b-pseudodifferential operators and b-Sobolev spaces

Passing from Diff_b(M) to the algebra ofb-pseudodifferential operators Ψ_b(M) amounts to allowing symbols to be more general functions than polynomials; apart from symbols being smooth functions on ^bT^∗M, rather than onT^∗M ifM was boundaryless, this is entirely analogous to the way one passes from differential to pseudodifferential operators, with the technical details being a bit more involved. One can have a rather accurate picture of b-pseudodifferential operators, however, by considering the following: fora∈

C^∞(^bT^∗M), we saya∈S^m(^bT^∗M) ifasatisfies

|∂_z^α∂^β_ζa(z, ζ)|6Cαβhζi^m−|β|for all multi-indices αandβ (A.6) in any coordinate chart, wherezare coordinates in the base andζcoordinates in the fiber;

more precisely, in local coordinates (τ, x) nearX, we takeζ=(σ, ξ) as above. We define the quantization Op(a) of a, acting on smooth functions u supported in a coordinate chart, by

Op(a)u(τ, x) = (2π)⁻ⁿ Z

e^{i(τ−τ0)˜σ+i(x−x0)ξ}φ τ−τ⁰

τ

×a(τ, x, τσ, ξ)u(τ˜ ⁰, x⁰)dτ⁰dx⁰d˜σ dξ,

(A.7)

where theτ⁰-integral is over [0,∞), andφ∈C_c^∞ −¹₂,¹₂

is identically 1 near 0. The cutoff φensures that these operators lie in the ‘small b-calculus’ of Melrose, in particular that such quantizations act on weighted b-Sobolev spaces, defined below; see also the explicit description of the Schwartz kernels below using blow-ups. For generalu, we define Op(a)u using a partition of unity. We write Op(a)∈Ψ^m_b(M); every element of Ψ^m_b(M) is of the form Op(a) for somea∈S^m(^bT^∗M) modulo the set Ψ^−∞_b (M) of smoothing operators. We say thatais asymbol of Op(a). The equivalence class ofainS^m(^bT^∗M)/S^m−1(^bT^∗M) is invariantly defined on^bT^∗M and is called theprincipal symbol of Op(a).

A different way of looking at Ψb(M) is in terms of H¨ormander’s uniform algebra, namely pseudodifferential operators onRⁿ arising as, say, left quantizations of symbols

˜a∈S^m_∞(Rⁿz˜;Rⁿz˜) satisfying estimates

|∂_z^α_˜∂^β_˜za(˜˜ z,z)|˜ 6C_αβh˜zi^m−|β|for all multi-indices αandβ. (A.8)

To see the connection, consider local coordinates (τ, x) onMnear∂Mas above, and write

˜

z=(t, x) witht=−logτ, with the region of interest being a cylindrical set (C,∞)t×Ωx

corresponding to (0, e^−C)τ×Ωx. Then, the uniform estimates (A.8) are equivalent to estimates (pulling back ˜avia the mapψ(τ, x)=(−logτ, x))

|(τ ∂_τ)^α0∂_x^α0∂_σ^β0∂_ξ^β0ψ^∗a(τ, x, σ, ξ)|˜ 6C_αβh(σ, ξ)i^m−|β|

for all multi-indicesα= (α₀, α⁰) andβ= (β₀, β⁰),

(A.9)

and the quantization map becomes Op(˜f a)u(τ, x)

= (2π)⁻ⁿ Z

e^{ilog(τ /τ0)σ+i(x−x0)ξ}ψ^∗˜a(τ, x,−σ, ξ)u(τ⁰, x⁰)(τ⁰)⁻¹dτ⁰dx⁰dσ dξ, Letting ˜σ=τ⁻¹σ, this reduces to an oscillatory integral of the form (A.7), taking into account that inτ /τ⁰∈(C⁻¹, C),C >0, the function log(τ /τ⁰) in the phase is equivalent to (τ−τ⁰)/τ. (Notice that, in terms oftandt⁰, the cutoffφin (A.7) is a compactly supported function oft−t⁰, identically 1 near zero.) Withz=(τ, x),ζ=(σ, ξ), these estimates (A.9) would be exactly the estimates (A.6), if (τ ∂τ)^α0 were replaced by∂^α_τ⁰. Thus, (A.9) gives rise to the space of b-ps.d.o’s conormal to the boundary, i.e. in terms of b-differential operators, the coefficients are allowed to be merely conormal to X=∂M, rather than smooth up to it. As in the setting of classical (one-step polyhomogeneous) symbols, for distributions smoothness up to the boundary is equivalent to conormality (symbolic estimates in the symbol case) plus an asymptotic expansion; thus, apart from the fact that we need to be careful in discussing supports, the b-ps.d.o. algebra is essentially locally a subalgebra of H¨ormander’s uniform algebra. Most properties of b-ps.d.o’s are true even in this larger, ‘conormal coefficients’ class, and indeed this perspective is very important when the coefficients are generalized to have merely finite Sobolev regularity as was done in [71] and [76]; indeed, the ‘only’ significant difference concerns the normal operator, which does not make sense in the conormal setting. We also refer to [141, Chapter 6] for a full discussion, including an introduction of localizers, far from diagonal terms, etc.

IfA∈Ψ^m_b¹(M) andB∈Ψ^m_b²(M), thenAB, BA∈Ψ^m_b^1+m2(M), while [A, B]∈Ψ^m_b^1+m2−1(M),

and its principal symbol is

1

iH_ab≡1 i{a, b}, withH_a as above.

We also recall the notion of b-Sobolev spaces: fixing a volume b-density ν on M, which locally is a positive multiple of

which one can extend to s∈R by duality and interpolation. Weighted b-Sobolev spaces are denoted

H_b^s,α(M) =τ^αH_b^s(M), (A.10) i.e. its elements are of the form τ^αuwith u∈H_b^s(M). Any b-pseudodifferential opera-torP ∈Ψ^m_b(M) defines a bounded linear map P:H_b^s,α(M)!H_b^s−m,α(M) for alls, α∈R. Correspondingly, there is a notion of wave front set WF^s,α_b (u)⊂^bS^∗M for a distribution u∈H_b^−∞,α(M)=S

s∈RH_b^s,α(M), defined analogously to the wave front set of distribu-tions on Rⁿ or closed manifolds: a point $∈^bS^∗M is not in WF^s,α_b (u) if and only if there existsP ∈Ψ⁰_b(M), elliptic at$(i.e. with principal symbol non-vanishing on the ray corresponding to$), such thatPu∈H_b^s,α(M). Notice, however, that wedoneed to have a-priori control on the weightα(we are assuming u∈H_b^−∞,α(M)), which again reflects the lack of commutativity of Ψ_b(M) to leading order in the sense of decay of coefficients at∂M.

The Mellin transform is also well behaved on the b-Sobolev spacesH_b^s(X×[0,∞)), and indeed gives a direct way of defining non-integer order Sobolev spaces. For s>0 (cf. [110, equation (5.41)]),

is an isomorphism, with the analogue of (A.5) also holding. Note that the right-hand side of (A.11) is equivalent to

h|σ|i^sv∈L²(R;H_h|σ|i^s −1(X;ν)), (A.12) where the space on the right-hand side is the standard semiclassical Sobolev space and h|σ|i=(1+|σ|²)^1/2; indeed, for s>0 integer, both are equivalent to the statement that, for allβ with|β|6s,h|σ|i^s−|β|D^β_xv∈L²(R;L²(X;ν)). Here, by equivalence we mean not only the membership in a set, but also that of the standard norms, such as

corresponding to these spaces. Note that, by dualization, (A.12) characterizes the Mellin transform ofH_b^s,α for alls∈R.

The basic microlocal results, such as elliptic regularity, propagation of singularities and radial point estimates, have versions in the b-setting; these are purely symbolic (i.e.

do not involve normal operators), and thus by themselves are insufficient for a Fredholm analysis, since the latter requires estimates with relatively compact errors. It is usually convenient to state these results in terms of wave front set containments, but by the closed graph theorem such statements are automatically equivalent to microlocalized Sobolev estimates, and are indeed often proved by such.

Let us first discuss microlocal elliptic regularity for a classical operatorP ∈Ψ^m_b(M).

We recall thatP elliptic at$∈^bS^∗M ifσ_b(P) is invertible at$; here, one renormalizes the principal symbol by using any non-degenerate homogeneous degree-m section of

bT^∗M, so that the restriction to^bS^∗M, considered as fiber infinity, makes sense.

Proposition A.1. Supposeu∈H_b^s0,α for somes⁰, α,and $ /∈WF^s−m,α_b (Pu). Then,

$ /∈WF^s,α_b (u). Quantitatively, the estimate kB₂uk_H^s,α

b 6C(kB₁Puk_Hs−m,α b

+kukH^s_b^0,α)

is valid for B1, B2∈Ψ⁰_b(M), with B1 elliptic on WF⁰_b(B2)and P elliptic on WF⁰_b(B2).

Next, to describe propagation of singularities for a classical operator P ∈Ψ^m_b(M) with real principal symbolp(scalar ifP is acting on vector bundles), we recall that the characteristic set Char(P) is the complement of its elliptic set (the set of points where P is elliptic). Then, we have the following.

Proposition A.2. Let u∈H_b^s0,α for some s⁰, α. Then WF^s,α_b (u)\WF^s−m+1,α_b (Pu) is the union of maximally extended (null) bicharacteristics, i.e. integral curves of Hp

inside the characteristic set Char(P)of P inside Char(P)\WF^s−m+1,α_b (Pu).

This statement is vacuous at points$, whereHpis radial, i.e. tangent to the dilation orbits in the fibers of^bT^∗M\o. Elsewhere, it again amounts to an estimate, which now is of the form

kB2uk_H^s,α

b 6C(kB1uk_H^s,α

b +kSPuk_H^s−m+1,α

b +kuk_Hs0,α b

), (A.13)

which is valid wheneverB₁, B₂, S∈Ψ⁰_b(M) withSelliptic on WF⁰_b(B₂), and every bichar-acteristic from WF⁰_b(B₂) reaching the elliptic set of B₁, say in the backward direction alongH_p, while remaining in the elliptic set inS; this estimate gives propagation in the forward direction alongH_p.

The estimate (A.13) remains valid, for propagation in the forward direction, ifpis no longer real, but Imp60, and in the backward direction if Imp>0. Such an operator is calledcomplex absorbing. Notice that one has a better, elliptic, estimate where Imp>0;

the point is that the propagation of singularities estimate works at the boundary of this region.

Radial points ofH_p come in many flavors, depending on the linearization ofH_p. In the present context, at the b-conormal bundle of the boundary of the event or cosmologi-cal horizon at infinity, the important type is saddle points, or more precisely submanifolds L of normally saddle points, introduced in [75, §2.1.1] in the b-setting. (See [11] for a source/sink case, which is relevant to the wave equation on Minkowski-type spaces.) More precisely, the saddle point is of the following type: within the characteristic set ofP, one of the stable/unstable manifolds lies in ^bS_X^∗M, and the other, call it L, is transversal to ^bS_X^∗M; the full assumptions are stated in [75, §2.1.1]; see also the discussion of the dynamics in§3.4. In fact, one should really consider at least the infinitesimal behavior of the linearization towards the interior of the cotangent bundle as well, i.e. work on

bT^∗M, with ^bS^∗M its boundary at fiber infinity; then, withL a submanifold of^bS_X^∗M still, we are interested in the setting in which the statement about stable/unstable man-ifolds still holds in^bT^∗M. Then, there is a critical regularity, s=¹₂(m−1)+βα, where α is the Sobolev weight order as above, andβ arises from the subprincipal symbol of P at L; in the case of event horizons, it is the reciprocal surface gravity. (If P 6=P^∗, there is a correction term to the critical regularity; see§5.1.) Namely, the theorem ([75, Proposition 2.1]) states the following facts:

• fors>¹₂(m−1)+βα, one can propagate estimates from a punctured neighborhood ofL∩^bS_X^∗M inL toX;

• ifs<¹₂(m−1)+βα, the opposite direction of propagation is possible.

Remark A.3. This is the redshift effect in the direction of propagation into the boundary, since one has then a source/sink within the boundary; that is, in the direction in which the estimates are propagated, the linearization at L infinitesimally shifts the frequency (where one is in the fibers of ^bT^∗M) away from fiber infinity, i.e. to lower frequencies. Dually, this gives a blue shift effect when one propagates the estimates out of the boundary.

When theH_p-flow has an appropriate global structure, e.g. when one has complex absorption in some regions, and theH_p-flow starts from and ends in these, potentially after ‘going through’ radial saddle points, see [75,§2.1], one gets global estimates

kuk_H^s,α

b 6C(kPuk_Hs−m+1,α b

+kukH_b^s0,α), (A.14)

withs⁰<s, provided of course the threshold conditions are satisfied when radial points are present (depending on the direction of propagation). One also has dual estimates for P^∗, propagating in the opposite direction.

Due to the lack of gain inα, these estimates do not directly give rise to a Fredholm theory, even ifM is compact, since the inclusion mapH_b^s,α!H_b^s0,α is not compact even ifs>s⁰. This can be done via the analysis of the (Mellin transformed) normal operator N(Pb )(σ)≡Pb(σ). Namely, whenP(σ) has no poles in the regionb −Imσ∈[r⁰, r], and when large|Reσ| estimates hold for Pb, which is automatic when P has the global structure allowing for theglobal estimates (A.14), then one can obtain estimates like

kuk_H^s,r

b 6C(kN(P)uk_H^s−m+1,r

b +kuk_Hs0,r0 b

),

which, when applied to the error term of (A.14) via the use of cutoff functions, gives kuk_H^s,r

b 6C(kPuk_Hs−m+1,r b

+kukH_b^s0,r0),

withs>s⁰,r>r⁰, and dual estimates forP^∗, whichdoes give rise to a Fredholm problem forP.

We are also interested in domains inM, more precisely ‘product’ or ‘p’ submanifolds with corners Ω inM. Thus, Ω is given by inequalities of the formtj>0,j=1,2..., m, such that at any pointpthe differentials of those of thetj, as well as of the boundary defining functionτ ofM, which vanish at p, must be linearly independent (as vectors in T_p^∗M).

For instance, ifm=2, andp∈X=∂M witht₁(p)6=0,t₂(p)=0, thendτ(p) anddt₂(p) must be linearly independent. The main example of interest is the domain Ω defined in§3.5 (see Figure3.4), in which case we can taket₁=Q

±(r−(rb0,±±εM)) andt₂=1−τ. On a manifold with corners, such as Ω, one can consider supported and extendible distributions; see [81, Appendix B.2] for the smooth boundary setting, with simple changes needed only for the corners setting, which is discussed e.g. in [138,§3]. Here we consider Ω as a domain inM, and thus its boundary faceX∩Ω is regarded as having a different character from theH_j∩Ω, H_j=t⁻¹_j (0), i.e. the support/extendibility consider-ations do not arise atX—all distributions are regarded as acting on a subspace ofC^∞ functions on Ω vanishing atXto infinite order, i.e. they are automatically extendible dis-tributions atX. On the other hand, at theH_j we consider both extendible distributions, acting onC^∞ functions vanishing to infinite order at H_j, and supported distributions, which act on allC^∞functions (as far as conditions atH_j are concerned). For example, the space of supported distributions atH₁ extendible at H₂ (and at X, as we always tacitly assume) is the dual space of the subspace ofC^∞(Ω) consisting of functions van-ishing to infinite order at H₂ and X (but not necessarily at H₁). An equivalent way

of characterizing this space of distributions is that they are restrictions of elements of the dual C^−∞(M) of ˙C_c^∞(M) with support in t1>0 toC^∞ functions on Ω which vanish to infinite order at X and H2; thus, in the terminology of [81], they are restrictions of elements ofC^−∞(M) with support int1>0 to Ω\(H2∪X).

The main interest is in spaces induced by the Sobolev spacesH_b^s,r(M). Notice that the Sobolev norm is of completely different nature at X than at the Hj, namely the derivatives are based on complete, rather than incomplete, vector fields: Vb(M) is being restricted to Ω, so one obtains vector fields tangent to X but not to the Hj. As for supported and extendible distributions corresponding toH_b^s,r(M), we have, for instance,

H_b^s,r(Ω)^,−,

with the first superscript on the right denoting whether supported () or extendible (−) distributions are discussed at H1, and the second the analogous property at H2; thus,H_b^s,r(Ω)^,− consists of restrictions of elements ofH_b^s,r(M) with support int1>0 to Ω\(H2∪X). Then, elements of C^∞(Ω) with the analogous vanishing conditions, so in the example vanishing to infinite order at H1 and X, are dense in H_b^s,r(Ω)^,−; further the dual ofH_b^s,r(Ω)^,−isH_b^−s,−r(Ω)^−,with respect to theL²(sesquilinear) pairing. For distributions extendible (resp. supported) at all boundary hypersurface, we shall write

H_b^s,r(Ω)≡H_b^s,r(Ω)^−,− and H˙_b^s,r(Ω)≡H_b^s,r(Ω)^,. (A.15) The main use of these spaces for the wave equation is that, due to energy estimates, one can obtain a Fredholm theory using these spaces, with the supported distributions corresponding to vanishing Cauchy data (from where one propagates estimates in the complex absorption setting discussed above), while extendible distributions correspond to no control of Cauchy data (corresponding to the final spacelike hypersurfaces, i.e.

with future timelike outward-pointing normal vector, to which one propagates estimates);

note that dualization reverses these, i.e. one starts propagating forP^∗ from the spacelike hypersurfaces towards which one propagated for P. We refer to [75, §2.1] for further details.