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LECTURES ON GEOMETRIC SINGULAR ANALYSIS, WITH
APPLICATIONS TO ELLIPTIC AND HYPERBOLIC PDE
PETER HINTZ
Abstract. These lectures provide an introduction to geometric singular analysis, a sys- tematic framework for the study of partial differential equations (PDE) on noncompact or singular spaces, and of families of PDE depending on a parameter which become singular as the parameter tends to 0. We illustrate this framework in a number of examples rang- ing from the Laplace operator on Euclidean space, possibly with a singular potential or with a family of potentials which become singular at a point, and the Dirichlet problem on polygonal domains to hyperbolic problems such as the wave equation in the interior of a Schwarzschild black hole or on Minkowski and de Sitter spacetimes. Various problems invite the reader to work out details and further examples.
Contents
- Introduction and motivation......................................... 2 1.1. The Laplacian on R^3................................................. 3 1.2. Singular PDE........................................................ 5 1.3. Singular limits....................................................... 6 1.4. Plan of these lectures................................................ 7 1.5. Further reading...................................................... 7 1.6. Problems............................................................ 8
- Manifolds with boundary............................................ 9 2.1. b-vector fields and b-differential operators............................ 10 2.2. Conormality and asymptotic expansions.............................. 12 2.3. b-Sobolev spaces..................................................... 15 2.4. Problems............................................................ 16
- Applications: I........................................................ 19 3.1. The Laplacian on R^3 revisited........................................ 19 3.2. Waves in the interior of a Schwarzschild black hole................... 21 3.3. The wave equation on de Sitter space................................ 24 3.4. Problems............................................................ 26
- Manifolds with corners and blow-ups............................... 26 4.1. Problems............................................................ 30
- Applications: II....................................................... 31
Date: March 28, 2023. 1
2 PETER HINTZ
5.1. The Laplacian with a singular potential.............................. 31 5.2. The Laplace equation on polygonal domains.......................... 32 5.3. The wave equation on Minkowski space.............................. 35 5.4. Problems............................................................ 39
- Singular limits via geometric singular analysis..................... 40 6.1. q-analysis............................................................ 42 6.2. Problems............................................................ 43
- Applications: III...................................................... 45 7.1. Laplacians with degenerating potentials.............................. 45 7.2. Quasinormal modes of Schwarzschild–de Sitter black holes in the vanishing mass limit................................................................ 47 7.3. Problems............................................................ 50
References................................................................. 50
- Introduction and motivation
The goal of these lectures^1 is to introduce a general-purpose perspective, called^2 geometric singular analysis, for the analysis of partial differential equations (PDE) in the following contexts.
(1) PDE on noncompact spaces such as Rn. A typical example is the Laplacian^3 ∆ = −
∑n j=1 ∂
2 xj^ , where we write^ x^ = (x
(^1) ,... , xn) ∈ Rn. (2) PDE on singular spaces, or PDE with singular (but structured ) coefficients. Exam- ples include the Laplacian with a Coulomb potential ∆− (^) |x^1 | , where we are interested in the behavior near |x| = 0, or the Laplacian on polygonal domains in R^2. (3) Families of PDEs depending on a parameter > 0 which degenerate or become singular as ↘ 0. An example is ∆x + W (x) + −^2 V (x/) on Rnx , where W, V ∈ C∞ c (Rn).
As we shall see, item (1) is really a (very important) special case of item (2) via the process of compactification of the noncompact space. Moreover, in the study of singular family of PDEs, the key work often lies in analyzing individual PDE on suitable singular spaces. We discuss this in the example of item (3) in §7.1.
(^1) The author produced these notes for his lectures at the PDE mini-school at the University of North
Carolina, Chapel Hill, on March 24–26, 2023. Many thanks to the organizers, Yaiza Canzani and Jian Wang, for their kind invitation, and to their NSF RTG grant “Partial Differential Equations on Manifolds” (DMS-2135998) for supporting the mini-school. (^2) The term ‘geometric microlocal analysis’ is also often used, especially in the context of constructing
(approximate) solutions for PDE on singular spaces. In these lectures, we shall not employ any microlocal tools however, as the PDE we shall study are not particularly complicated. (^3) The sign convention used in these notes makes ∆ a non-negative operator, i.e. 〈∆u, u〉L (^2) (Rn) ≥ 0 for all
u ∈ C c∞ (Rn).
4 PETER HINTZ
where ∆S 2 = − (^) sin^1 θ ∂θ sin θ ∂θ −(sin θ)−^2 ∂ φ^2 in standard coordinates θ ∈ (0, π), φ ∈ (0, 2 π) on
the sphere S^2. Thus, ρ−^2 ∆ is constructed from ρ∂ρ and spherical derivatives, and it is an elliptic operator of this type (i.e. its leading order part is a positive definite quadratic form in ρ∂ρ, ∂θ, ∂φ). We say that ∆ is a (weighted) b-differential operator. A natural notion of regularity for solutions of ∆u = f is thus regularity under precisely these derivatives (called b-regularity). One can check that this is the same as regularity under the vector fields |x|∂xj.
We are thus led to define weighted b-Sobolev spaces H bs,α for s ∈ N 0 and α ∈ R by
H bs,α =
u : (〈x〉∂x)β^ (〈x〉αu) ∈ L^2 ∀ β ∈ Nn 0 , |β| ≤ s
(Usage of 〈x〉∂x ensures that we are recovering the standard Sobolev spaces over compact subsets of R^3 , in the sense that if u ∈ H bs,α , then χu ∈ Hs(R^3 ) for all χ ∈ C c∞ (R^3 ).) This is a Hilbert space with squared norm
‖u‖^2 H bs,α =
|β|≤s
(〈x〉∂x)β^ (〈x〉αu)
L^2.
We have 〈x〉^2 ∆ : H bs,α → H bs− 2 ,αfor s ≥ 2, α ∈ R. (See Problem 1.5.) We might then hope that ∆ : H bs,α → Hbs−^2 ,α+
is Fredholm. This is almost true: the weight α needs to be chosen carefully (namely, it must avoid the discrete set 12 + Z). One can moreover show that for α ∈ (− 32 , − 12 ), this operator is invertible.
One can see why a condition on α is necessary by considering the action of ρ−^2 ∆ on separated functions of the form ρλv(ω), which is
ρ−^2 ∆(ρλv) = ρλN (ρ−^2 ∆, λ)v, N (ρ−^2 ∆, λ) := −λ^2 + λ + ∆S 2 ∈ Diff^2 (S^2 ). (1.4)
Restricting v further to be a spherical harmonic Ylm(ω) of degree l ∈ N 0 , N (ρ−^2 ∆, λ) becomes multiplication by −λ^2 + λ + l(l + 1) = −(λ + l)(λ − l − 1). This vanishes for λ = −l, l + 1, corresponding to the fact that u(ρ, ω) := ρλYlm(ω) ∈ ker ρ−^2 ∆. This function should be regarded as relevant only near ρ = 0; let thus χ ∈ C∞ c ([0, 1)ρ) be identically 1
near 0. For α = − 32 + λ, the function χ(ρ)u(ρ, ω) just barely fails to lie in H^2 b ,α, and then
u = χ(ρ)ρu(ρ, ω) blows up in H b^2 ,αas ↘ 0, while ∆u remains bounded in Hb^0 ,α +2. (See Problem 1.5.)
The computation (1.4) also tells us what to expect about the asymptotic behavior of solutions of ∆u = 0 (or ∆u = f where f ∈ S (R^3 ), say): they have asymptotic expansions as |x| → ∞, or equivalently ρ ↘ 0, into terms of the form ρλYlm(ω) where λ = −l, l + 1. Note, finally, that the operator family N (ρ−^2 ∆, λ), insofar as it controls the asymptotic behavior of solutions of ∆u = f at infinity, should be regarded as living ‘at (the sphere at) infinity’ in R^3 , i.e. at ‘ρ = 0’. Thus, we wish to add to R^3 this sphere at infinity; this is accomplished via the radial compactification R^3 , defined more generally for Rn^ as
Rn^ :=
Rn^ t
[0, ∞)ρ × Sn−^1
Rn^ \ { 0 } 3 x = rω ∼ (r−^1 , ω).
This is a compact manifold with boundary ρ−^1 (0); it is diffeomorphic to the closed unit ball. See Figure 1.1.
GEOMETRIC SINGULAR ANALYSIS 5
ρ = r−^1
Rn ∂Rn
Figure 1.1. The radial compactification R^3 of R^3 , obtained by attaching a sphere S^2 = ρ−^1 (0) ‘at infinity’ to R^3 , where ρ = r−^1.
Theme I.1. Infinity is closer than you might think. Working ‘locally near infinity’ (ρ = 0) in a systematic fashion allows one to see the structure of the PDE clearly, understand its mapping properties, and read off the expected asymptotic behavior of solutions.
Theme I.2. Infinity is a singular place: PDEs typically degenerate in some (controlled) fashion there. (Cf. the appearance of ρ∂ρ, not ∂ρ, in (1.2).)
As hyperbolic examples, we shall discuss the wave equation near a Schwarzschild singu- larity in §3.2 and the wave equation on de Sitter space in §3.3.
1.2. Singular PDE. We already encountered a singular PDE that did not look like one at first sight, namely the Laplace operator on R^3 which is singular at infinity ρ = 0, cf. (1.2). Other types of singularities are less hidden. One example is the following operator near the origin of R^3 (with Z ∈ R):
P := ∆ −
Z
|x|
= −∂ r^2 −
r
∂r + r−^2 ∆S 2 −
r
= r−^2
−(r∂r)^2 − r∂r + ∆S 2 − Zr
If we regard this as an operator on (0, ∞)r × S^2 , this looks quite similar to (1.2), except for the presence of the term r in parenthesis. If one seeks approximate elements in the nullspace of this operator of the form rλv(ω), one is led to the operator
N (r^2 P, λ) = −λ^2 − λ + ∆S 2 ∈ Diff^2 (S^2 )
which is the same as (1.4) except for a sign change in λ. This operator, on the 2-sphere, wants to live at r = 0. To make this happen, one regards polar coordinates as valid down to r = 0, and thus considers P as a (weighted) b-differential operator on
[R^3 ; { 0 }] := [0, ∞)r × S^2.
This manifold is called the blow-up of R^3 at { 0 }. The smooth map [R^3 ; { 0 }] 3 (r, ω) 7 → rω ∈ R^3 is a diffeomorphism on {r > 0 }, but the preimage of { 0 } is a 2-sphere. This perspective has the following benefits.
(1) The operator r^2 P , regarded as a second order b-differential operator (constructed from r∂r and angular derivatives) on [R^3 ; { 0 }], has smooth coefficients. Thus, we have exchanged analytic complexity (singular potential) on a simple space (R^3 ) for analytic simplicity (smooth coefficients) on a more complicated space ([R^3 ; { 0 }]).
GEOMETRIC SINGULAR ANALYSIS 7
ideas rigorous is to consider P as a single differential operator on [0, 1) × R^3 x and blow up = x = 0. See Figure 1.2 (albeit in two dimension less, for artistic reasons). In this manner, the limit of P as ↘ 0 is described by one operator ( Pˆ ) on a compactification of R^3 ˆx and another operator (P 0 ) on [R^3 x; { 0 }].
ˆx = x
x
Figure 1.2. Blow-up of [0, 1) × Rx at = x = 0, and some local coordinates.
For example then, if both Pˆ and P 0 are invertible, on function spaces which are related in an appropriate manner at the corner at the intersection of the two limiting regimes, then one can hope to prove the invertibility of P for small > 0. We will carry this out in detail in §7.1.
Theme III. Analyze families of PDE which depend on a parameter in a singular manner by studying all operators at once on a total space which incorporates both the underlying manifold and the parameter, and which is resolved (blown up) so as to exhibit all different asymptotic regimes.
As an example from general relativity, we sketch in §7.2, following [HX22], how to study the quasinormal mode spectrum of Schwarzschild–de Sitter black holes in the limit that the black hole mass is small.
1.4. Plan of these lectures. In §2, we describe the notions of b-geometry and b-analysis on general manifolds with boundary, including b-vector fields and b-differential operators and their normal operators and indicial families, b-Sobolev spaces, and asymptotic expan- sions (‘polyhomogeneity’). We discuss applications of this theory in §3.
The general procedure of blow-ups of suitable submanifolds of manifolds with corners is discussed in §4. We apply it to a variety of settings (mentioned in §1.2 above) in §5.
In §6 finally, we describe, following the motivation in §1.3, how to study (a certain class of) singular limits of families of PDE using techniques from geometric singular analysis, with examples gives in §7.
1.5. Further reading. Grieser’s notes on the b-calculus [Gri01] provide a detailed in- troduction to and motivation for many notions discussed here, including manifolds with corners, blow-ups, conormality, and polyhomogeneity. The focus of later parts of those notes is on distributions and pseudodifferential operators on manifolds with boundary. (By contrast, in these notes, we do not use any microlocal techniques.) For a textbook account of (microlocal) b-analysis, with applications to index theory, see Melrose’s book [Mel93]; see also [H¨or07, §18.3]. A systematic, extensive, and general treatment of analysis on manifolds with corners is Melrose’s book (in progress) [Mel96].
8 PETER HINTZ
On average, applications of geometric singular analysis techniques in the literature are skewed towards elliptic theory (spectral theory, Fredholm analysis, index theory) and re- lated topics (such as heat kernel asymptotics) on noncompact and/or singular spaces, often via the construction of parametrices (approximate inverses) and their integral (‘Schwartz’) kernels. Important examples concern the spectral theory on asymptotically hyperbolic manifolds [MM87] and low energy resolvent asymptotics on asymptotically conic spaces [GH08]. Spectral theory on asymptotically Euclidean spaces has a non-elliptic character when viewed through the lens of singular (‘scattering’) analysis on the radial compactifica- tion [Mel94]. Vasy’s lecture notes [Vas18] provide a detailed account of microlocal analysis in this setting.
Early non-elliptic applications include studies of local interactions of singularities for nonlinear wave equations [MR85]. Applications to the global theory of (non)linear wave have only emerged more recently [Vas10, MSBV14, Vas13, HV15], with applications to nonlinear stability problems in general relativity [HV18, Hin18]. In these lectures, we briefly describe two such examples using only ‘hands-on’^5 techniques on de Sitter space in §3.3 (following [Vas10, HX22]) and Minkowski space in §5.3 (following [HV20, Hin23]). We also re-interpret some results in the literature in geometric singular analysis terms, e.g. results by Fournodavlos–Sbierski [FS20] on the behavior of waves near the Schwarzschild singularity in §3.2.
Geometric singular analysis has featured implicitly in gluing constructions and singular perturbation theory; for explicit examples (and references to earlier gluing results), we refer the reader to [KS22, SS21, HX22, Hin21, Hin22]. We briefly sketch one such example in §7.2.
1.6. Problems.
Problem 1.1 (Utility of closed range). Let A : X → Y be an operator between two Banach spaces (X, ‖ · ‖X ), (Y, ‖ · ‖Y ). Show that A has closed range if and only if there exists a constant C < ∞ so that for all y ∈ ran A there exists x ∈ X with Ax = y and ‖x‖X ≤ C‖y‖Y.
Problem 1.2 (Derivative on R). Show that (^) ddx : H^1 (R) → L^2 (R) has trivial nullspace and dense but not closed range. Make the final statement explicit by finding a sequence fj ∈ L^2 (R) in the range so that fj → f ∈ L^2 (R) does not lie in the range.
Problem 1.3 (Laplacian on Rn). Let n ≥ 1. Show that ∆ : H^2 (Rn) → L^2 (Rn) has trivial nullspace and dense but not closed range. Show that for all s, α ∈ R,
∆ : Hs,α(Rn) = {u ∈ S ′(Rn) : 〈x〉αu ∈ Hs(Rn)} → Hs−^2 ,α(R)
has finite-dimensional kernel, but its range is not closed (but the closure of the range may be positive-dimensional). Here 〈x〉 = (1 + |x|^2 )^1 /^2. Hint. First prove this for α = 0. Deduce the general statement from this by showing that 〈x〉−α∆〈x〉α^ = ∆ + Rα where Rα : Hs^ → Hs−^2 is a compact operator (use the Rellich compactness theorem).
(^5) occasionally termed ‘physical space’, and in any case not microlocal
10 PETER HINTZ
M
H 1 H 2
Figure 2.1. A compact manifold M whose boundary ∂M has two con- nected components H 1 , H 2.
2.1. b-vector fields and b-differential operators. As we saw in (1.2) and (1.6), PDEs on M ◦^ viewed from the perspective of M degenerate at ∂M , and their solutions are typically not smooth, but rather have behavior like xz^ (log x)kv(y) in the best of cases. What are the vector fields with respect to which such functions are regular? The derivatives transversal to ∂M featuring in (1.2) and (1.6) are weaker (x∂x instead of ∂x). We phrase this invariantly:
Definition 2.1 (b-vector fields and b-differential operators). The space Vb(M ) ⊂ V(M ) of b-vector fields consists of all smooth vector fields which are tangent to ∂M. For m ∈ N 0 , the space Diffm b (M ) of b-differential operators consists of all locally finite sums of up to m-fold compositions of elements of Vb(M ).
These are sometimes called totally characteristic vector fields/operators. A b-differential operator defines continuous linear maps C∞(M ) → C∞(M ) and C˙∞(M ) → C˙∞(M ); thus, two operators Pi ∈ Diffm b i(M ), i = 1 , 2, can be composed, and we have P 1 ◦ P 2 ∈
Diffm b 1 +m^2 (M ).
Concretely, Vb(M ) consists of all V ∈ V(M ) which in local coordinates x ≥ 0, y ∈ Rn−^1 near a boundary point are of the form
a(x, y)x∂x +
n∑− 1
j=
bj (x, y)∂yj , a, bj ∈ C∞([0, ∞) × Rn−^1 ),
and b-differential operators are of the form
P =
j+|α|≤m
ajα(x, y)(x∂x)j^ ∂yα , ajα ∈ C∞. (2.2)
Near an interior point, there is no difference between differential and b-differential operators. We then have the notion of ellipticity, which in M ◦^ is the standard one^6 and which near ∂M and in terms of (2.2) requires
j+|α|≤m ajα(x, y)ξ
j (^) ηα (^6) = 0 for all (ξ, η) ∈ (R × Rn− (^1) ) \
{(0, 0)}.
In order to study b-differential operators globally near the boundary ∂M , and also to measure growth and decay at the boundary, we introduce:
Definition 2.2 (Boundary defining functions). A boundary defining function on M is a nonnegative function ρ ∈ C∞(M ) so that ∂M = ρ−^1 (0) and dρ(p) 6 = 0 for all p ∈ ∂M.
Lemma 2.3 (Conjugation by weights). Let P ∈ Diffm b (M ) and α ∈ R. Then ραP ρ−α^ : u 7 → ραP (ρ−αu) defines an element of Diffm b (M ).
(^6) Namely, if P = ∑ |α|≤m aα(x)∂xα , then the polynomial ∑ |α|≤m ξα^ does not vanish for ξ ∈ Rn^ \ { 0 }.
GEOMETRIC SINGULAR ANALYSIS 11
Proof. This is ultimately due to the fact that xαx∂xx−α^ = x∂x − α. See Problem 2.7.
This result allows us to define spaces of weighted b-differential operators,
ρ−αDiffm b (M ) = {ρ−αP : P ∈ Diffm b (M )}.
Operators of this class still map C˙∞(M ) → C˙∞(M ) continuously, but of course they are not continuous anymore as operators on C∞(M ).
Example 2.4 (Regular singular ODEs). On M = [0, ∞), b-differential operators are of the form P =
∑m j=0 aj^ (x)(x^
d dx )
j (^) where aj ∈ C∞([0, ∞)). If am(x) 6 = 0, the equation
P u = f (or P u = 0 with specified initial data) is a regular singular ODE of order m, and the standard method for analyzing and solving it involves the characteristic polynomial N (P, λ) =
∑m j=0 aj^ (0)λ
j (^). For example, a root λ of N (P, λ) gives rise to xλ (^) asymptotics of
u.
Example 2.5 (Laplace operator). The function^7 〈x〉−^1 on Rn^ is a boundary defining function on Rn, and ∆ ∈ 〈x〉−^2 Diff^2 b(Rn). Indeed, for bounded x this says nothing more than that ∆ is a second order differential operator with smooth coefficients, whereas near ∂Rn^ it follows from (1.2). The analogue of the characteristic polynomial of 〈x〉^2 ∆ at ρ = r−^1 = 0 is N (〈x〉^2 ∆, λ) = −λ^2 + λ + ∆S 2 from (1.4). Moreover, ∆ ∈ 〈x〉−^2 Diff^2 b(Rn) is elliptic (in the sense that 〈x〉^2 ∆ ∈ Diff^2 b is elliptic).
We proceed to define the ‘characteristic polynomial’ in general. First, recall that ∂M has a collar neighborhood, i.e. an open neighborhood of ∂M is diffeomorphic to
[0, ∞)ρ × ∂M.
In such a collar neighborhood, one can write P ∈ Diffm b (M ) as
P =
∑^ m
j=
Pj (ρ)(ρ∂ρ)j^ , Pj ∈ C∞
[0, ∞)ρ; Diffm−j^ (∂M )
Freezing coefficients at ρ = 0 gives the normal operator 8
N (P ) :=
∑^ m
j=
Pj (0)(ρ∂ρ)j^ ∈ Diffm b ([0, ∞) × ∂M ). (2.3)
Unlike P , which does not need to have any symmetries, the operator N (P ) is invariant under dilations in ρ. Just as the principal symbol of P captures P modulo operators of lower differential order, i.e. modulo Diffmb −^1 (i.e. the principal symbol of P ∈ Diffm b (M ) vanishes if and only if P ∈ Diffmb −^1 (M )), the normal operator captures P modulo operators with additional decay at ∂M : if P ∈ Diffm b (M ), then N (P ) = 0 if and only if P ∈ ρDiffm b (M ). It is useful to express this more quantitatively as follows: if χ ∈ C∞ c ([0, ∞) × ∂M ) is equal to 1 near { 0 } × ∂M , then χN (P )χ is a well-defined operator on M , and
P − χN (P )χ ∈ ρDiffm b (M ). (2.4)
(^7) or more precisely its continuous extension to Rn, which is then a smooth function on Rn (^8) One can define this in a manner that is independent of the choice of collar neighborhood, namely as a
dilation-invariant m-th order b-differential operator on the inward pointing normal bundle +N ∂M.
GEOMETRIC SINGULAR ANALYSIS 13
smooth functions u on M ◦^ for which, in a collar neighborhood [0, ∞)ρ × ∂M , there exist u(z,k) ∈ C∞(∂M ), (z, k) ∈ E, so that
u(ρ, y) ∼
(z,k)∈E
ρz^ (log ρ)ku(z,k)(y), ρ ↘ 0.
The notation ‘∼’ means the following: if χ ∈ C∞ c ([0, ∞) × ∂M ) is equal to 1 near { 0 } × ∂M , then u −
(z,k)∈E Re z≤C
χ(ρ, y)ρz^ (log ρ)ku(z,k)(y) ∈ AC^ (M ) ∀ C ∈ R. (2.7)
We say that u is polyhomogeneous with index set E, or E-smooth.
The requirements on index sets guarantee that all sums in (2.7) are finite and the space AE phg(M ) is independent of the choice of collar neighborhood. As a special case, we have
AE phg(M ) = C∞(M ) for E = N 0 × { 0 }, and more generally AE phg(M ) = ραC∞(M ) for E = (α + N 0 ) × { 0 }.
In Problem 2.13 you are tasked to show that solutions of regular singular ODEs P u = 0 (see Example 2.4) are polyhomogeneous. Here, we prove a slightly weaker result, namely that solutions which are conormal are necessarily polyhomogeneous. (This re-proves Prob- lem 2.13 once one shows that solutions of P u = 0 are necessarily conormal; see Prob- lem 2.14.) The key tool is the Mellin transform:
Definition 2.10 (Mellin transform). The Mellin transform of a function u on (0, ∞) is defined by
(Mu)(λ) =
0
x−λu(x)
dx x
The inverse Mellin transform of a function v = v(λ) is, for a choice of α ∈ R, defined by
(M− α 1 v)(x) :=
2 π
∫ (^) α+i∞
α−i∞
xλv(λ) dλ.
When u is a function on (0, ∞) × Y where Y is a smooth manifold (possibly with boundary or corners), we define its Mellin transform (Mu)(λ, y) parametrically in y ∈ Y.
Via the coordinate change x = e−t, λ = −iσ, we have (Mu)(λ) =
−∞ e
−iσtu(e−t) dt,
which is the Fourier transform of t 7 → u(e−t). Similarly, M− α 1 v is the same, in logarithmic coordinates, as the inverse Fourier transform, with integration contour Im σ = α. For example then, if u ∈ C c∞ ((0, ∞)), then (Mu)(λ) is well-defined for all λ ∈ C, and it is indeed holomorphic. When u ∈ L^2 ((0, ∞), dxx ), then (Mu)(λ) is well-defined as an L^2 - function of λ ∈ R.
That the Mellin transform is a good tool for b-analysis is evidenced by the fact that
M
x
d dx
u
(λ) = λ(Mu)(λ). (2.8)
This implies that N (P, λ) is the conjugation of N (P ) by the Mellin transform.
Lemma 2.11 (Mellin transform characterization of conormality and polyhomogeneity). Let C > 0 , and let u : (0, ∞) → C be a function with u(x) = 0 for x ≥ C and |u(x)| ≤ Cxβ for some C, β.^11
(^11) One can significantly weaken these assumptions to u an element of the dual space of C˙∞([0, ∞)).
14 PETER HINTZ
(1) We have u ∈
> 0 A
α−([0, ∞)) if and only if (Mu)(λ) is holomorphic in Re λ < α and satisfies |(Mu)(λ)| ≤ CN (1 + | Im λ|)−N^ for all N. (2) Let E ⊂ C × N 0 be an index set. Then u ∈ AE phg([0, ∞)) if and only if (Mu)(λ) extends from Re λ − 1 to a meromorphic function in λ ∈ C so that {(z, k) ∈ C × N 0 : (Mu)(λ) has a pole of order k + 1 at λ = z} ⊂ E,
and so that for all C there exists C′^ so that |(Mu)(λ)| ≤ CN | Im λ|−N^ for all λ ∈ C with | Re λ| < C and | Im λ| > C′.
Proof. The a priori assumptions on u ensure that u = M− γ 1 (Mu) for all γ < β, as follows
from the Fourier inversion formula applied to x−γ^ u ∈ L^2 ([0, ∞), dxx ). We only prove one direction for part (2). Namely, if (Mu)(λ) has the stated property, then in
u(x) =
2 π
∫ (^) γ+i∞
γ−i∞
xλ(Mu)(λ) dλ
we can shift the contour of integration from γ + iR to γ′^ + iR for any γ′^ > γ so that (Mu)(λ) has no poles on γ′^ + iR. The poles z of order k + 1 of (Mu)(λ) with γ < Re λ < γ′ produce terms ρz^ (log ρ)j^ , j ≤ k, by the residue theorem. The integral over the final contour lies in Aγ^ ([0, ∞)). (The conormal regularity follows from (2.8) and the rapid decay as | Im λ| → ∞.)
Proposition 2.12 (Conormal implies polyhomogeneous: regular singular ODE setting). Let P =
∑m j=0 pj^ (x)(x^
d dx )
j (^) , with pj ∈ C∞([0, ∞)) and pm(0) 6 = 0, be a regular singular
ordinary differential operator. Suppose that u ∈ Aα([0, ∞)) solves P u = 0. Then u is poly- homogeneous, with an index set depending only on the set of λ ∈ specb(P ) (with multiplicity, i.e. really on (λ, k) ∈ Specb(P )) with Re λ ≥ α.^12
Proof. Let χ ∈ C∞ c ([0, 1)) be equal to 1 near 0. Write the equation P u = 0 as
N (P )(χu) = f := [N (P ), χ]u − χ(P − N (P ))u. (2.9)
The first term on the right lies in C c∞ ((0, 1)); the second term (by (2.4)) lies in Aα+1([0, ∞)). Thus f ∈ Aα+1([0, ∞)), with supp f ⊂ [0, 1) compact. We now pass to the Mellin transform,
N (P, λ)(M(χu))(λ) = (Mf )(λ), Re λ < α.
Therefore, we have
M(χu)(λ) = N (P, λ)−^1 (Mf )(λ). (2.10)
The right hand side is meromorphic in the larger domain Re λ < α + 1, with poles at λ ∈ specb(P ). Lemma 2.11 implies that χu is the sum of a polyhomogeneous function (with index set determined by Specb(P )) and a conormal function in
> 0 A
α+1−([0, ∞)).
Plugging this improved information into (2.9) shows that f is the sum of a polyhomo- geneous function and a function in
> 0 A
α+2−, and thus so is χu in view of (2.10). We
can iterate this argument any finite number of times, and thus deduce that u is polyhomo- geneous (cf. the characterization (2.7)).
(^12) More generally, this holds when P u = f ∈ C˙∞([0, ∞)). More generally still, if f ∈ AF phg([0, ∞)) is itself
polyhomogeneous, then so is u, and the index set of u takes into account both the boundary spectrum of P and the index set of F. Careful inspection of the proof produces upper bounds on the index set of u.
16 PETER HINTZ
Proposition 2.15 (Compact inclusions). Let s, s′^ ∈ N 0 and α, α′^ ∈ R. Suppose that s > s′ and α > α′. Then the inclusion map
H bs,α (M ) → Hs
′,α′ b (M^ )
is compact.
Example 2.16 (b-Sobolev spaces on the radial compactification). If M = Rn, then μ = |dx| = rn−^1 |dr dgSn− 1 |; near ρ := r−^1 = 0, this is ρ−n+1| dρρ 2 dgSn− 1 |. Thus, μ is of the form
required in Definition 2.13 with w = −n. In the case n = 3, Proposition 2.14 gives (in the notation of (1.3)) for all β ∈ R and > 0 the inclusions
H∞,−^
3 2 +β b ⊂ A
β (^) (R (^3) ) ⊂ H∞,−^32 +β− b
Finally, we characterize b-Sobolev spaces on the ‘model space’ [0, ∞) × ∂M (where the normal operator of a b-differential operator lives as a dilation-invariant operator, cf. the expression (2.3)) using the Mellin transform:
Lemma 2.17 (Weighted b-Sobolev spaces and the Mellin transform). Let ν 0 denote a pos- itive density on ∂M , and define Hs b([0, ∞) × ∂M ; xw| dxx dν 0 |) to consist of all distributions
u so that (x∂x)j^ P u ∈ L^2 ((0, ∞) × ∂M ; xw| dxx dν 0 |) for all j ≤ s and P ∈ Diffs−j^ (∂M ). Then the Mellin transform is an isomorphism
M : xαH bs
[0, ∞) × ∂M ; xw
∣∣ dx x
dν 0
v ∈ L^2
Re λ = α −
w 2
; Hs(∂M )
〈λ〉j^ v ∈ L^2
Re λ = α −
w 2
; Hs−j^ (∂M )
∀ j ≤ s
Proof. One can easily reduce this to the case α = w = 0. For s = 0, this is then Plancherel’s theorem in logarithmic coordinates. For s ≥ 1, one uses the intertwining property (2.8).
2.4. Problems.
Problem 2.1 (Smoothness on the radial compactification). Show that C∞(Rn) consists of all smooth functions on Rn^ which have an asymptotic expansion (2.1).
Problem 2.2 (Projective coordinates on the radial compactification). Let n ≥ 1. For j = 1,... , n, define on
U˜ (^) (±j) =
x = (x^1 ,... , xn) ∈ Rn^ : ±xj^ >
max k 6 =j
|xk|
⊂ Rn
the functions ρ(j) = (^) |x^1 j (^) | and ˆxk (j) = x
k |xj^ | ,^ k^6 =^ j; set ˆx(j)^ = (ˆx
k (j))k^6 =j^ ∈^ R
n− (^1). (Thus, we have
ρ(j) ∈ (0, ∞) and ˆxk (j) ∈ BRn− 1 (0, 2) on U˜ (^) (±j).) Set
U (^) (±j) = [0, ∞) × BRn− 1 (0, 2),
and show that the map (0, ∞) × BRn− 1 (0, 2) 3 (ρ(j), xˆ(j)) 7 → x ∈ U˜ (^) (±j) extends to a smooth
map U (^) (±j) → Rn^ which is a diffeomorphism onto its image C (±j) ⊂ Rn. (In other words,
GEOMETRIC SINGULAR ANALYSIS 17
(ρ(j), xˆ(j)) defines a smooth coordinate system on C(j).) Show that
Rn^ = Rn^ ∪
⋃^ n
j=
±
C (±j).
Problem 2.3 (Linear maps on Rn). Let n ≥ 1, and let A ∈ GL(n, R) be an invertible linear map on Rn. Show that A extends, by continuity from the interior, to a diffeomorphism Rn^ → Rn. Conclude that the radial compactification of an n-dimensional real vector space is well-defined.
Problem 2.4 (Radial compactifications of vector bundles). Let E → M denote a smooth real vector bundle of rank n ∈ N over the smooth manifold M. Show that its fiberwise radial compactification (or ‘fiber-radial compactification’) E¯ =
p∈M Ep^ →^ M^ can be given a structure of a smooth fiber bundle which is uniquely determined by the requirement that a trivialization U ×Rn^ of E over an open set U ⊂ M extends by continuity from the interior of Rn^ ⊂ Rn^ to a smooth trivialization U × Rn^ of E¯ over U.
Problem 2.5 (Diffeomorphisms on manifolds with boundary). The goal of this problem is to show that b-vector fields are in essence the generators of families of diffeomorphisms of manifolds with boundary.
(1) Suppose (− 1 , 1) 3 s 7 → φs is a smooth family of diffeomorphisms of a manifold M with boundary, with φ 0 = Id. For p ∈ M , set V (p) := (^) dds φs(p)|s=0. Show that V ∈ Vb(M ). (2) Conversely, if M is compact and V ∈ Vb(M ), show that the time s flow of V defines a smooth (1-parameter) family of diffeomorphisms of M.
Problem 2.6 (Lie algebra). Show that Vb(M ) is a Lie algebra, where the Lie bracket is the vector field commutator. That is, show that [V, W ] ∈ Vb(M ) whenever V, W ∈ Vb(M ).
Problem 2.7 (Conjugation by weights). Prove Lemma 2.3.
Problem 2.8 (*-algebra of weighted b-differential operators). Let M be a manifold with boundary, and let ρ denote a boundary defining function.
(1) Let mi ∈ N 0 , αi ∈ R, and Pi ∈ ρ−αi^ Diffm b i(M ). Show that P 1 ◦ P 2 ∈ ρ−α^1 −α^2 Diffm b 1 +m^2 (M ). (2) Suppose M is equipped with a smooth positive density μ 0. If P ∈ ρ−αDiffm b (M ), show that P ∗^ ∈ ρ−αDiffm b (M ). Show that this remains true for densities μ = ρwμ 0 for any w ∈ R. Make this concrete in the special case M = Rn, μ = |dx|, and P = ∆.
Problem 2.9 (Ellipticity). Suppose P ∈ Diffm b (M ) is elliptic. Show that N (P, λ) is elliptic for all λ ∈ C, and its principal symbol is independent of λ.
Problem 2.10 (Boundary spectrum). Let n ≥ 1, and write ∆ = −
∑n j=1 ∂
2 xj^ ∈^ Diff
(^2) (Rn)
for the Laplacian.
(1) Compute specb(〈x〉^2 ∆). (2) Let V ∈ 〈x〉−^2 C∞(Rn) (i.e. V = 〈x〉−^2 V 0 where V 0 ∈ C∞(Rn)). Suppose that C := (|x|^2 V )|∂Rn is constant. (Thus, for |x| > 1 we have V (x) = (^) |xC| 2 + O(|x|−^3 ).) Compute specb(〈x〉^2 (∆ + V )).
GEOMETRIC SINGULAR ANALYSIS 19
- Applications: I
3.1. The Laplacian on R^3 revisited. We consider only the case of n = 3 dimensions here, and leave the general case to the problems. We already saw that
∑^3
j=
−∂^2 xj ∈ 〈x〉−^2 Diff^2 b(R^3 ),
and we computed its boundary spectrum to be specb(〈x〉^2 ∆) = Z.
Theorem 3.1 (Laplacian on weighted b-Sobolev spaces). Let s ≥ 3 and α /∈ 12 + Z. Fix
on R^3 the density |dx|. Then ∆ : H bs,α (R^3 ) → Hsb− 2 ,α+2(R^3 ) is Fredholm. For α > − 32 , it
is injective, and for α < − 12 it is surjective; in particular, it is invertible for α ∈ (− 32 , − 12 ).
Remark 3.2 (Generalization). The Fredholm statement holds also for every elliptic b-
operator which differs from ∆ by an element of 〈x〉−^3 Diff^2 b(R^3 ). (Adding to ∆ an element of 〈x〉−^3 Diff^2 b changes P by an element in 〈x〉−^1 Diff^2 b(R^3 ), and in particular leaves the normal operator of P unchanged.) This includes the Laplace operator on certain asymptotically Euclidean spaces, and also allows for the addition of terms a∂xj and V where a ∈ 〈x〉−^2 C∞ and V ∈ 〈x〉−^3 C∞.
As far the literature is concerned, a more general result was proved by Melrose in [Mel93, §5.17] (see also Theorem 3.3 below). The application to the Laplace operator appears to have been first given in Carron–Coulhon–Hassell in [CCH06, Lemma 3.2].
Proof of Theorem 3.1. We only consider P = 〈x〉^2 ∆ ∈ Diff^2 b(R^3 ).
- Step 1. Elliptic estimate. Near a point in ∂R^3 , we can pull back P via a logarithmic coordinate change to a uniformly elliptic operator on R × R^2 : schematically, P = −(ρ∂ρ)^2 − ∂ y^2 where ρ = r−^1 and y ∈ R^2 denotes local coordinates on S^2 , so with t = − log ρ this becomes P = −∂ t^2 − ∂^2 y. We can thus use (uniform) elliptic estimates on standard Sobolev
spaces on R × R^2 and pull them back to estimates on b-Sobolev spaces on R^3 to get the a priori estimate^15
‖u‖Hs,α b (R^3 )^
≤ C
‖P u‖Hs− 2 ,α b (R^3 )^
Since the inclusion H bs,α ↪→ H^0 b ,αis not compact, this estimate does not tell us much yet. (It neither implies the finite-dimensionality of ker P nor the closed range property.)
- Step 2. Inversion of the indicial family; semi-Fredholm estimate. With ρ = r−^1 , let now χ ∈ C c∞ ([0, ∞)ρ × S^2 ) be identically 1 near ρ = 0. Then
‖u‖H 0 ,α b
≤ ‖χu‖H 0 ,α b
for any N , since supp(1 − χ) is disjoint from ∂R^3. The norm on χu here is, by Lemma 2. (with s = 0)
‖χu‖ραL (^2) ([0,∞)×S (^2) ;ρ− (^3) | dρ ρ dgS^2 |)^
≤ C‖M(χu)‖L (^2) ({Re λ=α+ 3 2 };L^2 (S^2 ))
(^15) The weight α in this estimate is arbitrary. When α = − 3 2 , then this indeed follows from estimates on unweighted Sobolev spaces on R × R^2. One can reduce the case of general α to this special case by working with 〈x〉β^ P 〈x〉−β^ instead of P for appropriate β.
20 PETER HINTZ
But since α ∈/ 12 + Z, we have α + 32 ∈/ specb(P ) = {−l − 1 , l : l ∈ N 0 }. Therefore, N (P, λ) = −λ^2 + λ + ∆S 2 : H^2 (S^2 ) → L^2 (S^2 ) is invertible for Re λ = α + 32.^16 Moreover, there exists a constant C so that for v ∈ L^2 (S^2 ) we have^17
‖v‖L 2 ≤ C‖N (P, λ)v‖L 2 , Re λ = α +
Indeed, for any bounded set of λ this follows (with the L^2 -norm on the left replaced by the H^2 -norm) from the invertibility of N (P, λ). For λ = (α + 32 ) + is, s ∈ R, we have
−λ^2 + λ = s^2 + O(|s|),
and therefore N (P, λ) = ∆S 2 + 1 + c(α, s) where Re c(α, s) ≥ 0 when |s| = | Im λ| is sufficiently large, and the estimate (3.3) then follows (with C = 1 for such λ) from 〈N (P, λ)v, v〉L 2 ≥ ‖v‖^2 L 2.
We can thus estimate ‖M(χu)‖L (^2) ({Re λ=α+ 3 2 };L^2 (S^2 ))^
≤ C‖N (P, ·)M(χu)‖L (^2) ({Re λ=α+ 3 2 };L^2 (S^2 ))
and therefore
‖χu‖H 0 ,α b
≤ C‖N (P )(χu)‖H 0 ,α b
≤ C
‖χP u‖H 0 ,α b
‖χ(P − N (P ))u‖H 0 ,α b
‖[P, χ]u‖H 0 ,α b
Since χ(P − N (P )) ∈ ρDiff^2 b, the second term is bounded by C‖ρu‖H 2 ,α b
= ‖u‖H 2 ,α− 1 b
Moreover, [P, χ] is a first order operator whose coefficients are compactly supported in M ◦, and thus the third term is bounded by CN ‖u‖H 1 ,−N b
for any N. Plugging this, together
with (3.2), into (3.1) gives
‖u‖Hs,α b (R^3 )^
≤ C
‖P u‖Hs− 2 ,α b (R^3 )^
This is the semi-Fredholm estimate one is after in this business: it implies that P : Hs,α b (R^3 ) →
H bs −^2 ,α(R^3 ) has finite-dimensional kernel and closed range (see Problem 3.1).
- Step 3. Fredholm property. The L^2 -orthogonal complement of the range ∆(H bs,α (R^3 )) ⊂
Hbs −^2 ,α+2(R^3 ) is equal to the kernel of the formal adjoint ∆∗^ on Hb− s+2,−α+2(R^3 ).^18 By
elliptic regularity, elements of this kernel lie in Hs
′,−α+ b (R^3 ) for all^ s
′. By what we have
already shown, we deduce that the range of ∆ has finite codimension.
- Step 4. Kernel and cokernel. Suppose α > − 32 and u ∈ H bs,α (R^3 ) ∩ ker ∆. Then
u ∈ Hs
′,α b for all^ s
′ (^) by elliptic regularity, and thus u ∈ Aα+^32 (R (^3) ) by Sobolev embed-
ding (Proposition 2.14). That is, u decays at infinity, and by the maximum principle it
(^16) Here is one way to see this. Since N (P, λ) is elliptic, it is Fredholm as an operator H^2 (S^2 ) → L^2 (S^2 ). Elements of its kernel are automatically smooth; by decomposing an element in ker N (P, λ) into spherical harmonics and studying the action of N (P, λ) on each piece, one concludes that ker N (P, λ) = { 0 }, cf. the discussion after (1.4). Moreover, N (P, λ) is a compact perturbation of the operator ∆S 2 + 1 which is invertible (being symmetric and having trivial kernel); therefore, N (P, λ) has index 0, which due to its injectivity implies its surjectivity. (^17) This estimate is technically correct, but morally wrong: N (P, λ) is elliptic, so one expects Sobolev
spaces to appear on both side. However, one should not use standard Sobolev spaces, but rather ‘large parameter Sobolev spaces’, with Im λ the large parameter; these are the spaces which secretly feature on the right hand side of (2.12). (^18) We have not defined this space appearing here; but using logarithmic coordinates it can be defined in
terms of negative order Sobolev spaces on R × R^2.
