alpha',\beta}\bigl(1+|\zeta|^2\bigr)^{s-|\beta|} $$

where m=(x,y)∈ℝ⁺×ℝⁿ and ζ:=(ξ,ξ′,μ)∈ℝ×ℝ×ℝⁿ. The integral in (24) makes sense as an oscillatory integral: We integrate by parts a sufficient number N of times in ζ to get $\Delta_{\zeta}^{N} a(m,m';\zeta)$ uniformly L ¹ in ζ; of course, we pick up a singularity of the form (x ²+x′²+|y−y′|²)^−N by this process but the outcome still makes sense as an element in the dual of $\dot{C}^{\infty}(\bar{X}\times \bar{X})$. If $\tilde{X}$ is an open manifold extending $\bar{X}$, such a kernel can be extended to a kernel $\tilde{K}$ on the manifold $\tilde{X}\times\tilde{X}$ so that $\tilde{K}$ is classically conormal to the embedded closed submanifold Δ _∂ . Therefore, our kernels (which are extendible distributions on $\bar{X}\times\bar{X}$) can freely be considered as the restriction of distributional kernels acting on a subset of functions of $\tilde {X}\times \tilde{X}$, i.e., the set $\dot{C}^{\infty}(\bar{X}\times\bar{X})$ which corresponds to smooth functions with compact support included in $\bar{X}\times\bar{X}$. Standard arguments of pseudo-differential operator theory show that we can require that K _s in charts is, up to a smooth kernel, of the form

$$K_s\bigl(x,y,x',y'\bigr)= \frac{1}{(2\pi)^{n+2}}\int_\mathbb{R}\int_\mathbb {R}\int_{\mathbb{R} ^n} e^{-ix\xi-ix'\xi'-i(y-y')\mu}a\bigl(y;\xi, \xi',\mu\bigr)d\mu d\xi d\xi'. $$

Indeed, it suffices to apply a Taylor expansion of a(x,y,x′,y′;ζ) at Δ _∂ ={x=x′=y−y′=0} and use integration by parts to show that the difference obtained by quantizing these symbols and the symbols of the form a(y,ζ) is given by smooth kernels.

We say that the symbol a is classical of order s if it has an asymptotic expansion as ζ:=(ξ,ξ′,μ)→∞

$$ a(y;\zeta)\sim\sum_{j=0}^\infty a_{s-j}(y;\zeta) $$

(25)

where a _j are homogeneous functions of degree s−j in ζ. It is clear from their definition that operators in $I^{s}(\bar{X}\times \bar{X},\Delta_{\partial})$ have smooth kernels on $(\bar{X}\times\bar{X})\setminus\Delta_{\partial}$. Let us consider the diagonal singularity of K when its symbol is classical.

Lemma 26

An operator $K_{s}\in I^{s-n-2}(\bar{X}\times\bar{X},\Delta_{\partial})$ has a kernel which is the sum of a smooth kernel together with a kernel which is smooth outside Δ _∂ and has an expansion at Δ _∂ in local coordinates (x,y,x′,y′) of the form

(26)

where $R:=(x^{2}+{x'}^{2}+|y-y'|^{2})^{\frac{1}{2}}$, (x,x′,y−y′):=Rω and K ^j,K ^j,1 are smooth.

Proof

Assume K has a classical symbol a like in (25). First, we obviously have that for any N∈ℕ, $K\in C^{N}(\bar{X}\times\bar{X})$ if s<−N. Let us write t=s−n−2; then we remark that for all y, the homogeneous function a _t−j (y,.) has a unique homogeneous extension as a homogeneous distribution on ℝⁿ⁺² of order t−j if s∉j−ℕ₀ (see [31, Theorem 3.2.3]), and its Fourier transform is homogeneous of order −s+j. Clearly, K(x,y,x′,y′) can be written as the Fourier transform in the distribution sense in ζ of A _N +B _N where for N∈ℕ

Now |ζ|^−s+N B _N (y,ζ) is in L ¹(dζ) in |ζ|>1, thus $\mathcal{F}_{\zeta\to Z}((1-\chi(\zeta))B_{N}(y,\zeta))$ is in C ^[N−s] with respect to all variables if $\chi\in C_{0}^{\infty}(\mathbb{R}^{n+2})$ equals 1 near 0, while the Fourier transform $\mathcal{F}(\chi B_{N})$ and $\mathcal{F}(\chi A_{N})$ have the same regularity and are smooth since the convolution of $\mathcal{F}(\chi)$ with a homogeneous function is smooth. This implies the expansion of K at the diagonal when t∉ℤ.

For the case t∈ℤ, this is similar but a bit more complicated. We shall be brief and refer to Beals–Greiner [4, Chap. 3.15] for more details (this is done for the Heisenberg calculus there but their proof obviously contains the classical case). Let us denote by δ _λ the action of dilation by λ∈ℝ⁺ on the space $\mathcal{S}'$ of tempered distributions on ℝⁿ⁺²; then any homogeneous function f _k of degree −n−2−k∈−n−2−ℕ₀ on ℝⁿ⁺² can be extended to a distribution $\tilde {f}_{k}\in\mathcal {S}'$ satisfying

$$ \delta_\lambda(\tilde{f}_k)= \lambda^{-n-2-k} \tilde{f}_k + \lambda^{-n-2-k}\log (\lambda) P_k $$

(27)

for some $P_{k}\in\mathcal{S}'$ of order k supported at 0. This element P _k is zero if and only if f _k can be extended as a homogeneous distribution on ℝⁿ⁺², or equivalently

(28)

According to Proposition 15.30 of [4], the distribution $\tilde {f}_{k}$ has its Fourier transform which can be written outside 0 as

$$\mathcal{F}(\tilde{f}_k) (Z)=L_k(Z)+M_k(Z) \log|Z| $$

where L _k is a homogeneous function of degree k on ℝⁿ⁺²∖{0} and M _k a homogeneous polynomial of degree k. Thus reasoning as above when t∉ℤ, this concludes the proof. It can be noted from (28) that in the expansion at Δ _∂ in (26), one has K ^j,1=0 for all j=0,…,k for some k∈ℕ if the symbols satisfy the condition

(29)

for all j=0,…,k and all y∈M. Using the expression of the symbol expansion after a change of coordinates, it is straightforward to check that this condition is invariant with respect to the choice of coordinates. □

A consequence of this lemma (or another way to state it) is that if $K\in I^{s-n-2}(\bar{X}\times\bar{X},\Delta_{\partial})$ is classical, then its kernel lifts to a conormal polyhomogeneous distribution on the manifold with corners $\bar{X}\times_{0}\bar{X}$ obtained by blowing up Δ _∂ inside $\bar{X}\times\bar{X}$ and

$$ \beta^*K\in C^\infty(\bar{X}\times_0 \bar{X})+ \begin{cases} \rho_{\mathrm{ff}}^{-s}C^\infty(\bar{X}\times_0\bar{X}) & \mbox{if }s\notin\mathbb{Z},\\[4pt] \rho_{\mathrm{ff}}^{-s}C^\infty(\bar{X}\times_0\bar{X})+\log(\rho_{\mathrm{ff} })C^\infty(\bar{X}\times_0\bar{X}) & \mbox{if }s\in \mathbb{N}_0,\\[4pt] \rho_{\mathrm{ff}}^{-s}(C^\infty(\bar{X}\times_0\bar{X})+\log(\rho_{\mathrm{ff} })C^\infty(\bar{X}\times_0\bar{X})) & \mbox{if }s\in -\mathbb{N}. \end{cases} $$

(30)

Therefore, $I^{s}(\bar{X}\times\bar{X},\Delta_{\partial})$ is a subclass of the full 0-calculus of Mazzeo–Melrose [38], in particular with no interior diagonal singularity. Let us make this more precise:

Lemma 27

Let ℓ∈−ℕ; then a classical operator $K\in I^{\ell }(\bar{X}\times\bar{X},\Delta_{\partial})$ with a local symbol expansion (25) has a kernel which lifts to $\beta^{*}K\in\rho_{\mathrm{ff}}^{-\ell -n-2}C^{\infty}(\bar{X}\times_{0}\bar{X})+C^{\infty}(\bar{X}\times_{0}\bar{X})$ if the symbol satisfies the condition (29) for all j∈ℕ₀. Conversely, if $K\in C^{-\infty}(\bar{X}\times\bar{X})$ is a distribution which lifts to β ^∗ K in $\rho_{\mathrm{ff}}^{-\ell -n-2}C^{\infty}(\bar{X}\times_{0}\bar{X})+C^{\infty}(\bar{X}\times_{0}\bar{X})$, then it is the kernel of a classical operator in $I^{-n-2}(\bar{X}\times \bar{X},\Delta_{\partial})$ with a symbol satisfying (29) for all j∈ℕ₀.

Proof

Let us start with the converse: We can extend smoothly the kernel β ^∗ K to the blown-up space $[\tilde{X}\times\tilde{X}, \Delta_{\partial}]$ where $\tilde{X}$ is an open manifold extending smoothly $\bar{X}$. Then the extended function has an expansion to all order in polar coordinates (R,ω) at {R=0} (i.e., around Δ _∂ ) where $R=(x^{2}+{x'}^{2}+|y-y'|^{2})^{\frac{1}{2}}$ and Rω=(x,x′,y−y′)

for some smooth K ^j; in particular, using Fourier transform in Z=(x,x′,y−y′) one finds that for all k∈ℕ, there exists a classical symbol a ^k(y,ζ)

$$K\bigl(x,y,x',y'\bigr)-\frac{1}{(2\pi)^{n+2}}\int e^{ix\xi+ix'\xi'+i(y-y')\mu } a^k\bigl(y;\xi,\xi',\mu\bigr)d\xi d \xi' d\mu\in C^k(\bar{X}\times \bar{X}) $$

with a ^k being equal to $\sum_{j=0}^{k}a^{k}_{j}(y;\zeta)$ when |ζ|>1 for some homogeneous functions $a^{k}_{j}$ of degree ℓ−j. Moreover, the $a^{k}_{j}$ can be extended as homogeneous distributions on ℝⁿ⁺² since they are given by Fourier transforms of the homogeneous distributions K ^j(y,Z) in the variable Z. Using that a homogeneous function on ℝⁿ⁺²∖{0} which extends as a homogeneous distribution on ℝⁿ⁺² has no logλ terms in (27), or equivalently satisfies (28), this ends one way.

To prove the first statement, it suffices to consider the kernel in local coordinates, and locally β ^∗ K has the structure (30) with no log(ρ _ff) if the local symbol satisfies (29). Notice that having locally the structure $\rho_{\mathrm{ff}}^{-s}C^{\infty}(\bar{X}\times\bar{X})$ for a function is a property which is independent of the choice of coordinates. But from what we just proved above, this implies that in any choice of coordinates the local symbol satisfies (29). □

We shall call the subclass of operators in Lemma 27 the class of log-free classical operators of order ℓ∈−ℕ, and denote it $I_{\mathrm{lf}}^{\ell}(\bar{X}\times \bar{X},\Delta_{\partial})$.

For (log-free if s∈−ℕ) classical operators in $I^{s}(\bar{X}\times \bar{X},\Delta_{\partial})$, there is also a notion of principal symbol which is defined as a homogeneous section of degree s of the conormal bundle N ^∗Δ _∂ : If a has an expansion $a(y,\zeta)\sim\sum_{j=0}^{\infty}a_{s-j}(y,\zeta)$ as ζ→∞ with a _s−j homogeneous of degree s−j in ζ, then the principal symbol is given by σ _pr(K)=a _s . The principal symbol is actually not invariantly defined if one considers K as an extendible distribution on $\bar{X}\times \bar{X}$: If a(y,ζ) and a′(y,ζ) are two classical symbols for the kernel K, then if Z=(x,x′,z)

$$\mathcal{F}_{\zeta\to Z} \bigl(a_s(y,\zeta)-a_s'(y, \zeta)\bigr)=0 \quad\textrm{when }x>0 \textrm{ and } x'>0, $$

thus it is defined only up to this equivalence relation.

To make the correspondence with the 0-calculus of Mazzeo–Melrose [38], we recall that the normal operator of an operator $K\in C^{\infty}(\bar{X}\times_{0}\bar{X})$ is given by the restriction to the front face: If y∈Δ _∂ , $N_{y}(K):= K|_{\mathrm{ff}_{y}}$ where ff _y is the fiber at y of the unit interior pointing spherical normal bundle S ⁺ NΔ _∂ of Δ _∂ inside $\bar{X}\times\bar{X}$, then we remark that the normal operator at y∈Δ _∂ of an admissible operator $K\in I_{\mathrm{lf}}^{-n-2}(\bar{X}\times \bar{X};\Delta_{\partial})$ is given by the homogeneous function of degree 0 on ℝ⁺×ℝ⁺×ℝⁿ≃ff _y ×ℝ₊

$$N_y(K) (Z)=\mathcal{F}_{\zeta\to Z}\bigl(\sigma_\mathrm{pr}(K) (y,\zeta)\bigr). $$

2.2 B.2 Operators from $\bar{X}$ to $\partial\bar{X}$ and Conversely

We define operators in $I^{s}(\bar{X}\times\partial\bar{X},\Delta_{\partial})$ and $I^{s}(\partial\bar{X}\times\bar{X},\Delta_{\partial})$ by saying that their respective distributional kernels are the sum of a smooth kernel on $\bar{X}\times\partial\bar{X}$ (resp., $\partial\bar{X}\times\bar{X}$) and of a singular kernel $K_{s}\in C^{-\infty}(\bar{X}\times\partial\bar{X})$ (resp., $L_{s}\in C^{-\infty}(\partial\bar{X}\times\bar{X})$) supported near Δ _∂ of the form (in local coordinates)

(31)

with a and b some smooth symbols

for all α,β. We shall say they are classical if their symbols have an expansion in homogeneous functions at ζ→∞, just like above for operators on $\bar{X}$. It is easy to see that such operators map respectively $\dot{C}^{\infty}(\bar{X})$ to $C^{\infty}(\partial\bar{X})$ and $C^{\infty}(\partial\bar{X})$ to $C^{-\infty}(\bar{X})\cap C^{\infty}(X)$.

Using the exact same arguments as for operators on $\bar{X}$, we have the following lemma.

Lemma 28

Let ℓ∈−ℕ, then a classical operator $K\in I^{\ell }(\bar{X}\times\partial\bar{X},\Delta_{\partial})$ with a local symbol expansion $a(y,\zeta)\sim\sum_{j=0}^{\infty}a_{-n-1-j}(y,\zeta)$ has a kernel which lifts to $\beta_{1}^{*}K\in\rho_{\mathrm{ff}}^{-\ell -n-1}C^{\infty}(\bar{X}\times_{0}\partial\bar{X})+C^{\infty}(\bar{X}\times_{0}\partial \bar{X})$ if

(32)

for all j∈ℕ₀. Conversely, if $K\in C^{-\infty }(\bar{X}\times \bar{\partial}{X})$ is a distribution which lifts to $\beta_{1}^{*}K$ in $\rho_{\mathrm{ff}}^{-\ell-n-1}C^{\infty}(\bar{X}\times_{0}\partial\bar{X})+C^{\infty}(\bar{X}\times_{0}\partial\bar{X})$, then it is the kernel of a classical operator in $I^{\ell}(\bar{X}\times \partial\bar{X},\Delta_{\partial})$ with a symbol satisfying (29) for all j∈ℕ₀. The symmetric statement holds for operators in $I^{\ell}(\partial\bar{X}\times\bar{X},\Delta_{\partial})$.

We shall also call the operators of Lemma 28 log-free classical operators and denote this class by $I^{\ell}_{\mathrm{lf}}(\bar{X}\times\partial\bar{X},\Delta_{\partial})$ and $I^{\ell}_{\mathrm{lf}}(\partial\bar{X}\times\bar{X},\Delta_{\partial})$.

Notice that, since the restriction of a function in $C^{\infty}(\bar{X}\times_{0}\bar{X})$ to the right boundary gives a function in $C^{\infty}(\bar{X}\times_{0}\partial\bar{X})$, we deduce that an operator $I^{-n-2}(\bar{X}\times\bar{X},\Delta_{\partial})$ satisfying condition (29) induces naturally (by restriction to the boundary on the right variable) an operator in $I^{-n-1}(\bar{X}\times\partial\bar{X},\Delta_{\partial})$ satisfying (32). This can also be seen by considering the oscillatory integrals restricted to x′=0 but it is more complicated to prove.

2.3 B.3 Compositions

We start with a result on the composition of operators mapping from $\bar{X}$ to M with operators mapping M to M or M to $\bar{X}$. This will be done using the push-forward theorem of Melrose [39, Theorem 5].

Proposition 29

Let $A:C^{\infty}(M;{{}^{0}\varSigma}\otimes\varOmega^{\frac{1}{2}})\to C^{\infty}(M;{{}^{0}\varSigma}\otimes\varOmega^{\frac{1}{2}})$ be a pseudo-differential operator of negative order with lifted kernel in $\mathcal{A}_{\mathrm{phg}}^{{E_{\mathrm{ff}}}}(M\times_{0} M; \mathcal{E}\otimes\varOmega_{b}^{\frac{1}{2}})$. Let $B: \dot{C}^{\infty}(\bar{X};{{}^{0}\varSigma}\otimes\varOmega_{b}^{\frac {1}{2}}) \to C^{\infty}(M;{{}^{0}\varSigma}\otimes\varOmega^{\frac{1}{2}})$ be an operator with lifted kernel in $\mathcal{A}_{\mathrm{phg}}^{F_{\mathrm{ff}},F_{\mathrm{rb}}} (M\times_{0}\bar{X};\mathcal{E}_{r}\otimes\varOmega_{b}^{\frac{1}{2}})$ and let $C:C^{\infty}(M;{{}^{0}\varSigma}\otimes\varOmega^{\frac{1}{2}})\to C^{-\infty}(\bar{X};{{}^{0}\varSigma}\otimes\varOmega_{b}^{\frac{1}{2}})$ be an operator with lifted kernel on $\mathcal{A}_{\mathrm{phg}}^{G_{\mathrm{ff}},G_{\mathrm{lb}}}(\bar{X}\times_{0}{M}; \mathcal{E}_{l}\otimes\varOmega_{b}^{\frac{1}{2}})$. Then the Schwartz kernels of A∘B and C∘B lift to polyhomogeneous conormal kernels

and the index sets satisfy

Proof

The proof is an application of the Melrose push-forward theorem. Let us discuss first the composition A∘B. We denote by Δ both the diagonal in M×M and the submanifold $\{(m,m')\in M\times\bar{X};m=m'\}$, by (π _j ) _j=l,c,r the canonical projections of $M\times M\times\bar{X}$ obtained by projecting-off the j factor (here l,c,r mean left, center, right), and let

The triple space $M\times_{0}M\times_{0}\bar{X}$ is the iterated blow-up

$$ M\times_0M\times_0\bar{X}:=[M \times M\times\bar{X};\Delta_3,\Delta_{2,l}, \Delta_{2,c},\Delta_{2,r}]. $$

(33)

The submanifolds to blow up are p-submanifolds; moreover, Δ₃ is contained in each Δ_2,j and the lifts of Δ_2,j to the blow-up $[M\times M\times \bar{X};\Delta_{3}]$ are disjoint. Consequently (see, for instance, [28, Lemma 6.2]) the order of blow-ups can be commuted and the canonical projections π _j lift to maps

which are b-fibrations. The manifold $M\times_{0}M\times_{0}\bar{X}$ has 5 boundary hypersurfaces, the front face ff′ obtained by blowing up Δ₃, the faces $\operatorname{lf},\operatorname{cf}, \operatorname{rf}$ obtained from the respective blow-up of Δ_2,l ,Δ_2,c ,Δ_2,r and finally the face rb′ obtained from the lift of the original face $M\times M\times M\subset M\times M\times\bar{X}$. We denote by ρ _f a smooth boundary defining function of the face $f\in\{\mathrm{ff}',\operatorname{rf},\operatorname{cf},\operatorname{lf},\mathrm{rb}'\}$. If k _A and k _B are the lifted kernel of A and B to, respectively, M×₀ M and $M\times_{0}\bar{X}$ then it is possible to write the composition as a push-forward

$$k_{A\circ B}.\mu={\beta_c}_* \bigl(\beta_r^*k_A. \beta_l^*k_B.\beta_c^*\mu \bigr) $$

if $\mu\in C^{\infty}(M\times_{0}\bar{X};\mathcal{E}_{r}\otimes\varOmega_{b}^{\frac{1}{2}})$. An easy computation shows that a smooth b-density ω on $M\times M\times\bar{X}$ lifts through β to an element

$$\beta^*\omega\in\rho_{\mathrm{ff}'}^{2n}(\rho_\mathrm{lf} \rho_\mathrm{rf}\rho_\mathrm{cf})^n C^\infty(M \times_0M\times_0\bar{X}; \varOmega_b) $$

so by considering the lifts through β _l ,β _c ,β _r of boundary defining functions in $M\times_{0}\bar{X}$, $M\times_{0}\bar{X}$ and $\bar{X}\times_{0}\bar{X}$, respectively, we deduce that there is some index set K=(K _ff′,K _rb′,K _lf,K _rf,K _cf) such that

Then from the push-forward theorem of Melrose [39, Theorem 5], we obtain that

and this shows the first composition result for A∘B. Remark that to apply [39, Theorem 5], we need the index of K _rf>0, i.e., E _ff+n/2>0, but this is automatically satisfied with our assumption that A is a pseudo-differential operator of negative order on M.

The second composition result is very similar, except that there are more boundary faces to consider. One defines $\Delta_{3}:=\{(m,m',m'')\in\bar{X}\times M\times \bar{X}; m=m'=m''\}$ and let

$$\Delta_{2,j}=\bigl\{(m_l,m_c,m_r) \in\bar{X}\times M\times\bar{X}; m_i=m_k \textrm{ if }j\notin\{i,k\}\bigr\} $$

similarly as before. The triple space is defined like (33); it now has 6 boundary faces which we denote as in the case above but with the additional face, denoted lb′, obtained from the lift of the original boundary $M\times M\times \bar{X}$. The same arguments as above show that the canonical projections from $\bar{X}\times_{0} M\times_{0}\bar{X}$ obtained by projecting-off one factor lift to b-fibrations β _r ,β _l ,β _c from the triple space to $\bar{X}\times_{0}M$, $M\times_{0}\bar{X}$, and $\bar{X}\times_{0}\bar{X}$. Like for the case above, one has to push forward a distribution $\beta_{r}^{*}k_{C}.\beta_{l}^{*}k_{B}.\beta_{c}^{*}\mu$, and a computation gives that there is an index set L=(L _ff′,L _rb′,L _lb′,L _lf,L _rf,L _cf) such that

and by pushing forward through β _c using Melrose [39, Theorem 5], we deduce that the result is polyhomogeneous conormal on $\bar{X}\times_{0}\bar{X}$ with the desired index set. □

In order to analyze the composition K ^∗ K in Sect. 3.2, we use the symbolic approach since it is slightly more precise (in terms of log terms at the diagonal) than the push-forward theorem in this case, and a bit easier to compute the principal symbol of the composition. We are led to study the composition between classical operators K and L where $K:C^{\infty}(\bar{X}) \to C^{\infty}(\partial\bar{X})$ is an operator in $I^{-1}(\bar{X}\times\partial\bar{X})$ and $L:C^{\infty}(\partial\bar{X}) \to C^{\infty}(\bar{X})$ is in $I^{-1}(\partial\bar{X}\times\bar{X})$. We show

Lemma 30

Let $K\in I^{-1}(\bar{X}\times\partial\bar{X})$ and $L\in I^{-1}(\partial\bar{X}\times\bar{X})$ with principal symbols σ _K (y;ξ,μ) and σ _L (y;ξ,μ). The composition L∘K is a classical pseudo-differential operator on $\partial\bar{X}$ in the class $L\circ K\in\varPsi^{-1}(\partial\bar{X})$. Moreover, the principal symbol of LK is given by

$$ \sigma_\mathrm{pr}(L\circ K) (y;\mu)=(2 \pi)^{-2}\int_{0}^\infty\hat { \sigma}_{L}(y;-x,\mu). \hat{\sigma}_{K}(y;x,\mu) dx $$

(34)

where $\hat{\sigma}$ denotes the Fourier transform of σ in the variable ξ.

Proof

Since the composition with smoothing operators is easier, we essentially need to understand the composition of singular kernels like (31). Writing the kernel of K and L as a sum of elements K _j ,L _j of the form (31), we are reduced to analyze in a chart U

where dΩ:=dy′dx′dξdξ′dμdμ′, $\chi\in C_{0}^{\infty}(U)$ and a,b are compactly supported in U in the y and y″ coordinates. If U intersects the boundary $\partial\bar{X}$, then χ is supported in x′≥0. The kernel of the composition L _j K _j in the chart U is then

where

We want to prove that c(y,y″;μ) is a symbol of order −1 with an expansion in homogeneous terms in μ as μ→∞. We shall only consider the case where $U\cap\partial\bar{X}\not=\emptyset$ since the other case is simpler. First, remark that in U the function χ can be taken of the form χ(x,y)=φ(x)ψ(y) with $\psi\in C_{0}^{\infty}(\mathbb{R}^{n})$ and $\varphi\in C_{0}^{\infty}([0,1))$ equal to 1 in [0,1/2]; therefore, $\hat{\chi}(\xi,\mu)=\hat{\varphi}(\xi)\hat{\psi }(\mu)$ with $\hat{\psi}$ Schwartz and by integration by parts one also has

$$\hat{\varphi}(\xi)= \frac{1}{i\xi}\bigl( 1+ \hat{\varphi'}( \xi)\bigr). $$

with $\hat{\varphi'}$ Schwartz. We first claim that $|\partial_{y}^{\alpha}\partial^{\beta}_{y''}\partial_{\mu}^{\gamma}c(y,y'';\mu)|\leq C \langle\mu\rangle^{-1-|\gamma|}$ uniformly in y,y″: Indeed, using the properties of $\hat{\chi}$ and the symbolic assumptions on a,b, we have that for any N≫|β|, there is a constant C>0 such that

where j+k=|γ|. Using polar coordinates iξ+ξ′=re ^iθ in ℂ≃ℝ², the integral above is bounded by

$$C\int \biggl(\frac{1}{1+r|\cos(\theta)|+|\mu|} \biggr)^{1+k} \biggl( \frac{1}{1+r|\sin\theta|+|\mu|} \biggr)^{1+j}\frac{1}{1+r|\cos \theta-\sin\theta|}r drd\theta $$

which, by a change of variable r→r|μ| and splitting the θ integral in different regions, is easily shown to be bounded by C〈μ〉^−1−|β|.

To prove that LK is a classical operator of order −1 (with an expansion in homogeneous terms), we can modify slightly the usual proof of composition of pseudo-differential operators, like in Theorem 3.4 of [24]. Let $\theta\in C_{0}^{\infty}(\mathbb{R})$ be an even function equal to 1 near 0. We write for μ=λω with ω∈S ⁿ⁻¹

with

where Ω=(σ,s,ζ,x′). Let us denote the phase by Φ:=x′ζ+σ.s. The last integral can be dealt with by integrating by parts in x′:

(35)

We can extend φ′=∂ _x φ by 0 on (−∞,0] to obtain a $C_{0}^{\infty}(\mathbb{R})$ function which vanishes near 0. Since φ′ now vanishes near 0, one easily proves that the first integral in (35) is O(λ ^−N) for all N, uniformly in y,y″ by using integration by parts N times in x′ and ∂ _x′(e ^−iλx′ζ)=−iλζe ^−iλx′ζ. Now for the second integral in (35), we use stationary phase in (σ,s); one has for any N∈ℕ

(36)

with |S _N (y,y″;ξ′,ζ,μ)|≤C〈(ξ′,μ)〉⁻¹〈(ξ′+|μ|ζ,μ)〉^−1−N. Now, both a and b can be written in the form a=a _N +a _h and b=b _h +b _N where a _N (y;ξ,μ),b _N (y;ξ,μ) are bounded in norm by C〈(ξ,μ)〉^−N and a _h (y;ξ,μ),b _h (y,ξ,μ) are finite sums of homogeneous functions $a_{h}^{-j},b_{h}^{-j}$ of order −j in |(ξ,μ)|>1 for j=1,…,N−1. Replacing a,b in (36) by their decomposition a _N +a _h and b _N +b _h we get that c(y,y″,μ) is the sum of a term bounded uniformly by C〈μ〉^−N+2 and some terms of the form

$$\lambda\int\frac{1-\theta(\zeta)}{\zeta}b_h^{-j}\bigl(y; \xi',\mu \bigr)\partial^\alpha\psi\bigl(y'' \bigr)\partial_\mu^\alpha a_h^{-k} \bigl(y'';\xi'+\lambda \zeta,\mu \bigr)d \zeta d\xi'. $$

The integral is well defined and is easily seen (by changing variable ξ′→λξ′) to be homogeneous of order −k−j−|α|+1 for λ=|μ|>1 . This shows that c ₂(y,y″;μ) has an expansion in homogeneous terms. It remains to deal with c ₁. We first apply stationary phase in the (σ,s) variables and we get

for some $S_{N}'$ which will contribute O(λ ^−N−2) like for c ₂ above. Decomposing a(y,ξ,μ) and b(y,ξ,μ) as above in homogeneous terms outside a compact set in (ξ,μ), it is easy to see that up to a O(λ ^−N) term, we can reduce the analysis of c ₁(y,y″;μ) to the case where a,b are replaced by terms $a_{h}^{-j},b_{h}^{-k}$ homogeneous of orders −j,−k outside compacts. We then have

(37)

and we write by Taylor expansion at ζ=0

$$ \theta(\zeta)\partial_\mu^\alpha a_h^{-k}\bigl(y''; \xi'+\zeta,\omega \bigr)=\theta(\zeta)\partial_\mu^\alpha a_h^{-k}\bigl(y''; \xi',\omega\bigr)+ \zeta\theta(\zeta) a' \bigl(y'',\xi',\zeta,\omega\bigr) $$

(38)

for some a′(y″;ξ′,ζ,μ) smooth in y″ and homogeneous of degree −k−1 in |(ξ,ζ,μ)|>1. For the term with a′, we have by integration by parts in x′

(39)

and the first term is O(λ ^−∞) by non-stationary phase while the second one is homogeneous of order −1 in λ (the integrals in all variables are converging). It remains to deal with the first term in (38). We notice that θ is even and so

Since $\hat{\theta}$ is Schwartz, the last line clearly has an expansion of the form πλ ⁻¹+O(λ ^−∞) for some constant C, and combining with (39), we deduce that (37) is thus homogeneous of degree λ ^−j−k−1 modulo O(λ ^∞). This ends the proof of the fact that KL is a classical pseudo-differential operator on M.

Now we compute the principal symbol. According to the discussion above, it is given by

where $\zeta\sigma_{K}'(y;\xi',\zeta,\mu):=\sigma_{K}(y;\xi'+\zeta ,\mu)-\sigma_{K}(y;\xi',\mu)$. It is straightforward to see that this is equal to (34) by using the fact that the Fourier transform of the Heaviside function is the distribution πδ−i/ζ. Notice that the integral (34) makes sense since σ _K ,σ _L are L ² in the ξ′ variable. □