An elliptic theory of indicial weights and applications to non-linear geometry problems
aa r X i v : . [ m a t h . DG ] F e b An elliptic theory of indicial weights and applications tonon-linear geometry problems
Yuanqi Wang ∗ Abstract
Given an elliptic operator P on a non-compact manifold (with proper asymptoticconditions), there is a discrete set of numbers called indicial roots. It’s known that P is Fredholm between weighted Sobolev spaces if and only if the weight is not indicial.We show that an elliptic theory exists even when the weight is indicial. We also discusssome simple applications to Yang-Mills theory and minimal surfaces. The elliptic theories based on weighted Sobolev (Schauder) spaces usually concern adiscrete set of real numbers. If a number is in the set, we say that it is indicial (or is anindicial root). A classical fact says that on a non-compact complete manifold, an ellipticoperator (with proper asymptotic conditions) is Fredholm between weighted Sobolev spacesif and only if the weight is not indicial. For earlier pioneering work, please see [10], [11], and[14]. For more recent work, please see [12].Following elementary ideas, we show that there is an elliptic theory even if the weightis indicial: first, we add polynomial weights { compare (6) to [11, (1.3)] } to refine the space;second, we consider graph norms with respect to the model operator [see (2)].In this note we only consider first and second-order operators modelled on the following. Definition 1.1.
Let Y be a ( n − − dimensional Riemannian manifold without boundary(which does not have to be connected). Let E, F be smooth vector-bundles over Y equippedwith smooth Hermitian metrics. Given arbitrary bundle isomorphisms σ : E → F, σ : E → E , we say that an operator P is TID (translation-invariant and diagonal) if P = σ ( − B P − a ∂∂t + a ∂ ∂t ) σ and the following holds . (1) • a = 0 or 1. a = − a = 0 (always achievable by normalization). • When a = 0, B P is a first-order self-adjoint elliptic differential operator C ∞ ( Y, E ) → C ∞ ( Y, E ). When a = 1, B P is second-order, simple, elliptic, and self-adjoint C ∞ ( Y, E ) → C ∞ ( Y, E ) (see Definition 2.1).
Remark . For any TID operator P , SpecB P is real and discrete. Moreover, there is acomplete eigen-basis of B P . Definition 1.3.
Let P be TID, and ( β, Λ) be a pair of real numbers such that Λ ∈ Spec ( B P ). When P is first-order, we say that ( β, Λ) is P − indicial if β = Λ. When P issecond-order, we say that ( β, Λ) is P − indicial if1. β = a and Λ − β + a β = 0, or if2. Λ ≤ − a and β = a . ∗ Department of Mathematics, Stony Brook University, NY, USA. [email protected].
1n the second case above, we say that ( β, Λ) is P − super indicial. We say that β is P − indicial (super indicial) if there is a Λ ∈ SpecB P such that ( β, Λ) is P − indicial (superindicial). This is consistent with the ” D A ” in [11, page 417], translated to our setting.Let N be a complete Riemannian manifold with finite many cylindrical ends, we considerasymptotically TID operators P : C ∞ ( N, E ) → C ∞ ( N, F ). This class should include mostof the Dirac and Laplace-type operators in geometry. In the setting as Theorem 1.4, P : c W k + m ,p − β,γ,b − ( N, E ) −→ W k,p − β,γ,b ( N, F ) ( (cid:3) | Sobolev,p − β,γ,b ) (see Definition 2 . , . P : b C k + m ,α − β,γ,b − ( N, E ) −→ C k,α − β,γ,b ( N, F ) ( (cid:3) | Schauder − β,γ,b ) for the norms) . (2)are bounded operators. Moreover, when β is not indicial and γ = 0, as subspaces of L loc , c W k + m ,p − β,γ,b − ( N, E ) = W k + m ,p − β ( N, E ) , b C k + m ,α − β,γ,b − ( N, E ) = C k + m ,α − β ( N, E ) ,W k,p − β,γ,b ( N, F ) = W k,p − β ( N, F ) , C k,α − β,γ,b ( N, F ) = C k,α − β ( N, F ) . (3)Thus our theory generalizes the one in [11] { for first and second-order operators, c.f [11,(1.3)] } . Assuming the weights are the same on the ends, our main result states as follows. Theorem 1.4.
Suppose P is a β − AT ID elliptic operator (see Definition 2.8), and β isnot P − super indicial. Then for any ≤ k ≤ k − , α ∈ (0 , , p ≥ , • ( (cid:3) | Sobolev,p − β,γ,b ) is Fredholm if b = 1 − p or β is not P − indicial; • ( (cid:3) | Schauder − β,γ,b ) is Fredholm if b = 1 or β is not P − indicial.Remark . The super-indicial roots are essentially different from the ordinary ones. For-tunately, they don’t exist for first-order operators, and they barely appear on second-orderoperators. For example, any super-indicial root in the setting of Corollary 1.9 must bepositive, but we only need β to be non-positive therein. Remark . Theorem 5.2 and Proposition 4.5 give reasonably general index formulas forfirst-order operators (see Remark 5.3). As a by-product, we prove an obvious identity(Proposition 5.4) on the eta-invariant defined in [3]. It can also be proved by the Fredholmtheory in [3]. However, the author is not able to find Proposition 5.4 in the literature.
Remark . Though the indicial roots do not prevent Fredholmnness, the index still changeswhen β goes across any of them (c.f. [11, Last 5 lines in Page 433]).Our theory still works when the weights are not the same on the ends (see Theorem2.11).Computations indicate that the our local inverses (Theorem 3.2) are different from thoseof Lockhart-McOwen [11, (2.3)] (by Fourier-transform in the t − direction). When k <
0, ourlocal inverses do not work for the W k,p ( C k,α ) theories. Remark . Assuming that P is translation invariant on each end, Theorem 1.4, 2.11 arestill true with ”if” replaced by ”if and only if” (see the proof in the Appendix). However,only assuming β − ATID, when b = 1, we don’t know whether ( (cid:3) | Schauder − β,γ,b ) is not Fredholm.The same doubt applies to the Sobolev theory.By simple conformal changes as in [11, Section 9], Theorem 1.4 is equivalent to a theoryin the conic setting (and hopefully the asymptotic conic setting). Please also see [15] andthe discussion above Lemma 7.2.Under stronger asymptotic conditions than Definition 2.8, we have a theory for super-indicial roots (on second-order operators), and a theory for powers of Laplace-type operatorsi.e. ∆ m , ( m ≥ .2 Simple applications Geometric objects with isolated conic singularities usually converge to their tangent conespolynomially (see [16]). Let r be the distance to the singular point, and t = − log r be thecylindrical coordinate. Our work implies a general phenomenon: the rate of convergence tothe tangent cone is either exponential or not faster than t (or − log r ).We first do minimal sub-manifolds. In the cylindrical setting, we say that a minimalgraph sub-manifold is asymptotic to a cone at a certain rate, if the section ” u ” in (63)converges to 0 at the rate (see Definition 6.1). Corollary 1.9.
Suppose Σ is a n − dimensional closed minimal sub-manifold in S N , n ≥ .Let be the negative number in Definition 3.1 with respect to the L Σ in (63). Then there isa δ depending on Σ with the following property.Suppose b Σ is a (locally defined) embedded minimal sub-manifold in R N +1 with isolatedcone singularity at O . Suppose b Σ is a graph over Cone (Σ) , and in the cylindrical setting,it converges to
Cone (Σ) at least at the rate δ t (see Definition 6.1). Then b Σ converges to Cone (Σ) exponentially at the rate O ( e −| | t ) .Remark . By Definition 2.8 and Remark 2.9, we can not make δ small by scaling.Adam-Simon [1] showed that there are singular minimal sub-manifolds converging to a coneat a rate comparable to ( − log r ) − . This suggests that in general, the assumption on therate in Corollary 1.9 can not be weaken.Similar results hold for Yang-Mills connections as well. Corollary 1.11.
Let n ≥ . Suppose g is a smooth metric on B nO ( R ) and g ( O ) = g E (theEuclidean metric). Suppose A O is a U ( m ) or SO ( m ) Yang-Mills connection on (the unitround) S n − . Let be the negative number in Definition 3.1 with respect to the B in (60).Then there is a δ > depending on A O with the following properties.Suppose A is a smooth Yang-Mills connection on B O ( R ) \ O . In the cylindrical settingas Section 6.1, suppose A converges to Cone ( A O ) at least at the rate δ t (see Definition 6.1). I : Suppose A is in Coulomb gauge relative to A O (with respect to g or the Euclideanmetric). Then A converges to Cone ( A O ) exponentially at the following rate. O ( e −| | t ) when | | < ,O ( e − t ) when | | > ,O ( te −| | t ) when | | = 1 . (4) II : When A O is irreducible, there exists a gauge s such that s ( A ) converges to Cone ( A O ) exponentially as (4).Remark . By (59), when n = 4, the weight 0 is super-indicial unless B is positivelydefinite. Organization of this note: the norms can be found in Section 2. We construct thelocal inverses in Section 3. In Section 4, we study regularity of harmonic sections, and com-plete the proof for Theorem 1.4. We give the index formula (for first-order operators) andstudy the eta-invariant in Section 5. We prove Corollary 1.9, 1.11 in Section 6.
Acknowledgement:
The author would like to thank Professor Simon Donaldson,Thomas Walpuski, and Lorenzo Foscolo for helpful discussions.
Definition 2.1.
In the setting of Definition 1.1, we say that an operator H : C ∞ ( Y, E ) → C ∞ ( Y, E ) is admissible, if there is a linear first-order differential operator H , and sections b Γ k , Γ k ∈ C ∞ ( Y, E ) such that Hξ = H ξ + Σ k k =1 b Γ k Z Y < ξ, Γ k > dV. (5)3e say that a second-order operator B : C ∞ ( Y, E ) → C ∞ ( Y, E ) is simple, if there is asmooth connection A on E and a smooth metric b g on Y (which does not need to be thegiven one), such that B − ∇ ⋆ b g A ∇ A is admissible. Definition 2.2. (Strips) Let kS m = ( m − k, m + k ), and { ξ m } ∞ m =3 be a partition of unityof Cyl subordinate to the cover { S m } ∞ m =3 i.e. ξ m is supported in 2 S m and is ≡ S m . Definition 2.3.
Let
Cyl t denote Y × ( t , ∞ ) , t ≥ .
1, we define the L − β,b [ Cyl t ] − spaceof sections to the underlying bundle by the norm | ξ | L p − β,b ( Cyl t ) , | e − βt t b ξ | L p ( Cyl t ) = ( Z Cyl t | e − βt t b ξ | p ) p . (6)We define | ξ | W k,p − β,b ( Cyl t ) , Σ kj =0 |∇ jA ξ | L p − β,b ( Cyl t ) . For the Schauder theory, we define | ξ | C k,α − β,b ( Cyl t ) , sup m ≥ t +1 m b e − βm | ξ | C k,α ( S m ) .Let ξ k β denote the projection of ξ onto Ker { B P − βId } (for all t ), and ξ ⊥ β = ξ − ξ k β be the perpendicular vector. When P is first-order, we define | σ − ξ | c W k,p,P − β,γ,b − ( Cyl t ) , | ξ ⊥ β | W k,p − β,γ ( Cyl t ) + | ξ k β | W k,p − β,b − ( Cyl t ) + | ∂ξ k β ∂t − βξ k β | W k − ,p − β,b ( Cyl t ) | σ − ξ | W k,p,P − β,γ,b ( Cyl t ) , | ξ ⊥ β | W k,p − β,γ ( Cyl t ) + | ξ k β | W k,p − β,b ( Cyl t ) | σ − ξ | b C k,α,P − β,γ,b − ( Cyl t ) , | ξ ⊥ β | C k,α − β,γ ( Cyl t ) + | ξ k β | C k,α − β,b − ( Cyl t ) + | ∂ξ k β ∂t − βξ k β | C k − ,α − β,b ( Cyl t ) . | σ − ξ | C k,α,P − β,γ,b ( Cyl t ) , | ξ ⊥ β | C k,α − β,γ ( Cyl t ) + | ξ k β | C k,α − β,b ( Cyl t ) . When P is second-order elliptic, let Λ β = β − a β , we define | σ − ξ | c W k,p,P − β,γ,b − ( Cyl t ) , | ξ ⊥ Λ β | W k,p − β,γ ( Cyl t ) + | ξ k Λ β | W k,p − β,b − ( Cyl t ) + | ∂ξ k Λ β ∂t − βξ k Λ β | W k − ,p − β,b ( Cyl t ) + | ∂ ξ k Λ β ∂t − β ξ k Λ β | W k − ,p − β,b ( Cyl t ) | σ − ξ | W k,p,P − β,γ,b ( Cyl t ) , | ξ ⊥ Λ β | W k,p − β,γ ( Cyl t ) + | ξ k Λ β | W k,p − β,b ( Cyl t ) | σ − ξ | b C k,α,P − β,γ,b − ( Cyl t ) , | ξ ⊥ Λ β | C k,α − β,γ ( Cyl t ) + | ξ k Λ β | C k,α − β,b − ( Cyl t ) + | ∂ξ k Λ β ∂t − βξ k Λ β | C k − ,α − β,b ( Cyl t ) + | ∂ ξ k Λ β ∂t − β ξ k Λ β | C k − ,α − β,b ( Cyl t ) . | σ − ξ | C k,α,P − β,γ,b ( Cyl t ) , | ξ ⊥ Λ β | C k,α − β,γ ( Cyl t ) + | ξ k Λ β | C k,α − β,b ( Cyl t ) . Remark . The σ of the adjoint operator L ⋆ in (52) is usually not identity, but it neveraffects the index or kernel.For all first and second-order TID-operators, we abuse notation and denote the corre-sponding operators on the link as B P . We need to solve the equations( ∂∂t − B P ) σ ξ = σ − h, ( ∂ ∂t − a ∂∂t − B P ) σ ξ = σ − h respectively . (7)Let u = ( σ ξ ) e − βt , f = ( σ − h ) e − βt , (7) become ∂u∂t − B P β u = f and ∂ u∂t − ( a − β ) ∂u∂t − B P β u = f respectively , (8)where B P β , (cid:26) B P − βId when P is first-order B P + a β − β when P is second-order elliptic . (9)4or any Λ ∈ Spec ( B P ) (repeated by multiplicity). Let λ [ ∈ Spec ( B P β )] , (cid:26) Λ − β when P is first-order , Λ + a β − β when P is second-order elliptic . (10)Let [ φ Λ , Λ ∈ Spec ( B P )] denote the orthonormal eigen-basis of L [ Y, E ] with respect to B P .Abusing notation, we let φ λ = φ Λ . In terms of the Fourier series u ( f ) = Σ Λ u λ φ Λ (Σ Λ f λ φ Λ ),(8) is equivalent to the ODE’s du λ dt − λu λ = f λ , d u λ dt − ( a − β ) du λ dt − λu λ = f λ for all λ ∈ SpecB P β respectively . (11) Remark . Let σ = Id . In terms of the Fourier-coefficients, when P is first-order, | ξ | c W , ,P β ,γ,b − ( Cyl t ) = Σ λ ∈ Spec ( B P β ) ,λ =0 [(1 + λ ) Z ∞ t ξ λ s γ ds + Z ∞ t | dξ λ ds | s γ ds ]+Σ λ ∈ Spec ( B P β ) ,λ =0 [ Z ∞ t ξ λ s b − ds + Z ∞ t | dξ λ ds | s b ds ] . When P is second-order elliptic, using the usual W , − elliptic estimate on strips, we rou-tinely verify the following for any ξ compactly supported in Cyl t + ǫ . | ξ | c W , ,P β ,γ,b − ( Cyl t ) ≤ C ( ǫ ) { Σ λ ∈ Spec ( B P β ) ,λ =0 Z ∞ t [(1 + λ ) ξ λ + (1 + | λ | ) | dξ λ dt | + | d ξ λ dt | ] t γ dt +Σ λ ∈ Spec ( B P β ) ,λ =0 [ Z ∞ t ξ λ t b − ds + Z ∞ t ( | dξ λ dt | + | d ξ λ dt | ) t b dt ] } Remark . Multiplying by e − βt is a linear isomorphism: b C k,α,P − β,γ,b − −→ b C k,α,P β ,γ,b − , c W k,p,P − β,γ,b − −→ c W k,p,P β ,γ,b − , C k,α − β,γ,b −→ C k,α ,γ,b , L − β,γ,b −→ L ,γ,b . Definition 2.7.
Suppose −→ β = ( β , ....., β l ), −→ γ = ( γ , ......, γ l ), −→ b = ( b , ......, b l ) arevectors of l − entries. Given an AT ID operator P over a manifold N with l cylindricalends, we denote the ends by U j , j = 1 ....l . We add the interior U to obtain an open coverof N . Using a partition of unity χ j , j = 0 ....l subordinate to the cover, we define | ξ | c W k,p,P − β,γ,b − ( N ) = | χ ξ | W k, ( U ) + Σ l j =1 | χ j ξ | c W k,p,P ,j − βj,γj,bj − ( U j ) , (12)where P ,j is the limit T ID operator of P on the j − th end. The same definition as (12)applies to all the other norms in Definition 2.3 (including b C k,α,P − β,γ,b − ( N ), C k,α,P − β,γ,b ( N ) etc).When the domain is the whole manifold, we usually hide the N in the norm symbols. Important Convention:
When β = ... = β l = β , we denote −→ β (a vector) as β (number). The same applies to −→ b and −→ γ . This makes the notations consistent. Definition 2.8.
Let δ > t , such that the Neumann-Series in Lemma 4.4 and Theorem 3.3 converge as desired.Let k ≥ | · | C k ( t, y ) , Σ ≤ i + j ≤ k | ∂∂t i ∇ j · | ( t, y ). We say that P satisfies the s β ( l , l ) | Cyl t − condition, if the following holds for any k ≤ k , t ≥ t + 1, α ∈ [0 , ξ ,and a δ small enough with respect to the data in Theorem 3.2. t l | ( P − P ) ξ ⊥ β | C k,α ( S t ) ≤ δ | ξ ⊥ β | C k + m ,α ( S t ) , t l | ( P − P ) ξ ⊥ β | C k ( t, y ) ≤ δ | ξ ⊥ β | C k + m ( t, y ) t l | ( P − P ) ξ k β | C k,α ( S t ) ≤ δ | ξ k β | C k + m ,α ( S t ) , t l | ( P − P ) ξ k β | C k ( t, y ) ≤ δ | ξ k β | C k + m ( t, y )We say that P satisfies s ( l ) | Cyl t if it satisfies s β ( l , l ) | Cyl t for all β and l = l = l .We say that P is −→ β − AT ID on N if for any i , it satisfies s β i (0 , | Cyl t for some t on the i − th end. 5 emark . By our definition, δ depends on P , B P , γ, β, b etc. Remark . It’s easy to check s ( l ) | Cyl t for differential operators. In an arbitrary coor-dinate neighbourhood, write P and P as P = a , ( y ) ∂∂t + Σ nγ =2 a ,γ ( y ) D γ ; P = a ( y, t ) ∂∂t + Σ nγ =2 a γ ( y, t ) D γ . (13)Let δ be small enough with respect to the data in Theorem 3.2 (even smaller than δ ), then P satisfies s ( l ) | Cyl t if the following holds for all y, k ≤ k + 1 , t ≥ t . t l | a γ − a ,γ | C k ( t, y ) ≤ δ c.f. [11 , (6 . . (14)A simple example of an s (0 , | Cyl t − operator which does not satisfy (14) is P + δ sin t ∂∂t .Defining ( (cid:3) | Sobolev,p −−→ β , −→ γ , −→ b ) and ( (cid:3) | Schauder −−→ β , −→ γ , −→ b ) as (2), Theorem 1.4 naturally generalizes to Theorem 2.11.
Let m , k , k , α , p be as in Theorem 1.4. Suppose P is −→ β − AT ID elliptic,and β j is not P ,j − super indicial for any j . Then • ( (cid:3) | Sobolev,p −−→ β , −→ γ , −→ b ) is Fredholm if for any j , b j = 1 − p or β j is not P ,j − indicial; • ( (cid:3) | Schauder −−→ β , −→ γ , −→ b ) is Fredholm if for any j , b j = 1 or β j is not P ,j − indicial. Dependence of the Constants : we follow the convention in [15, Definition 2.16,2.17]:the ” C ” in a result (and the proof) depends on the data in the result, except the ” t ” (initialtime for the cylinders). We will add subscripts when C depends on t or other parameters. Remark . From now on, we hide the P (or P ) in the c W ′ s ( b C ′ s ) . The underlyingoperator should be clear from the context. When γ = b , we abbreviate c W k,p,P − β,γ,b − , b C k,α,P − β,γ,b − , W k,p,P − β,γ,b , C k,α,P − β,γ,b to c W k,p − β,b − , b C k,α − β,b − , W k,p − β,b , C k,α − β,b respectively. When γ = b = 0, wefurther abbreviate W k,p − β,b , C k,α − β,b to W k,p − β , C k,α − β . Definition 3.1.
Let β < β be the indicial root adjacent to β from below (but not equal to β ), and ¯ β > β be the indicial root adjacent to β from above. Theorem 3.2.
Let P be a TID-operator, and β be not P − super indicial. The followingholds in view of Definition 1.1.( i ): When b = or β is not P − indicial, P : c W m , − β,γ,b − ( Cyl t ) → L − β,γ,b ( Cyl t ) admits a bounded linear right inverse. Let Q P ,t β, + ( Q P ,t β, − ) denote the right inverse when b > ( b < ) respectively when β is indicial, and Q P ,t β denote the right inverse when β isnot indicial (When β is not indicial, Q P ,t β, ± both mean Q P ,t β ).( ii ): The following (regularity) estimates hold. | Q P ,t β, + h | b C k + m ,α − β,γ,b − ( Cyl t ) ≤ C | h | C k,α − β,γ,b ( Cyl t ) when b > and β is indicial ; (15) | Q P ,t β, − h | b C k + m ,α − β,γ, ( Cyl t ) ≤ C | h | C k,α − β,γ,b ( Cyl t ) when b > and β is indicial ; (16) | Q P ,t β, − h | b C k + m ,α − β,γ,b − ( Cyl t ) ≤ C | h | C k,α − β,γ,b ( Cyl t ) when b < and β is indicial ; (17) | Q P ,t β h | C k + m ,α − β,γ ( Cyl t ) ≤ C | h | C k,α − β,γ ( Cyl t ) when β is not indicial ; (18) | Q P ,t β, + h | c W k + m ,p − β,γ,b − ( Cyl t ) ≤ C | h | W k,p − β,γ,b ( Cyl t ) when b > − p and β is indicial ;(19) | Q P ,t β, − h | c W k + m ,p − β,γ,b − ( Cyl t ) ≤ C | h | W k,p − β,γ,b ( Cyl t ) when b < − p and β is indicial ;(20) | Q P ,t β h | W k + m ,p − β,γ ( Cyl t ) ≤ C | h | W k,p − β,γ ( Cyl t ) when β is not indicial . (21)6 iii ): Suppose β is not super-indicial, then Q P ,t β, + = Q P ,t β, − on L − β,b ( Cyl t ) for any b . Important Convention : through-out the article, we say that h is in (or not in) aspace if and only if the norm of h is < ∞ (= ∞ ), respectively. Therefore all the estimatesin Theorem 3.2 are regularity estimates. Theorem 3.3.
Suppose P satisfies s β (0 , | Cyl t and β is not P − super-indicial. Thenexcept (16), P also satisfies ( i ), ( ii ), ( iii ) in Theorem 3.2 (with P replaced by P , and Q P ,t β, + , Q P ,t β, − , Q P ,t β replaced notationally by Q P,t β, + , Q P,t β, − , Q P,t β ).Remark . All the bounds in Theorem 3.2, 3.3 are independent of t . Proof of Theorem 3.3 assuming 3.2:
We momentarily hide t , β, ± in Q P ,t β, ± in each case ofTheorem 3.2. Theorem 3.2 and the β − AT ID condition (Definition 2.8) implies when δ issufficiently small, the Neumann-Series (c.f. [17, Theorem 2, page 69])[ Id − Q P ( P − P )] − , Σ ∞ j =0 [ Q P ( P − P )] j (22)converges to a two-sided inverse of Id − Q P ( P − P ). Hence Q P , (Σ ∞ j =0 [ Q P ( P − P )] j ) Q P is a right-inverse of P i.e. P Q P = Id , where we write P = P [ Id − Q P ( P − P )]. Lemma 3.5. (Hardy’s inequality) For any p ≥ , Z ∞ ( t b − Z ∞ t | f | ds ) p dt ≤ C p,b Z ∞ ( t b | f | ) p dt when b > − p ; (23) Z ∞ ( t b − Z t | f | ds ) p dt ≤ C p,b Z ∞ ( t b | f | ) p dt when b < − p . (24) For all b ∈ R , p ≥ , ϑ ≥ , and µ = 0 , there exists a constant C l µ ,b which depends only on b and the lower bound on | µ | with the following properties. µ p (1+ ϑ ) Z ∞ ( e µt t b Z ∞ t e − µs | f | ( s − t ) ϑ ds ) p dt ≤ C l µ ,p,b Z ∞ ( | f | t b ) p dt when µ > µ p (1+ ϑ ) Z ∞ ( e µt t b Z t e − µs | f | ( t − s ) ϑ ds ) p dt ≤ C l µ ,p,b Z ∞ ( | f | t b ) p dt when µ < . (26)(23) and (24) are special cases of [9, Theorem 330]. The proof for (25), (26) is elementary,we defer it to the Appendix.The proof of [8, Lemma 6.37, Theorem 7.25] (reflection about the boundary) yields Claim 3.6.
Let t ≥ . For any section h ∈ C k,α ( Y × [ t , t + 1]) or W k,p [ Y × ( t , t + 1)] ,there exists an extension h E,t such that • h E,t = 0 over (0 , t − . , and h E,t = h when t ≥ t ; • | h E,t | C k,α [ Y × (0 ,t +1)] ≤ C | h | C k,α [ Y × ( t ,t +1)] , | h E,t | W k,p [ Y × (0 ,t +1)] ≤ C | h | W k,p [ Y × ( t ,t +1)] ; • h E,t is translation-invariant in t i.e. h E,t ( f )( t ) = h E, ( f t )( t − t + 2) ,where f t ( t ) = f ( t + t − . We need to construct a linear operator ˙ Q P β λ for each of the equations in (11), such that u λ , ˙ Q P β λ f λ solves them respectively with required estimates. Summing the λ ’s up, weobtain the desired right inverse: e Q P β , ± f , Σ λ ∈ Spec ( B P β ) ( ˙ Q P β λ f λ ) φ λ . (27)When β = 0, it suffices to take b Q P β, ± , σ − · e βt · e Q P β , ± · e − βt · σ − . (28)7 roof of Theorem 3.2 (i) for first-order operators : We construct the ˙ Q P β λ as u λ ( , ˙ Q P β λ f λ ) u λ , ˙ Q P β λ f λ λ = 0, b > − R ∞ t f λ ds λ >
0, all b − e λt R ∞ t e − λs f λ ds λ = 0, b < R t f λ ds λ <
0, all b e λt R t e − λs f λ ds (29) By Remark . and completeness of the spaces in Definition . , it suffices to assume f ∈ C ∞ c ( Cyl ) , and only has finitely many non-zero Fourier coefficients . Withoutloss of generality, we only consider first-order operators, and assume that β = 0[ see the derivation of (32)] . The proof for second-order operators is similar . (30)Applying the 4 inequalities in Lemma 3.5 to the 4 cases in (29) respectively, we find Z ∞ u λ t b − dt ≤ C Z ∞ f λ t b dt (in cases 1, 2) , λ Z ∞ u λ t γ dt ≤ C Z ∞ f λ t γ dt (cases 3, 4) . Using the equation (11) to estimate du λ dt , we trivially obtain Z ∞ | du λ dt | t b dt = Z ∞ f λ t b dt (in cases 1, 2) , Z ∞ | du λ dt | t γ dt = Z ∞ f λ t γ dt (cases 3, 4) . The above 4 estimates in the 4 cases yield | e Q P β ,b f ⊥ | W m , ,γ ( Cyl ) ≤ C | f ⊥ | L ,γ ( Cyl ) ; | e Q P β ,b f | c W m , ,γ,b − ( Cyl ) ≤ C | f | L ,γ,b ( Cyl ) . (31)Using (28) ( σ − i are smooth) and the notation in Theorem 3.2 i , let f = σ − h E,t ∈ C ∞ c ( Cyl ), we obtain | b Q P β, ± h E,t | c W m , − β,γ,b − ( Cyl ) ≤ C | h E,t | L − β,γ,b ( Cyl ) . Let Q P ,t β, ± h , b Q P β, ± h E,t , (32)the following holds by Claim 3.6. | Q P ,t β, ± h | c W m , − β,γ,b − ( Cyl t ) ≤ | b Q P ,t β, ± h E,t | c W m , − β,γ,b − ( Cyl ) ≤ C | h E,t | L − β,γ,b ( Cyl ) ≤ C | h | L − β,γ,b ( Cyl t ) . (33)The above means Q P ,t β, ± is bounded.Similar ideas apply to second-order equations, we defer the detail to the Appendix. Byour constructions in (29), (64), (67), (28), we routinely verify Theorem 3.2 ( iii ). Proof of Theorem 3.2 (18), (21) : It suffices to apply Maz’ya-Plamenevskii’s trick ([13,Lemma 1.1, 4.1]). We adopt (30) and assume k = 0. Theorem 3.2 ( i ) yields Z l +1 l − | e Q P β ( ξ m f ) | L ( Y ) t γ dt ≤ Ce µ l Z l +1 l − | e Q P β ( ξ m f ) | L ( Y ) e − µ t t γ dt ≤ Ce µ l Z ∞ | ξ m f | L ( Y ) e − µ t t γ dt ≤ Ce µ ( l − m ) Z m +2 m − | f | L ( Y ) t γ dt ≤ Ce − | µ || l − m | sup m | t γ f | L (2 S m ) , (34) | e Q P β f | L ,γ ( Cyl ) ≤ C | f | L ,γ ( Cyl ) , (35)8here we let | µ | be small enough with respect the spectrum gap, and the sign of − µ bethe same as that of l − m (when l = m either sign works). Summing the m in (34) over allintegers ≥
3, we obtain | e Q P β f | L ,γ ( S l ) ≤ C sup m | t γ f | L (2 S m ) Σ m ≥ e −| µ || l − m | ≤ C | f | C ,γ ( Cyl ) for any l ≥ . (36)We recall | e Q P β f | L ( S l ) ≤ Cl − γ | e Q P β f | L ,γ ( S l ) , | f | C α ( S l ) ≤ Cl − γ | f | C α ,γ ( S l ) , | f | L p ( S l ) ≤ Cl − γ | f | L p ,γ ( S l ) , and the following standard (Schauder and L p ) estimate on S l | ξ | C ,α ( Sl ) ≤ C | P β ξ | C α ( S l ) + C | ξ | L ( S l ) , | ξ | W ,p ( Sl ) ≤ C | P β ξ | L p ( S l ) + C | ξ | L ( S l ) . (37)Then we obtain from (34) and (35) that l γ | e Q P β f | C ,α ( Sl ) ≤ C [ | f | C α ,γ ( Cyl ) + | f | C ,γ ( S l ) ] ≤ C | f | C α ,γ ( Cyl ) , (38) l γ | e Q P β f | W ,p ( Sl ) ≤ C [ | f | L p ,γ ( S l ) + | e Q P β f | L ,γ ( S l ) ] . (39)Taking sup l ≥ of (38) and Σ l ≥ of (39), we obtain by Definition 2.3 and (35) that | e Q P β f | C ,α ,γ ( Cyl ) ≤ C | f | C α ,γ ( Cyl ) , | e Q P β f | W ,p ,γ ( Cyl ) ≤ C | f | L p ,γ ( Cyl ) . (40)By the same argument in (33) [using (40) instead of 32)], we obtain (18) and (21). Proof of Theorem 3.2 (15), (16), (17), (19), (20) : We adopt (30). Using Lemma3.5 and (29), we find the simple estimates | Q P β + f k | ≤ C | Z ∞ t f k ds | ≤ C | f k | C ,b ( Cyl t ) | Z ∞ t s − b ds | ≤ C | f k | C ,b ( Cyl t ) t − b when b > , | Q P β − f k | ≤ C | Z t f k ds | ≤ C | f k | C ,b ( Cyl ) | Z t s − b ds | ≤ ( C | f k | C ,b ( Cyl ) t − b when b < ,C | f k | C ,b ( Cyl ) when b > . ( Z ∞ ( t b − | Q P β ± f k | ) p dt ) p ≤ C ( Z ∞ ( t γ | f k | ) p dt ) p when b > ( < ) 1 − p respectively . (41)Combining ∂∂t Q P β ± f k = f k , we find | Q P β + f k | b C ,α ,b − ( Cyl ) ≤ C | f k | C α ,b ( Cyl ) when b > b <
1) respectively , | Q P β ± f k | c W ,p ,b − ( Cyl ) ≤ C | f k | L p ,b ( Cyl ) when b > − p ( b < − p ) respectively , | Q P β − f k | b C ,α , ( Cyl ) ≤ C | f k | C α ,b ( Cyl ) when b > . (42)Because Q P β ± f ⊥ is perpendicular to the kernel, (31) and the proof of (18), (21) yield | e Q P β , ± f ⊥ | C ,α ,γ ( Cyl ) ≤ C | f ⊥ | C α ,γ ( Cyl ) , | e Q P β , ± f ⊥ | W ,p ,γ ( Cyl ) ≤ C | f ⊥ | L p ,γ ( Cyl ) . (43)(42), (43) amount to (the special cases of) (15), (16), (17), (19), (20) with P replaced by P β , β by 0. The argument in (32), (33) yields the desired five estimates in general. Remark . Without loss of generality, in the proof of Claim 4.2, Lemma 4.4, and Proposi-tion 4.5, we only consider first-order operators, and assume k = t = 1, γ = b (see Remark2.12). The proof for the other cases is absolutely the same. Though second-order elliptic op-erators are more complicated [there are 2 homogeneous solutions to the second-order ODEsin (11)], the desired regularity still follows in the same way.9 laim 4.2. Given any TID-operator P , suppose β, β are not P − super indicial. Then forany t , k ≥ , ǫ > , the following estimate holds uniformly in h ∈ kerP | L − β,γ, −
12 + ǫ ( Cyl t ) : | h | C k,α − β ( Cyl t ǫ ) ≤ C t | h | L − β,γ, −
12 + ǫ ( Cyl t ) . Proof : We adopt Remark 4.1. The condition | h | L − β, −
12 + ǫ ( Cyl ) < ∞ ( h ∈ KerP ) implies h k β = 0, then h = Σ λ ∈ SpecB P ,λ<β h λ e λt φ λ = Σ λ ≤ β h λ e λt φ λ . The condition dimEigen β B P < ∞ implies | h k β | C ,α − β ( Cyl ǫ ) ≤ C | h k β | L − β, −
12 + ǫ ( Cyl ) . Since h ⊥ β ∈ Ker b B P β , using the Schauder estimate in (37) like (39), the rate of decay isimproved i.e. | h ⊥ β | C ,α − β ( Cyl ǫ ) ≤ C | h ⊥ β | L − β ( Cyl ǫ ) ≤ C | h ⊥ β | L − β, −
12 + ǫ ( Cyl ) . Thus | h | C ,α − β ( Cyl ǫ ) ≤ | h ⊥ β | C ,α − β ( Cyl ǫ ) + | h k β | C ,α − β ( Cyl ǫ ) ≤ C ( | h ⊥ β | L − β, −
12 + ǫ ( Cyl ) + | h k β | L − β, −
12 + ǫ ( Cyl ) ) ≤ C | h | L − β, −
12 + ǫ ( Cyl ) . Definition 4.3. S P ,t β, ± , Q P ,t β, ± P − Id is bounded from c W m , − β,γ, − ± ǫ ( Cyl t ) to itself. Weverify S P,t β, ± , Q P,t β, ± P − Id = (Σ ∞ j =0 [ Q P ,t β, ± ( P − P )] j ) S P ,t β, ± . Lemma 4.4.
Let P be an operator on Cyl t as in Definition 2.8. Suppose β, β are not P − super indicial, then the following hold for any ǫ > , t ≥ , γ , and k ≤ k + m − . I : | S P,t β, + ξ | b C k,α − β, ( Cyl t ) ≤ C t | ξ | b C k,α − β,γ,ǫ ( Cyl t ) when P satisfies s ( l ) | Cyl t with l > , II : | S P,t β, + ξ | c W m , − β, − − ǫ ( Cyl t ) ≤ C t | ξ | c W m , − β,γ, −
12 + ǫ ( Cyl t ) when P satisfies s β | Cyl t , III : | S P,t β, + ξ | b C k,α − β, − ǫ ( Cyl t ) ≤ C t | ξ | b C k,α − β,γ,ǫ ( Cyl t ) when P satisfies s β | Cyl t , IV : | S P,t β, + ξ | C k,α − β, − − ǫ ( Cyl t ǫ ) ≤ C t | ξ | c W m , − β,γ, −
12 + ǫ ( Cyl t ) when P satisfies s β | Cyl t . Proof.
We adopt Remark 4.1 and only prove I and IV . II and III are similar (to I ). First,we deal with the model operator S P ,t β, + . Because S P , β, + ξ ∈ KerP , Claim 4.2 yields | S P , β, + ξ | b C ,α − β, ( Cyl ) ≤ C | S P , β, + ξ | b C ,α − β,ǫ ( Y × [2 , + | S P , β, + ξ | b C ,α − β, ( Cyl ) ≤ C | S P , β, + ξ | b C ,α − β,ǫ ( Cyl ) ≤ C | Q P , β, + P ξ | b C ,α − β,ǫ ( Cyl ) + C | ξ | b C ,α − β,ǫ ( Cyl ) ≤ C | ξ | b C ,α − β,ǫ ( Cyl ) . (44)When P satisfies s ( l ) | Cyl with l > | ( P − P ) η | C α − β,l ( Cyl ) ≤ Cδ | η | C ,α − β ( Cyl ) for any η. Hence we obtain the following (by Theorem 3.2 (16) and the above). | Q P , β, + ( P − P ) η | b C ,α − β, ( Cyl ) = | Q P , β, − ( P − P ) η | b C ,α − β, ( Cyl ) ≤ Cδ | η | C ,α − β ( Cyl ) . (45)Let δ be small enough such that Cδ ≤ in the above, the Neumann series converges i.e. | (Σ ∞ j =0 [ Q P , β, + ( P − P )] j ) η | b C ,α − β, ( Cyl ) ≤ C | η | C ,α − β ( Cyl ) . (46)Let η = S P , β, + ξ in (46), Lemma 4.4 I follows from (44) and Definition 4.3.On IV , using the Schauder estimate (37) and II (note S P, β, + ξ ∈ KerP ), we obtain | S P, β, + ξ | C k,α ( ǫSl ) ≤ C | S P, β, + ξ | L ( ǫS l ) ≤ Cl + ǫ e βl | S P, β, + ξ | L − β, − − ǫ ( ǫS l ) . (47)By Definition 2.7, the proof of IV is complete.10 roposition 4.5. Under the same conditions on P , ǫ , k , γ , β, β in Lemma 4.4, thefollowing hold uniformly for any h ∈ KerP | c W m , − β,γ, −
12 + ǫ ( Cyl t ) . | h | b C k,α − β, − ǫ ( Cyl t ǫ ) ≤ C t | h | c W m , − β,γ, −
12 + ǫ ( Cyl t ) when P satisfies s β | Cyl t , | h | b C k,α − β, ( Cyl t ǫ ) ≤ C t | h | c W m , − β,γ, −
12 + ǫ ( Cyl t ) when P satisfies s ( l ) | Cyl t with l > . Consequently, for any k ≤ k − , suppose the signs of µ and µ are the same, we have ( ker | Coker | Index )( (cid:3) | Sobolev,p − β,γ, − p + µ ) = ( ker | Coker | Index )( (cid:3) | Schauder − β,γ, µ ) respectively . (48) Moreover, suppose β < β are 2 adjacent indicial roots which are not P − super indicial, then ( ker | Coker | Index )( (cid:3) | Sobolev,p − β,γ , − p + ǫ ) = ( ker | Coker | Index )( (cid:3) | Sobolev,p − β,γ ,b )= ( ker | Coker | Index )( (cid:3) | Sobolev,p − β,γ , − p − ǫ ) . ( ker | Coker | Index )( (cid:3) | Schauder − β,γ , ǫ ) = ( ker | Coker | Index )( (cid:3) | Schauder − β,γ ,b )= ( ker | Coker | Index )( (cid:3) | Schauder − β,γ , − ǫ ) . for any p ≥ , b , γ i ( i = 1 , , ), β ∈ ( β, β ) , and ǫ , ǫ > .Proof : It’s a direct corollary of Lemma 4.4. We only prove the second assertion, the first iseasier. We adopt Remark 4.1. We note that h ∈ KerP implies h = − S P,tβ, + h for all t ≥ t . (49)Let t = t , Lemma 4.4 (IV) says | h | b C k,α − β, − ( Cyl t ǫ ) ≤ C | h | c W m , − β, −
12 + ǫ ( Cyl t ) . Let t = t + ǫ in(49), Lemma 4.4 (I) and the above imply | h | b C k,α − β, ( Cyl t ǫ ) ≤ C | h | b C k,α − β, − ( Cyl t ǫ ) ≤ C | h | c W m , − β, −
12 + ǫ ( Cyl t ) . Proof of Theorem 1.4, 2.11 : With the help of Lemma 4.4, it is standard. We only dothe argument for 1.4, and only show the c W ( b C ) theory for b > − p ( b > Q β, + ,N such that (cid:26) Q β, + ,N P = Id + S left ,P Q β, + ,N = Id + S right , S right is compact from W k,p − β,γ,b ( C k,α − β,γ,b ) to itself . (50)Lemma 4.4 IV says S left is bounded from c W m , − β,γ,b − to b C k + m − ,α − β, − . Since the weight isimproved, by the (cylindrical analogue of) [15, proof of Lemma 4.10], b C k + m − ,α − β, − embedscompactly into c W k + m ,p − β,γ,b − and b C k + m ,α − β,γ,b − when k ≤ k −
2. Then S left is compact from c W k + m ,p − β,γ,b − ( b C k + m ,α − β,γ,b − ) to itself. Hence [18, Theorem 4.6.5] implies P is Fredholm. Definition 5.1.
Let B be an operator as in Proposition 5.4. Let d λ , dimker ( B − λId ).Let Eta ( B ) denote the eta-invariant defined in [3, Theorem 3.10 (iii)], and H β,B , − d − Σ >λ ≥ β d λ when β < , − d when β = 0 , d + Σ <λ<β d λ when β > . , H β,B,b , (cid:26) H β,B when b > ,H β,B + d β when b < . heorem 5.2. Under the conditions in Theorem 2.11, suppose in addition that P is first-order elliptic, σ in (1) is an isometry, and P is translation invariant on each end. Then indexP | c W , −−→ β , −→ b − → L −−→ β , −→ b = Z X α dvol − Eta ( B P )2 + Σ l j =1 H β j ,B jP ,j ,b j , where (51) α is the − th order term in the expansion of the kernel of e − t e P ⋆ e P − e − t e P e P ⋆ , e P is thedouble of P on the double of N ( N ♯ ¯ N ).Remark . When P satisfies proper conditions as Definition 2.8, under the operator norm,we can usually deform it continuously to be translation-invariant on each end. Thus theindex of P can still be computed by deformation invariance. Proof of Theorem 5.2:
Without loss of generality, we assume σ = Id . We recall the ”Ex-tended L − sections” defined in [3, first paragraph of page 58], and note that the dual of L ,b is isomorphic to L , − b . Using Claim 4.2 for both L and L ⋆ = − ( ∂∂t + B ) σ − , we have KerL | c W , ,b − = KerL | L ( N,E ) , KerL ⋆ | L , − b = KerL ⋆ | Extended L ( N,F ) when b > ; KerL | c W , ,b − = KerL | Extended L ( N,E ) , KerL ⋆ | L , − b = KerL ⋆ | L ( N,F ) when b < . (52) IndexL | c W , ,b − → L ,b = (cid:26) h ( E ) − h ( F ) − h ∞ ( F ) when b > ; h ( E ) − h ( F ) + h ∞ ( E ) when b < , (53)where h ( E ) , h ( F ) , h ∞ ( F ) , h ∞ ( F ) are defined in [3, Corollary (3.14)]. Assuming −→ β = 0for all j , (51) follows from [3, Corollary (3.14), (3.25)].By Proposition 4.5 (c.f. Remark 1.7), Index ( P | Sobolev, − β,b ) = ( Index ( P | Sobolev, − β + ǫ ) when b > ; Index ( P | Sobolev, − β − ǫ ) when b < . ǫ > . (54)The index change formula of Lockhart-McOwen [11, Theorem 8.1] means for any i , let β i , i = i be unaltered, and β i go across an eigen-value λ of B i (from λ + ǫ to λ − ǫ ), theindex decreases by dimker ( B i P − λId ). The proof for general −→ β is complete with the helpof (54) [and (51) for −→ β = 0]. Proposition 5.4.
In the setting of Definition 1.1, suppose B is a self-adjoint first-orderelliptic differential operator on E → Y . Then Eta ( B − βId ) = Eta ( B ) − H β,B − d β .Proof. We form the full cylinder Y × ( −∞ , + ∞ ) t and consider P = ∂∂t − B . We considerthe open cover End + , Y × ( − , End − , where End + = Cyl and End − = Y × ( −∞ , − End − , under the coordinate s = − t , P = − ∂∂s − B = − [ ∂∂s − ( − B )]. Let −−→ β = ( − β, e − βt on End + and 1 on End − ), Theorem 5.2 says when b > that indexP | c W , −−→ β ,b − → L −−→ β ,b = Z X α dvol − Eta ( B )2 − Eta ( − B )2 − d H β,B . (55)On the other hand, let ρ be a smooth function which is equal to e βt on End + , and 1 on End − , the conjugation P ρ = ρ · P · ρ : c W , ,b − → L ,b has the same index as P . Noting P ρ = (cid:26) ∂∂t − ( B − βId ) on End + P on End − , then Theorem 5.2 says the following for P ρ . indexP ρ | c W , ,b − → L ,b = Z X α dvol − Eta ( B − βId )2 − Eta ( − B )2 − d β − d . (56)The equality between (55) and (56) yields the desired identity.12 Applications
Definition 6.1.
Let Γ be a section or connection of E → Cyl . Suppose Γ satisfies anelliptic equation of order m . For any C > τ ≥
0, and real number b , we say that Γconverges to Γ (at least) at the rate C e − τt t b [or O ( e − τt t b )], if Γ is a section or connection of E → Y respectively and lim sup t →∞ e − τt t b | Γ − Γ | C m ,α ( S t ) < C (or < ∞ ) respectively.When τ >
0, We say that the convergence is exponential. When τ = 0, We say that it’spolynomial. We only prove 1.11 in full detail, 1.9 follows similarly. Denoting A − A O by a , the YM-equation is 0 = d ⋆A F A = d ⋆A O + a ( d A O a + F A O + [ a, a ]).Thus ∆ A O ,Hodge a + ( − n +1 ⋆ [ a, ⋆F A O ] = Q Y M ( a ) − d ⋆A O F A O assuming d ⋆A O a = 0 , (57)where ∆ A O ,Hodge = d ⋆A O d A O + d A O d ⋆A O is the Hodge Laplacian, and Q Y M ( a ) = − d ⋆A O [ a, a ]2 + ( − n ⋆ [ a, ⋆d A O a ] + ( − n ⋆ [ a, ⋆ [ a, a ]] . (58)We note Q Y M is quadratic in a . We routinely verify( − n +1 ⋆ [ a, ⋆F A O ] = F A O ⊗ a ([15 , Def inition . , ∆ A O ,Hodge,g E = ∇ ⋆ E A O ∇ A O + F A O ⊗ g E a. Then in cylindrical coordinates ( r = e − t ), let a = vdt + θ ( θ does not contain dt ), we find∆ A O ,Hodge,E a + ( − n +1 ⋆ E [ a, ⋆ E F A O ] = ∇ ⋆ E A O ∇ A O + 2 F A O ⊗ g E a = − e t (cid:12)(cid:12)(cid:12)(cid:12) dt Id (cid:12)(cid:12)(cid:12)(cid:12) { ∂ ∂t − ( n − ∂∂t − B } (cid:20) vθ (cid:21) . Let Y = S n − , B is (59) B (cid:20) vθ (cid:21) , (cid:20) ∇ ⋆ Y ∇ v − d ⋆ b − n − v ∇ ⋆ Y ∇ θ − dv − F A O ⊗ Y θ − ( n − θ (cid:21) (c.f. [15 , (22) , (24)]) . (60)As an usual strategy for non-linear equations, we view Q Y M as a linear operator defining b Q Y M,a ( b ) = − d ⋆A O [ b, a ]2 + ( − n ⋆ [ b, ⋆d A O a ] + ( − n ⋆ [ b, ⋆ [ a, a ]] . (61)Hence Q Y M ( a ) = b Q Y M,a ( a ), and we can write (57) in cylindrical coordinates as P Y M ( a ) , { e − t (∆ A O ,Hodge + ( − n +1 ⋆ [ · , ⋆F A O ]) − e − t b Q Y M,a } a = − e − t d ⋆A O F A O . (62)The conditions on g , a , (59), and (60) implies P Y M is ATID in the cylindrical coordinates.Moreover, that d ⋆ E A O F A O = 0 (tangent connection is Yang-Mills) implies − e − t d ⋆A O F A O ∈ C α , ( Cyl − log R ) (exponential decay). Applying Q P Y M , − R , + (= Q P Y M , − R , − ) to both sidesof (62), Lemma 4.4 III implies that a decays exponentially. This in turn means P Y M satisfies s ( l ) for all l >
1. Then Corollary 1.11 I follows from applying Proposition 4.5 to a + Q P Y M , − R , + e − t d ⋆A O F A O ∈ KerP
Y M and Theorem 3.3 to Q P Y M , − R , + e − t d ⋆A O F A O .Combining I and Lemma 7.2, the proof of II is complete. By [1, Section 5, page 247], in the cylindrical coordinate t = log | x | , the graph typeminimal sub-manifold equation can be written as (63) in terms of a section u to T ⊥ Σ | S N (the normal bundle of Σ in S N ). We note (the transition functions of) T ⊥ Σ | S N does notdepend on t (or | x | ), then this bundle is in the case considered by Definition 1.1. P MSM u , Lu + R ( u ) + ( M Σ − L Σ ) u = 0 , where (63)13 Lu = u ′′ − ( n + 1) u ′ + L Σ u (see [16, Page 565] and [1, (1.8)]), • R ( u ) satisfies [16, (1.9)], L Σ is the linearisation of M Σ (see [1, (5.2) and Page 248].By the idea in (61), we can view R + M Σ − L Σ as a linear operator i.e.( R + M Σ − L Σ )( v ) , Q MSM ( v ) = a ( x, t, u, ∇ u, ∇ u, u ′ ) · ∇ v + b ( x, t, u, ∇ u, ∇ u ) · v ′ + c ( x, t, u, ∇ u, ∇ u ) · ∇ v ′ + d ( x, t, u, ∇ u, ∇ u ) · v ′′ where a, b, c, ∇ can be found in [16, the paragraph enclosing (1.9); between line 1 in page 565and (7.35)]. Exactly as the proof of Corollary 1.11, u decays as δ t implies P MSM is 0 − ATID.Thus Proposition 4.5 [and the discussion below (62)] yields the desired improvement.
Proof of Theorem 3.2 (i) for second-order operators : We mainly focus on the casewhen a − β >
0. We solve the second-order ODE in (11) according to the following. m , a − β .Assume m >
0. Corresponding solution u λ , ˙ Q P β λ f λ and the derivative. Det , m + 4 λ , µ , √ | Det | , µ + , m + µ , µ − , m − µ Det > λ = 0 u λ = − √ Det [ e µ + t R ∞ t e − µ + s f λ ds + e µ − t R t e − µ − s f λ ds ], u ′ λ = − √ Det [ µ + e µ + t R ∞ t e − µ + s f λ ds + µ − e µ − t R t e − µ − s f λ ds ]2 λ = 0, b > u λ = √ Det {− e µ + t R ∞ t e − µ + s f λ ds + R ∞ t f λ ds } , u ′ λ = − µ + √ Det e µ + t R ∞ t e − µ + s f λ ds λ = 0, b < u λ = √ Det {− e µ + t R ∞ t e − µ + s f λ ds − R t f λ ds } , u ′ λ = − µ + √ Det e µ + t R ∞ t e − µ + s f λ ds Det = 0 u λ = e mt R ∞ t e − ms f λ ( s − t ) ds Det < u λ = e mt µ { R ∞ t e − ms f λ [cos µt sin( µs ) − sin µt cos( µs )] ds } (64) Case 1, 5 in (64):
When λ = 0, we have √ | λ | C ≤ µ + ≤ C p | λ | , − √ | λ | C ≤ µ − ≤ − C p | λ | .We use (25), (26) to estimate u λ and u ′ λ termwise, then use the second-order equation in(11) to estimate u ′′ λ . Then we obtain the following for Case 1 and 4. Z ∞ | u ′′ λ | t γ dt + | λ | Z ∞ | u ′ λ | t γ dt + λ Z ∞ u λ t γ dt ≤ C Z ∞ f λ t γ dt. (65)Since | cos x | , | sin x | ≤ x , and √ λ ≤ C when λ = 0 [ Spec ( B ) is discrete], (65) holdsfor Case 5 in similar way. Cases 2,3 in (64):
We still do the termwise estimates. Using (25) for e µ + t R ∞ t e − µ + s f λ ds ,(26) for e − µ + t R t e µ + s f λ ds , and (23) for R ∞ t f λ ds , we obtain Z ∞ | u ′′ λ | t b dt + Z ∞ | u ′ λ | t b dt + Z ∞ u λ t b − dt ≤ C Z ∞ f λ t b dt. (66)Remark 2.5, (65), (66) amount to | e Q P β ,b f | c W , ,γ,b − ( Cyl ) ≤ C | f | L ,γ,b ( Cyl ) . The argumentbetween (31) and (33) gives the desired local right inverse Q P ,t β,b with the desired bound.The proof when a − β < m = a − β < Det > λ = 0 the same as Case 1 in (64)2 λ = 0, b > u λ = √ Det {− R ∞ t f λ ds − e µ − t R t e − µ − t f λ ds } λ = 0, b < u λ = √ Det {− e µ − t R t e − µ − s f λ ds + R t f λ ds } Det = 0 u λ = e mt R t e − ms f λ ( t − s ) ds Det < u λ = − e mt µ { R t e − ms f λ [cos µt sin( µs ) − sin µt cos( µs )] ds } (67)14he proof when m = a − β = 0 and λ > ∇ ⋆A O ∇ A O denote the rough Laplacian onΩ ( adE ). By [15, (17)], in cylindrical coordinate t = − log r , ∇ ⋆A O ∇ A O χ = e t { d χdt − ( n − dχdt + ∆ A O , S n − χ } , (68)where ∆ A O , S n − is the rough Laplacian on the link. Let C k,αγ,b,cone denote the weightedSchauder-spaces in the cone setting defined in [15, Definition 2.10]. On 0 − forms, we notethat pulling back is an isomorphism C k,αγ,b,cone [ B O ( R )] → C k,α − γ,b ( Cyl − log R ). Definition 7.1.
We say that A O is irreducible if every parallel section to Ω ( adE ) is 0everywhere. This means 0 is not an eigenvalue of ∆ A O , S n − . Lemma 7.2. (Existence of Coulomb gauge) Under the setting in the first paragraph ofCorollary 1.11 (for any n ≥ ), suppose A O is irreducible. Suppose A is a C − connectionon B O ( R ) \ O such that | A − A O | C ,α ,b,cone [ B O ( R )]) < δ for some b ≥ . (69) Then there exists a gauge s on B O ( R ) such that • | s − Id | C ,α ,b,cone [ B O ( R )] ≤ C | A − A O | C ,α ,b,cone [ B O ( R )]) ≤ Cδ , • d ⋆A O [ s ( A ) − A O ] = 0 in a smaller (truncated) ball B O ( R ′ ) \ O , and | S ( A ) − A O | C ,α ,b,cone [ B O ( R ′ ) − O ]) ≤ C | A − A O | C ,α ,b,cone [ B O ( R ) − O ]) ≤ Cδ . (70) Proof.
Let a , A − A O . By Theorem 3.3 and paragraph enclosing (68), for any b ≥ R ′ small enough, ∇ ⋆A ∇ A is invertible: C ,α ,b,cone [ B ( R ′ )] | Ω ( adE ) −→ C α ,b,cone [ B ( R ′ )] | Ω ( adE ) .Writing s = e − χ , then d ⋆A O [ s ( A ) − A O ] = d ⋆A O [ s − as + s − d A O s ] = d ⋆A O [ e − χ ae χ + e − χ d A O e χ ] (71)is a continuously differentiable map: C ,α ,b,cone [ B O ( R )] | Ω ( adE ) × C ,α ,b,cone [ B O ( R )] | Ω ( adE ) −→ C α ,b,cone [ B O ( R )] | Ω ( adE ) . Then the proof is complete by the standard argument in [6, Proposition 2.3.4].
Proof of (25), (26) : For any s ≥ , leaving the proof to the readers, we have | Z s e µt t d dt | ≤ C l µ ,d e µs s d µ when µ > , Z ∞ s e µt t d dt ≤ C l µ ,d e µs s d − µ when µ < . (72)For (25), H¨older’s inequality and the change of variable z = s − t yield( Z ∞ t e − µs f ( s − t ) ϑ ds ) p ≤ ( Z ∞ t e − µs f p ds )( Z ∞ t e − µs ( s − t ) pϑp − ds ) p − ≤ C ( Z ∞ t e − µs f p ds ) e − µ ( p − t µ ϑp + p − ) . (73)Then (72) and (73) yield Z ∞ ( e µt t b Z ∞ t e − µs f ( s − t ) ϑ ds ) p dt ≤ µ pϑ + p − ( Z ∞ e − µs f p ds )( Z s e µt t pb dt ) ≤ C l µ ,p,b µ p (1+ γ ) Z ∞ f p s pb ds. The proof of (25) is complete . The proof of (26) is similar. 15 roof of Remark 1.8 : We only have to show the ”only if”. Without loss of generality, weonly consider the Schauder theory, and assume m = 1, γ = b = 1 , −→ β = 0, σ = σ = Id .For any φ ∈ KerB P , let f = φ t . When P = ∂∂t − B (translation-invariant on the end)and t ≥
10, the general solution to
P u = φ t is (log t + C ) φ / ∈ b C ,α , . Hence any extension e f of f to the whole N is not in RangeP | b C ,α , .On the other hand, let f k = φ t
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