aa r X i v : . [ m a t h . A P ] M a r SCHAUDER ESTIMATES ON SMOOTH AND SINGULARSPACES
YAOTING GUI AND HAO YIN
Abstract.
In this paper, we present a proof of Schauder estimate on Eu-clidean space and use it to generalize Donaldson’s Schauder estimate on spacewith conical singularities in the following two directions. The first is that weallow the total cone angle to be larger than 2 π and the second is that wediscuss higher order estimates. Contents
1. Introduction 12. H¨older space on R n
53. A proof of the Schauder estimates on R n
94. Preliminaries about X β X β X β -polynomial is still harmonic 166. Generalized H¨older space on X β C ,αβ space 217. Schauder estimates for X β Introduction
In this paper, we discuss the classical Schauder estimate on Euclidean space R n and on some singular space with conical type singularities. The discussioncontained in this paper should apply with minor modification to a class of conicaltype singular spaces, however, for simplicity, we restrict ourselves to a special case,namely, R × R n − with a singular metric(1) g β = | z | β − dz + dξ , β > . where z is in C (identified with R ) and ξ is in R n − . The geometry is nothing butthe product of a 2-dimensional cone with cone angle 2 πβ and R n − . Our attention is drawn to this space because of the recent study of conical K¨ahler geometryproposed by [14] and [5]. In the rest of this paper, we denote this space togetherwith the metric by X β .We shall restrict ourselves to the interior estimates only and hopefully, theboundary value problem will be discussed in the future. Hence by Schauder es-timate in R n , we mean the inequality k u k C ,α ( B / ) ≤ C ( n, α )( k f k C α ( B ) + k u k C ( B ) )if △ u = f on B . If f is in C k,α for k ∈ N , we can bound k u k C k +2 ,α by successivelytaking derivatives and applying the above C ,α estimate.Besides the classical proof of potential theory, there are many different proofs byCampanato [3], Peetre [10], Trudinger [15], Simon [13], Safonov [11, 12], Caffarelli[1, 2] and Wang [16]. We refer the readers to [16] for brief comments on these proofs.Many of the above proofs have important applications to the study of nonlinear(or even fully nonlinear) elliptic and parabolic equations. The proof given belowis motivated by the study of regularity problem on spaces with conical singularity.The ideas used here are related to the above mentioned proofs, for example, we shalluse a characterization of H¨older continuous function known to Campanato and weshall compare the solution to polynomials as Caffarelli did in [1, 2]. Moreover,the idea of pointwise Schauder estimate, due to Han [7, 8], is particularly usefuland effective for conical singularities. In the first part of this paper, we give aproof of the Schauder estimate on R n . The proof is by far not as simple as theabove mentioned ones. We need it, first to illustrate the basic idea of this paper,and second to prove some theorem that will be needed for the proof of Schauderestimate on X β .We start with an equivalent formulation of the H¨older space and the H¨older normon R n . Thanks to the Taylor expansion theorem, if u is C k,α in a neighborhood of x , then there exists a polynomial P x of degree k such that u ( x + h ) = P x ( h ) + O x ( | h | k + α ) , for | h | < δ x . It is natural to ask about the reverse: if a function u has the above expansionaround each point x in an open set Ω, is it true that u ∈ C k,α (Ω)? As shown bythe function u ( x ) = x sin 1 x , x ∈ R , which is not C , we can not expect a positive answer without putting more re-strictions to the expansion. It turns out that we need to ask the expansion to beuniform in the following sense: for some positive constants Λ and δ independent of x ∈ Ω, • the coefficients of the polynomial P x are bounded by Λ; • the constant in the definition of O x ( | h | k + α ) is bounded by Λ; • δ x > δ. The function u that satisfies the above assumption is then said to have uniformlybounded expansion , or UBE for simplicity. It will be shown in Section 2 that theset of UBE functions is the same as C k,α functions if one does not mind shrinkingthe domain a little, which is not a problem since we are only concerned with interiorestimate in this paper. This allows us to translate the classical Schauder estimateon R n into a theorem about UBE functions. CHAUDER ESTIMATES ON SMOOTH AND SINGULAR SPACES 3
A feature of the UBE characterization is that it seems to be a pointwise property.The proof of the Schauder estimate then reduces to showing that if △ u = f and f has an expansion of order k + α at 0 bounded by Λ in the above sense, then u hasan expansion at 0 up to order k + α + 2 bounded by a constant multiple of Λ andits own C norm. This is exactly what we do in Section 3.Similar to what Han did in [7, 8], an important step (Lemma 3.2) is the following:let f be O ( | x | k + α ), then there exists u that is O ( | x | k +2+ α ) satisfying △ u = f on B and sup B \{ } | u || x | k +2+ α ≤ C sup B \{ } | f || x | k + α . This was proved by using the potential in [7, 8] and it is our intention to avoid usingthe potential, because the analysis of Green’s function on X β could be complicated.Hence, we provide a proof of Lemma 3.2 using only the fact that harmonic functionsare polynomials. This argument generalizes well on X β .We then move on to the discussion of the singular space X β . If n is even andwe identify R × R n − with C × C n − , X β together with g β is also a (noncomplete)K¨ahler manifold. For β ∈ (0 ,
1) and α ∈ (0 , min n , β − o ), one can define C α function, using the Riemannian distance as usual. Donaldson [5] observed that ifone defines C ,αβ space by requiring the function u , its gradient (in Riemanniangeometric sense), its complex Hessian (in the above mentioned K¨ahler structure)to be C α , there is still a Schauder estimate. This estimate plays an important rolein the study of conical K¨ahler geometry. Its original proof due to Donaldson is bypotential theory and recently, there is another proof (without potential theory) ofthe same estimate by Guo and Song [6]. Moreover, there is also a parabolic versionof Donaldson’s estimate due to Chen and Wang [4].It is the main purpose of this paper to generalize the above Schauder estimatedue to Donaldson in two directions (with a different proof). First, we allow any β > β ∈ (0 , β > can be conic with integer β >
1. Applications along this line will be pursuedin a separate paper.This generalization is achieved by defining a new function space U q on X β (inSection 6). Here q is some positive number, taking the place of k + α for the usual C k,α space. Briefly speaking, the definition is a combination of two ideas: first werequire the function to be C k,α for q = k + α away from the singularities; second,we use the idea in the first part of this paper, namely, we use uniformly boundedexpansion up to order q to describe the regularity of f at the singular points;finally, we need to take care of the transition between the two point of views. See(H1-H3) in Definition 6.4 for details.We need to be clear about the type of expansion that is used near a singularpoint of X β , because the Taylor expansion is not available here. This is the topic YAOTING GUI AND HAO YIN of Section 6.1. On one hand, we need the expansion to be general so that it canbe used to describe the regularity of the solution that we care; on the other hand,we want the expansion to be very special so that it contains as much informationas possible. A choice of the expansion is a balance of the above two considerations.Our previous experience on the regularity issue of PDE’s on conical spaces [17, 18]suggests that the good choice depends both on the parameter β and on the typeof equations that we are interested in. In order not to distract the attention of thereaders, we give one particular choice in Section 6.1 by defining the T -polynomial.This choice is sufficient to present the idea of our proof and it is general enough tohave Donaldson’s Schauder estimate as a special case. Remark 1.1.
For future applications, we list a family of properties (P1-P4). Aslong as the definition of T -polynomial satisfies (P1-P4), the Schauder estimateholds. Our choice is justified by the following theorems. They are the main results ofthis paper. The first is the Schauder estimate. Here D is a countable and discreteset of positive numbers (see Section 6), ˆ B r is the metric ball in X β with the originas its center and U q is the new space of functions given by Definition 6.4. Theorem 1.2.
Let q > and q, q + 2 / ∈ D . Suppose f ∈ U q ( ˆ B ) and u is a boundedweak solution to △ β u = f. Then u is in U q +2 ( ˆ B ) and k u k U q +2 ( ˆ B ) ≤ C (cid:16) k u k C ( ˆ B ) + k f k U q ( ˆ B ) (cid:17) . The second is a comparison between the newly defined space U q and the Don-aldson’s C ,αβ space, whose definition we recall in Section 6.3. Theorem 1.3.
Suppose < β < and < α < min n , β − o . If we write X for C α ( U α , C ,αβ and U α respectively) and Y for U α ( C α , U α and C ,αβ respectively), then u ∈ X ( ˆ B ) implies that u ∈ Y ( ˆ B ) and k u k Y ( ˆ B ) ≤ C k u k X ( ˆ B ) . With Theorem 1.3, Theorem 1.2 implies the Schauder estimate of Donaldson in[5].The rest of the paper is organized as follows. In Section 2, we give a characteriza-tion of the C k,α space on R n using uniformly bounded expansion. This was knownto Campanato back to the 1960’s. We include a proof for completeness, which maybe omitted for a first reading. In Section 3, we prove the Schauder estimate on R n .These two sections form the first part of the paper. We then move on to the studyon X β . We first set up some notations and recall some easy facts about Poissonequations on X β in Section 4. In Section 5, we study bounded harmonic functionson X β , which is key to the proof of Theorem 1.2. In Section 6, we define the space U q and prove Theorem 1.3. In the final section, we prove Theorem 1.2. Acknowledgement.
The second author would like to thank Professor XinanMa for bringing the references [7, 8] to his attention.
CHAUDER ESTIMATES ON SMOOTH AND SINGULAR SPACES 5 H¨older space on R n In this section, we define a new space of functions that satisfy the uniform Taylorexpansion condition and prove that it is equivalent to the usual H¨older space C k,α .As remarked in the introduction, this result is not new and the proofs are includedfor completeness.We write B r for the ball of radius r centered at the origin in R n . Definition 2.1.
Suppose r and δ are two positive real numbers. For a function u defined on B r + δ , we say that it has uniformly bounded expansion (or UBE forsimplicity) up to order q on B r with scale δ if there exists some Λ > such thatfor any x ∈ B r and h ∈ B δ u ( x + h ) = p x ( h ) + O x ( q ) , where p x ( h ) is a polynomial of h whose coefficients (depending on x ) are uniformlybounded by Λ and O x ( q ) is also a function of h satisfying (2) | O x ( q )( h ) | ≤ Λ | h | q , ∀ | h | < δ, ∀ x ∈ B r . Related to the above definition, we define the following notations:(a) The infimum of all Λ satisfying (2) is denoted by [ O x ( q )] O q ,B δ , which is nothingbut sup h ∈ B δ ,h =0 | O x ( q ) || h | q . (b) The set of all functions that have UBE up to order q on B r with scale δ isdenoted by U q,δ ( B r ).(c) For u ∈ U q,δ ( B r ), the infimum of Λ in the above definition is defined to be thenorm of u , denoted by k u k U q,δ ( B r ) .It turns out that U q,δ is equivalent to the usual H¨older space C k,α with q = k + α in the following sense. Proposition 2.2.
Suppose q = k + α for some k ∈ N ∪ { } and α ∈ (0 , .(i) If u ∈ C k,α ( B r + δ ) , then u ∈ U q,δ ( B r ) and k u k U q,δ ( B r ) ≤ C ( δ, q, r, n ) k u k C k,α ( B r + δ ) . (ii) If u ∈ U q,δ ( B r + η ) for some η > , then u ∈ C k,α ( B r ) and k u k C k,α ( B r ) ≤ C ( η, q, δ, r, n ) k u k U q,δ ( B r + η ) . The rest of this section is devoted to the proof of this proposition. The first partfollows trivially from the Taylor theorem with integral remainder.The proof of (ii) is by induction and we assume without loss of generality that r = 1. The starting point of the induction is the observation that the claim holdstrivially true when k = 0. For k >
0, the expansion in Definition 2.1 implies that u is differentiable for each x ∈ B η . Therefore, the proof of Proposition 2.2 reducesto Claim 1: If u ∈ U q,δ ( B η ), then for i = 1 , · · · , n , some η ′ ∈ (0 , η ) and some δ ′ > ∂u∂x i ∈ U q − ,δ ′ ( B η ′ ) and (cid:13)(cid:13)(cid:13)(cid:13) ∂u∂x i (cid:13)(cid:13)(cid:13)(cid:13) U q − ,δ ′ ( B η ′ ) ≤ C ( δ, q, η, n ) k u k U q,δ ( B η ) . YAOTING GUI AND HAO YIN
For the proof of the claim, we recall some notations. Let ǫ = ( ǫ , · · · , ǫ n ) be amulti-index. For h ∈ R n , we write h ǫ = h ǫ · · · h ǫ n n . For some δ ′ > fix h = ( h , · · · , h n ) satisfying | h | < δ ′ and h i = 0 for all i . Given this h and a multi-index ǫ with | ǫ | < q + 2, wedefine the difference quotient operator P ǫ,h which maps a function defined on B η to a function defined on B η ′ as follows. For ǫ = (0 , · · · , , · · · , i -th position, P ǫ,h [ f ]( y ) := f ( y + h i e i ) − f ( y ) h i where e i is the natural basis of R n . For ǫ = ǫ ′ + e i , we define P ǫ,h [ f ]( y ) = P ǫ ′ ,h [ f ]( y + h i e i ) − P ǫ ′ ,h [ f ]( y ) h i . Since | ǫ | is bounded by q + 2, by choosing δ ′ small (say, δ ′ = η − η ′ q +2) , so that | h | small) depending on η and η ′ , P ǫ,h [ f ] is a function defined on B η ′ . Lemma 2.3. P ǫ,h is well defined, i.e., it is independent of the order of inductionin its definition. Moreover, we have (3) P ǫ,h [ f ]( y ) = 1 h ǫ X ≤ γ ≤ ǫ ( − | γ | +1 C γǫ f ( y + γh ) . Here(a) γ is a multi-index and ≤ γ ≤ ǫ means that for each i = 1 , · · · , n , we have ≤ γ i ≤ ǫ i ;(b) γh = ( γ h , · · · , γ n h n ) ;(c) C γǫ := C γ ǫ · · · C γ n ǫ n . The proof is very elementary and omitted. We shall also need the followinglemma about combinatorics.
Lemma 2.4.
For any multi-index σ , set Q σǫ = X ≤ γ ≤ ǫ ( − | γ | +1 C γǫ γ σ . If σ i < ǫ i for some i = 1 , · · · , n , then Q σǫ = 0 . As a consequence, if we denote themulti-index ( γ − , · · · , γ n − by γ − , then for the same σ and ǫ above Q σǫ = X ≤ γ ≤ ǫ ( − | γ | +1 C γǫ ( γ − σ . Proof.
We only prove the first claim of the lemma. An easy observation is that Q σǫ = − n Y i =1 X ≤ γ i ≤ ǫ i ( − γ i C γ i ǫ i γ σ i i . To show the product is zero, it suffices to show that the i -th factor is zero if σ i < ǫ i .Consider the polynomial f ( y i ) = (1 − y i ) ǫ i = X ≤ γ i ≤ ǫ i C γ i ǫ i ( − y i ) γ i . CHAUDER ESTIMATES ON SMOOTH AND SINGULAR SPACES 7
For each j = 0 , · · · , σ i , since j < ǫ i , we have( ∂ y i ) j f | y i =1 = 0 , which gives X j ≤ γ i ≤ ǫ i ( − γ i C γ i ǫ i γ i !( γ i − j )! = 0 . By setting F ( γ ; j ) = γ · ( γ − · · · · · ( γ − j + 1) , we have(4) X ≤ γ i ≤ ǫ i ( − γ i C γ i ǫ i F ( γ i ; j ) = 0 .F ( γ ; j ) is a polynomial of γ of degree j . Since σ i < ǫ i , γ σ i i is then a linear com-bination of F ( γ i ; j ), j = 0 , · · · , ǫ i . With (4), this concludes the proof of Lemma2.4. (cid:3) Remark 2.5.
The above lemma is related to the fact that difference quotient killspolynomials.
Now, we come back to the proof of Claim 1, which consists of two steps. In thefirst step, we restrict ourselves to a special type of h satisfying(5) | h i | ≥ √ n | h | , for i = 1 , · · · , n. We denote the set of such h by Ω. The reason will be clear in a minute. Definition 2.6.
Suppose r and δ are two positive real numbers. A function u : B r + δ → R is said to have partially uniformly bounded expansion with respectto Ω , up to order q and with scale δ , if the assumptions in Definition 2.1 hold with h ∈ B δ replaced by h ∈ Ω ∩ B δ .The space of these functions is denoted by U p,δ Ω ( B r ) and its norm by k·k U p,δ Ω . The goal of the first step is the following claim, which is a partial version ofClaim 1.
Claim 2: If u ∈ U q,δ Ω ( B η ), then for i = 1 , · · · , n , some η ′ ∈ (0 , η ) and some δ ′ > ∂u∂x i ∈ U q − ,δ ′ Ω ( B η ′ ) and (cid:13)(cid:13)(cid:13)(cid:13) ∂u∂x i (cid:13)(cid:13)(cid:13)(cid:13) U q − ,δ ′ Ω ( B η ′ ) ≤ C ( δ, q, η, n, Ω) k u k U q,δ Ω ( B η ) . In the second step, we shall derive Claim 1 from Claim 2. For the proof of Claim2, recall that(6) P ǫ,h [ f ]( x ) = 1 h ǫ X ≤ γ ≤ ǫ ( − | γ | +1 C γǫ f ( x + γh ) . Using the partial UBE assumption, we may expand f ( x + γh ) into a polynomialcentered at x ,(7) f ( x + γh ) = f ( x ) + X | σ | We may also use the expansion centered at x + h to get(8) f ( x + γh ) = f ( x + h ) + X | σ | In summary, we have proved that ∂f∂x i is partially UBE with respect to A Ω upto order q − 1. Now, we take orthogonal matrices A , · · · , A l such that R n = A Ω [ · · · [ A l Ω . Then the partial UBE conditions for each k combine to be UBE if we can justifythat ˜ a ǫ,A k ( x )is independent of k = 1 , · · · , l . This is true because we can choose A k so that A k Ω ∩ A k Ω is either empty or has non-empty interior.3. A proof of the Schauder estimates on R n We give another proof to the well-known interior Schauder estimate in this sec-tion.Given Proposition 2.2, it suffices to prove Theorem 3.1 (Schauder estimate) . Suppose that f ∈ U q,δ ( B ) for some q = k + α with k ∈ N and α ∈ (0 , . If u is a bounded solution to △ u = f on B , then u liesin U q +2 ,δ ( B − δ ) and (12) k u k U q +2 ,δ ( B − δ ) ≤ C ( n, q, δ )( k f k U q,δ ( B ) + k u k C ( B ) ) . For the proof, we need the following lemma (see Lemma 2.1 in [8]), Lemma 3.2. If f : B r → R is O ( q ) and q is not an integer, then there exists some u ∈ O (2 + q ) satisfying △ u = f on B r . Moreover, for some C > depending on n, q, r , [ u ] O q +2 ,B r ≤ C [ f ] O q ,B r . Before the proof of Lemma 3.2, we show how Theorem 3.1 follows from it. Forany x ∈ B − δ fixed, there exists a polynomial p f,x ( h ) (of order k ) such that f ( x + h ) = p f,x ( h ) + e f,x ( h ) on B δ , where e f,x is O ( q ). By the definition, all the coefficients of p f,x are bounded by k f k U q,δ ( B ) . Hence there exists another polynomial p u,x ( h ) (of order k + 2, notunique) whose coefficients are bounded by C ( n, q ) k f k U q,δ ( B ) such that △ p u,x = p f,x on B δ . By Lemma 3.2, there is some e u,x ∈ O ( q + 2) such that △ e u,x = e f,x on B δ .Therefore,(13) △ ( u ( x + h ) − p u,x ( h ) − e u,x ( h )) = 0 on B δ . Moreover, also by Lemma 3.2,[ e u,x ] O q +2 ,B δ ≤ C [ e f,x ] O q ,B δ . In particular, k e u,x k C ( B δ ) is bounded by a multiple of k f k U q,δ ( B ) .By (13), u ( x + h ) − p u,x ( h ) − e u,x ( h ) is a bounded harmonic function that isbounded on B δ by the right hand side of (12). By well-known properties of harmonicfunctions, u ( x + h ) − p u,x ( h ) − e u,x ( h ) = ˜ p ( h ) + ˜ e ( h ) , where ˜ p ( h ) is a polynomial of order k + 2 and ˜ e ( h ) is O q +2 ,B δ and again, thecoefficients of ˜ p and [˜ e ] O q +2 ,Bδ is bounded by the right hand side of (12).By setting ˜ p u,x = p u,x + ˜ p and ˜ e u,x = e u,x + ˜ e , we have u ( x + h ) = ˜ p u,x ( h ) + ˜ e u,x ( h ) . Notice that the constants in the above argument are independent of x ∈ B − δ ,hence we have verified that u ∈ U q +2 ,δ ( B − δ ) with the desired bound.The rest of this section is devoted to the proof of Lemma 3.2. Without loss ofgenerality, we assume δ = 1.We decompose B into the union of a sequence of annulus A l := B − l \ B − l − for l = 0 , , , · · · . Set f l = f · χ A l , where χ A l is the characteristic function of A l . For simplicity, in the rest of thisproof, we write Λ f for [ f ] O q ,B δ . By definition, | f l ( x ) | ≤ Λ f | x | q χ A l ≤ Λ f − lq on R n . Let w l be the unique solution (vanishing at the infinity) to the Poisson equation △ w l = f l on R n . Obviously, w l is harmonic in the complement of A l . Moreover,we have the uniform bound(14) sup R n | w l | ≤ C Λ f − l ( q +2) . Since w l is harmonic in B − l − , it is a converging power series there and let P l bethe polynomial that is the part of this series with order strictly smaller than q + 2,namely, if w l ( x ) = X ǫ a ǫ x ǫ then(15) P l ( x ) = X | ǫ | There exists a constant C q depending on n and q such that(i) on B − l − , | u l ( x ) | ≤ C q Λ f (1+[ q ] − q ) l | x | [ q ]+3 ; (ii) on A l , | u l ( x ) | ≤ C q Λ f − ( q +2) l CHAUDER ESTIMATES ON SMOOTH AND SINGULAR SPACES 11 or equivalently | u l ( x ) | ≤ C q Λ f | x | q +2 ; (iii) on B \ B − l , | u l ( x ) | ≤ C q Λ f − l ( q +2) + X | ǫ | First, we estimate P l ( x ) in B − l as follows | P l ( x ) | ≤ X | ǫ | With these preparations, we claim that(18) u ( x ) = ∞ X l =0 u l ( x )converges on B and gives the desired solution u in Lemma 3.2.Since w l is harmonic in a neighborhood of 0 ∈ R n , P l defined in (15) is a harmonicpolynomial on the entire R n . As a consequence, △ u l = f l on R n . Hence, to show Lemma 3.2, it suffices to prove(19) ∞ X l =0 | u l | ( x ) ≤ C q Λ f | x | q +2 on B , which not only implies the convergence of (18), but also gives the expected boundof u in Lemma 3.2. For each x ∈ B \ { } , all but finitely many u l ’s are harmonic ina neighborhood of x , hence, the convergence is smooth and u satisfies the Poissonequation △ u = f . For each fixed x ∈ B \ { } , let l be given by the condition that x ∈ A l . In other words, | x | < − l ≤ | x | .To estimate the left hand side of (19), we compute l − X l =0 | u l | ( x ) ≤ C q Λ f l − X l =0 (1+[ q ] − q ) l ! | x | [ q ]+3 ≤ C q Λ f (1+[ q ] − q ) l | x | [ q ]+3 ≤ C q Λ f | x | q +2 , where we used (i) in Lemma 3.3. (ii) of Lemma 3.3 implies | u l ( x ) | ≤ C q Λ f | x | q +2 . Similarly, using (iii) of Lemma 3.3, we have X l>l | u l | ( x ) ≤ C q Λ f X l>l − l ( q +2) + X | ǫ | Preliminaries about X β In this section, we first define some notations and then recall some basic prop-erties about the Poisson equation on X β whose proofs are omitted.4.1. Notations. Aside from the natural coordinates ( x, y, ξ ) of X β = R × R n − ,there is a global coordinate system ( ρ, θ, ξ ) on the smooth part of X β given by ρ = 1 β r β , x = r cos θ, y = r sin θ. In terms of ( ρ, θ, ξ ), the metric g β in (1) becomes g β = dρ + ρ β dθ + dξ . Here is a list of notations that are useful.(i) The singular set, denoted by S , corresponds to { ρ = 0 } and can be parametrizedby ξ .(ii) d ( x, y ) is the Riemannian distance (given by g β ) between x and y in X β .(iii) S δ is the set of points whose distance to S is smaller than δ > δ = X β \ S δ .(v) For a point x ∈ X β , ˆ B r ( x ) is the set of points whose distance to x is smallerthan r .(vi) x is the origin of X β , i.e. the point with ρ = 0 and ξ = 0. ˆ B ( x ) is the unitball, which for simplicity is often denoted by ˆ B .(vii) d ( x, x ) is usually denoted by d ( x ). CHAUDER ESTIMATES ON SMOOTH AND SINGULAR SPACES 13 (viii) ˜ x is the point in X β with ρ = 1, θ = 0 and ξ = 0. Then there is a constant c β depending only on β such that ˆ B c β (˜ x ) is topologically a ball and thatthe restriction of g β to it is comparable with the flat metric on B c β (0) ⊂ R n .Throughout this paper, we fix this c β and denote ˆ B c β (˜ x ) by ˜ B , which alsoserves as a unit ball. We also write ˜ B r for ˆ B c β r (˜ x )For each x ∈ X β \ S , there is a natural scaling map Ψ x which maps ˆ B c β ρ ( x ) ( x )to ˜ B , where ρ ( x ) is the ρ coordinate of x , i.e. the distance to S . If x = ( ρ , θ , ξ ),then Ψ x ( ρ, θ, ξ ) = ( ρρ , θ − θ , ξ − ξ ρ ) . Here θ − θ is understood as the natural minus operation of the group S . Thescaling Ψ x induces a pushforward of functions, which we denote by S x . Moreprecisely, if u is a function defined on ˆ B c β ρ ( x ) ( x ), then S x ( u )( y ) = u (Ψ − x ( y )) ∀ y ∈ ˜ B. Basics on the Poisson equation. We collect a few basic facts about PDEson X β .We start with an observation that is known and utilised by many authors. Con-sider another copy of R n , whose coordinates are given by ( w, v, ξ ), where w, v ∈ R and ξ ∈ R n − . The Euclidean metric on R n is given by dw + dv + dξ . One can check by direct computation that the mapping( ρ, θ, ξ ) ( ρ cos θ, ρ sin θ, ξ ) ∈ R n is bi-Lipschitzian from X β to R n . Hence, the Sobolev space W , ( X β ) ( L p ( X β )) isthe same set of functions as W , ( R n ) ( L p ( R n )). Moreover, the Sobolev inequalityon X β holds with a different constant.One can prove the following by the usual variation method and Moser iteration.Please note that we state it in a scaling invariant form. Lemma 4.1. Let f be an L ∞ function supported in ˆ B r . There exists a solution u ∈ W , loc ( X β ) ∩ L ∞ ( X β ) to the Poisson equation △ β u = f with the bound k u k L ∞ ( X β ) ≤ Cr k f k L ∞ ( ˆ B r ) . Bounded harmonic functions on X β Suppose that u is a bounded harmonic function on ˆ B ⊂ X β , i.e., △ β u = 0 . We discuss in this section the regularity of u in ˆ B . Before our discussion on X β ,we recall that if u is a harmonic function on B ⊂ R n , then we can bound anyderivatives of u on B / by the C norm of u on B . Equivalently, there is theTaylor expansion of u at 0,(20) u ( x ) = X | σ |≤ k a σ x σ + O ( | x | k +1 ) , where σ is a multi-index and the a σ ’s and the constant in the definition of O ( | x | k +1 )are bounded by the C norm of u on B . The first goal of this section is to provea generalization of this result for harmonic functions on X β . More precisely, weprove an analog of (20) for harmonic function on X β and provide estimates for thecoefficients in the expansion.Although it is very likely that the expansion we prove below (see Proposition5.2) gives a converging series if we trace the bound for coefficients in the expansion,we do not pursue it here. However, we need the fact that the approximationpolynomial of a harmonic function (as given in Proposition 5.2) is still harmonicas in the case of R n . This is the second goal of this section and is contained in thesecond subsection.5.1. The expansion. The first thing we need to do is to generalize the conceptof polynomial (or monomial) used in the Taylor expansion. For the time being, weconcentrate on the expansion describing the regularity of harmonic functions. Interms of the polar coordinates ( ρ, θ, ξ ), the (monic) X β -monomials are defined tobe ρ j + kβ cos kθξ σ , ρ j + kβ sin kθξ σ , for each multi-index σ of R n − and any j, k ∈ N ∪ { } . Then sum 2 j + kβ + | σ | iscalled the degree of the X β -monomial and an X β -polynomial is a finite linearcombination of monomials. Remark 5.1. Throughout this paper, unless stated otherwise, the range of j, k and σ in a summation is understood as above. With these definitions, we can now state the main result of this subsection. Proposition 5.2. If u is a bounded harmonic function in ˆ B , then for any q > and d ( x ) < / , (21) u ( x ) = X j + kβ + | σ | Suppose that u is a weak harmonic function on ˆ B . Then for anymulti-index σ , ∂ σξ u is also a weak harmonic function on ˆ B and (cid:13)(cid:13) ∂ σξ u (cid:13)(cid:13) C ( ˆ B ) ≤ C ( | σ | ) k u k C ( ˆ B ) . When ξ is fixed, we write u ( ξ ) for u as a function of ρ, θ . The regularity of u ( ξ )is essentially a two dimensional problem that has been studied in [17]. The proofin [17] yields Lemma 5.4. For | ξ | < / , (22) u ( ξ ) = X j + kβ For any multi-index σ , (23) ∂ σξ u ( ξ ) = X j + kβ Remark 5.5. Please note the difference between a j,k,σ in this lemma and a σj,k in Proposition 5.2. Lemma 5.4 does not claim any relation between a j,k ( ξ ) and a j,k,σ ( ξ ) . It will be clear in a minute that a j,k,σ ( ξ ) = ∂ σξ a j,k . The same applies to b j,k and b j,k,σ . Given Lemma 5.4 and Lemma 5.3, we claim that: Claim: the a j,k ( ξ ) and b j,k ( ξ ) in (22) are smooth functions of ξ on {| ξ | < / } .Moreover, for any multi-index σ ,(24) sup | ξ | < / (cid:12)(cid:12) ∂ σξ a j,k (cid:12)(cid:12) + (cid:12)(cid:12) ∂ σξ b j,k (cid:12)(cid:12) ≤ C ( j, k, σ ) k u k C ( ˆ B ) . Proposition 5.2 follows from the claim, because we can expand a j,k ( ξ ) and b j,k ( ξ )(in (22)) into the sum of a Taylor polynomial of ξ and a remainder O ( | ξ | q ). Thenthe O ( d q ) in Proposition 5.2 is the sum of O ( ρ q ) in (22), a sum of X β -monomialswith degree no less than q and ρ j + kβ O ( | ξ | q ) , j + kβ < q in the expansion of a j,k ( ξ ) and b j,k ( ξ ).For the proof of the claim, we start with a , ( ξ ). By (22), a , ( ξ ) = lim ρ → u. By Lemma 5.3 (cid:12)(cid:12)(cid:12) ∂ σξ u (cid:12)(cid:12)(cid:12) is uniformly bounded for any | ξ | < , hence the convergenceis in fact in C l for any l , then our claim for a , follows.Now for a fixed σ , (23) gives a , ,σ ( ξ ) = lim ρ → ∂ σξ u. As before, since lim ρ → u is in C l topology for any l > 0, we havelim ρ → ∂ σξ u = ∂ σξ ( lim ρ → u ) , which implies that(25) a , ,σ ( ξ ) = ∂ σξ a , ( ξ ) . Since our claim for a , is proved, we may assume that it is zero at the verybeginning of the proof by replacing u with u − a , ( ξ ). Thanks to (25), a consequenceof this assumption is that(26) a , ,σ ≡ , ∀ σ. Next, suppose that 2 j + k β is the next (smallest) nonzero power in the expansion,namely, j = 1 and k = 0 when β < / j = 0 and k = 1 when β ≥ / (26)and (23) imply ∂ σξ uρ j + k β is uniformly bounded (w.r.t. ξ ) by a constant depending on σ . Hence, the conver-gence a j ,k ( ξ ) = lim ρ → π Z π uρ j + k β cos k θdθ is uniform in any C l topology. This implies that our claim for a j ,k holds, whichenables us to assume a j ,k ( ξ ) = 0 at the beginning. Notice that we also have (bythe same reason) ∂ σξ a j ,k ( ξ ) = a j ,k ,σ ( ξ ) , so that we can repeat the argument to prove the claim for any a j,k and b j,k .5.2. The approximating X β -polynomial is still harmonic. We prove in thissection Proposition 5.6. Suppose that u satisfies the assumptions of Proposition 5.2 andtherefore has an expansion given by (21) . Then (27) △ β X j + kβ + | σ | To prove this proposition, it suffices to justify that the △ β of the remainder O ( d q ) (in (21)) is an O ( d q − ). In fact, letting T be the X β -polynomial in (27) (orequivalently, in (21)), we have(28) △ β u = △ β T + O ( d q − ) = 0 , if our claim for O ( d q ) holds. One can check that △ β T is an X β -polynomial ofdegree smaller than q − 2, then it vanishes due to (28).For the claimed property of O ( d q ), recall that, by the proof of Proposition 5.2,it is the sum of(1) ρ j + kβ cos kθξ σ for 2 j + kβ + | σ | > q ; (there is a similar term with sin replacingcos)(2) ρ j + kβ cos kθO ( | ξ | q );(3) the O ( ρ q ) term in (22).It is trivial that the desired property is true for functions in (1). For (2), notice that O ( | ξ | q ) comes from the Taylor expansion of a j,k ( ξ ) and b j,k ( ξ ) that are smooth in ξ and hence ∂ ξ O ( | ξ | q ) is O ( | ξ | q − ). Given this, it is straightforward to check that △ β (cid:16) ρ j + kβ cos kθO ( | ξ | q ) (cid:17) = O ( d q − ) , where d = ρ + | ξ | . Think about β = 1 / 2. Annoying discussion. CHAUDER ESTIMATES ON SMOOTH AND SINGULAR SPACES 17 It remains to study the remainder O ( ρ q ) in (22). In the proof of Proposition 5.2,we have shown that a j,k ( ξ ) and b j,k ( ξ ) in (22) are smooth functions and for anymulti-index σ , ∂ σξ a j,k ( ξ ) = a j,k,σ ( ξ ) and ∂ σξ b j,k ( ξ ) = b j,k,σ ( ξ ) . By comparing (22) and (23), we notice that any ξ -derivative of the O ( ρ q ) in (22)is still a O ( ρ q ).Since u is a harmonic function, it is not difficult to prove by the interior Schauderestimate and Lemma 5.3 that(29) sup ˆ B \S (cid:12)(cid:12) ( ρ∂ ρ ) k ( ∂ θ ) k u (cid:12)(cid:12) ≤ C ( k , k ) . Using equation (22), the estimate (29) holds for O ( ρ q ), which implies that (cid:18) ∂ ρ + 1 ρ ∂ ρ + 1 β ρ ∂ θ (cid:19) O ( ρ q ) = O ( ρ q − ) . This concludes the proof of Proposition 5.6 by noticing that ρ ≤ d and O ( ρ q − ) is O ( d q − ). (cid:3) Generalized H¨older space on X β In this section, we define the X β counterpart of C k,α function on R n and compareit with Donaldson’s C ,αβ space when 0 < β < < α < min n β − , o .6.1. Formal discussion. The basic idea of our definition as illustrated by Sec-tion 2 is to use generalized ‘polynomial’ to describe the regularity near a singularpoint. In this subsection, we are concerned with the question of what is the correct‘polynomial’ for X β .Recall that in Proposition 5.2, for the expansion of harmonic functions, we de-fined X β -polynomials, which are finite linear combinations of ρ j + kβ cos kθξ σ , ρ j + kβ cos kθξ σ where j, k ∈ N ∪ { } and σ is a multi-index of R n − . X β -polynomials are not enough for the study of more complicated PDE solutions,because the product of two X β -polynomials is not X β -polynomial. This motivatesthe following definition. Definition 6.1. Suppose that j, k, m ∈ N ∪ { } satisfy k − m ∈ N ∪ { } and σ is amulti-index of dimension n − . The functions ρ j + kβ cos mθξ σ , ρ j + kβ sin mθξ σ are called (monic) T -monomials of degree j + kβ + | σ | . A T -polynomial is afinite linear combination of T -monomials. It is elementary to check that the product of T -polynomials is T -polynomial. Remark 6.2. Our previous experience in PDE’s with conical singularity suggeststhat the regularity of solutions near a singular point (like the cone singularity in X β ) depends both on the singularity of space and on the type of PDE that we areworking with. While the above definition of T -polynomial suffices (see [17] ) for the study ofnonlinear equations like △ β u = F ( u ) . It is not enough for the conical complex Monge-Ampere equation studied in [18] . Tominimize the difficulty of understanding this paper, we refrain from working in thatgenerality. Instead, we will list below a family of properties that should be satisfiedby T -polynomials. It will be clear in the proof that follows, if these properties hold,the main result of this paper remains true for other definitions of T -polynomials. Our definition of T -polynomial satisfies a family of properties, which we sum-marize in the form of a lemma. Lemma 6.3. (P1) Let f be a monic T -monomial of degree q . For any q ′ < q , l ∈ N ∪ { } and any point x ∈ X β satisfying ρ ( x ) < / and ξ ( x ) = 0 , k S x ( f ) k C l ( ˜ B ) ≤ C ( q, l, q ′ ) ρ ( x ) q ′ . (P2) Let f be a monic T -monomial of degree q . There is a constant dependingonly on q and δ > such that for any l ∈ N ∪ { } , k f k C l (Ω δ ∩ ˆ B ) ≤ C ( q, l, δ ) . (P3) Let f be a T -polynomial of degree q . There is a constant C depending onlyon q such that k f k C l ( ˜ B / ( z )) ≤ C ( l, q ) k f k C ( ˜ B / ( z )) for any l ∈ N ∪ { } and all z ∈ ˜ B / .(P4) Let f be any T -polynomial of degree q . There exists a T -polynomial u ofdegree q + 2 (not unique) such that △ β u = f. Moreover, the coefficients of u are bounded by a multiple (depending on q ) of thecoefficients of f .Proof. (P1) and (P2) can be checked by direct computation. Notice that (P1) istrue even if q ′ = q and we have stated it in this weaker form, because we have inour mind monomials involving log terms (for example ρ log ρ ), which may appearin applications.For (P3), recall that the number of (monic) T -monomials with degree no morethan q is finite and that they are linearly independent functions so that the C norm of a T -polynomial on any open set bounds every coefficient.The proof of (P4) is an induction argument based on the fact that △ β ρ γ +2 cos mθξ σ is a linear combination of ρ γ cos mθξ σ and a T -polynomial whose order in ξ issmaller than | σ | by 2. (cid:3) The definition. In this section, we define the generalized H¨older space U q .We assume that q is any positive number that is not in D = (cid:26) j + kβ | j, k ∈ N ∪ { } (cid:27) , which is the set of degrees of T -monomials.The overall idea is to require that u is C k,α in the usual sense in Ω δ ∩ ˆ B andfor each x ∈ S ∩ ˆ B , u has some uniform expansion (using T -polynomials) in a ballof size δ . CHAUDER ESTIMATES ON SMOOTH AND SINGULAR SPACES 19 Definition 6.4. Suppose that q = k + α for some k ∈ N ∪ { } and α ∈ (0 , . u issaid to be in U q ( ˆ B ) if and only if there is some Λ > such that(H1) u is C k,α on ˆ B ∩ Ω δ and k u k C k,α ( ˆ B ∩ Ω δ ) ≤ Λ . (H2) For each x ∈ S ∩ ˆ B , there is a T -polynomial P x such that (i) the degreeof P x is smaller than q and the coefficients of P x is bounded by Λ and (ii) u ( x + y ) = P x ( y ) + O ( d ( y ) q ) , ∀ y ∈ ˆ B δ , where O ( d ( y ) q ) above satisfies | O ( d ( y ) q ) | ≤ Λ d ( y ) q . (H3) For each x ∈ S δ ∩ ˆ B , let ˜ x be the projection of x to S . (30) k S x ( u − P ˜ x ( · − ˜ x )) k C k,α ( ˜ B / ) ≤ Λ ρ ( x ) q . The infimum of Λ such that (H1-H3) hold for some u ∈ U q ( ˆ B ) is defined to bethe norm of u , denoted by k u k U q ( ˆ B ) .(H1-H2) is in line with the overall idea mentioned before Definition 6.4. (H3) isnecessary to describe the behavior of the functions in U q ( ˆ B ) at those points thatare closer and closer to the singular set. Definition 6.4 may look unusual, but it willbe justified when we show that Donaldson’s C ,αβ is a special case of U q in Section6.3 and when we prove a Schauder estimate in Section 7.Several remarks are helpful in understanding the defnition. Remark 6.5. We need to check that for < q < q ( q , q / ∈ D ), U q ( ˆ B ) ⊂U q ( ˆ B ) , which is not totally trivial from the definition above. To see this, let u bein U q ( ˆ B ) , it suffices to check (H2) and (H3) in the definition of U q ( ˆ B ) . For(H2), let x ∈ S ∩ ˆ B , then there is some T -polynomial P x such that u ( x + y ) = P x ( y ) + O ( d ( y ) q ) ∀ y ∈ ˆ B δ . Let P ′ x be the part of P x that involves only monomials of degree smaller than q .Let Q x = P x − P ′ x . Obviously, u ( x + y ) = P ′ x ( y ) + O ( d ( y ) q ) ∀ y ∈ ˆ B δ . For (H3), it suffices to check that for each x ∈ S δ ∩ ˆ B with ˜ x being its projectionto S , Q ˜ x satisfies (31) k S x ( Q ˜ x ( · − ˜ x )) k C k ,α ( ˜ B / ) ≤ C Λ ρ ( x ) q , if q = k + α . In fact, Q ˜ x is a finite linear combination of T -monomials of degreestrictly larger than q and (H2) implies that the coefficients of the combination arebounded by Λ . Hence, (31) follows from (P1). Remark 6.6. Due to (P2) above, the definition of U q ( ˆ B ) is independent of theconstant δ . A different choice of δ yields an equivalent norm k·k U q . Remark 6.7. Finally, we remark that (30) in (H3) of Definition 6.4 can be replacedby (32) k S x ( u − P ˜ x ( · − ˜ x ))( z ) k C k,α ( ˜ B / ) ≤ Λ ρ ( x ) q . This may look plausible, however, we give a detailed proof, because we shallneed it explicitly in the proof of Theorem 7.1. In the sequel, we shall use (H3’) forthe assumption (H3) with (30) replaced by (32). Assume that we have a function u satisfying (H1), (H2) and (H3’).Since (H3) only matters when ρ ( x ) is small, we fix x ∈ S δ/ ∩ ˆ B . By (H2),(33) sup ˜ B | S x ( u − P ˜ x ( · − ˜ x )) | = sup ˆ B cβρ ( x ) ( x ) | u − P ˜ x ( · − ˜ x ) | ≤ C Λ ρ ( x ) q . For any y ∈ ˆ B c β ρ ( x ) / ( x ) = ˜ B / ( x ), (H3’) implies that k S y ( u − P ˜ y ( · − ˜ y ))( z ) k C k,α ( ˜ B / ) ≤ Λ ρ ( y ) q ≤ C Λ ρ ( x ) q . By the definition of S x and S y , S x ( u − P ˜ y ( · − ˜ y ))( z ) = ( u − P ˜ y ( · − ˜ y ))(Ψ − x ( z ))= ( u − P ˜ y ( · − ˜ y ))(Ψ − y ◦ Ψ y ◦ Ψ − x ( z ))= S y ( u − P ˜ y ( · − ˜ y ))(Ψ y ◦ Ψ − x ( z )) . (34)If z = ( ρ z , θ z , ξ z ), we compute explicitlyΨ y ◦ Ψ − x ( z ) = Ψ y ( ρ x ρ z , θ x + θ z , ξ x + ρ x ξ z )= ( ρ x ρ y ρ z , θ z + θ x − θ y , ρ x ξ z + ξ x − ξ y ρ y ) . (35)Obviously when z = Ψ x ( y ) ∈ ˜ B / , Ψ y ◦ Ψ − x ( z ) = ˜ x . Since y ∈ ˆ B c β ρ ( x ) / ( x )(so that 1 / < ρ x /ρ y < y ◦ Ψ − x is a Lipschitz map with Lipschitzconstant smaller than 2, which implies thatΨ y ◦ Ψ − x ( ˜ B / ( z )) ⊂ ˜ B / . Noticing that the map Ψ y ◦ Ψ − x is uniformly bounded (for all y ∈ ˜ B / ( x )) in any C k norm on ˜ B / ( z ), (34) implies that(36) k S x ( u − P ˜ y ( · − ˜ y )) k C k,α ( ˜ B / ( z )) ≤ C Λ ρ ( x ) q . We claim that(37) k S x ( u − P ˜ x ( · − ˜ x )) k C k,α ( ˜ B / ( z )) ≤ C Λ ρ ( x ) q . (H3) follows from (37) because y is any point in ˜ B / ( x ), z = Ψ x ( y ) and˜ B / ⊂ [ y ∈ ˜ B / x ) ˜ B / ( z ) . By comparing (36) and (37), the proof of the above claim reduces to(38) k S x ( P ˜ y ( · − ˜ y ) − P ˜ x ( · − ˜ x )) k C k,α ( ˜ B / ( z )) ≤ C Λ ρ ( x ) q . Since S x ( P ˜ y ( · − ˜ y ) − P ˜ x ( · − ˜ x )) is a T -polynomial with degree smaller than q , (P3)implies that (38) follows from(39) k S x ( P ˜ y ( · − ˜ y ) − P ˜ x ( · − ˜ x )) k C ( ˜ B / ( z )) ≤ C Λ ρ ( x ) q . Recall that z = Ψ x ( y ), hence Ψ x maps ˜ B / ( z ) to ˜ B / ( y ) ⊂ ˜ B ( x ). (H2) impliesrespectively (cid:13)(cid:13) u − P ˜ y ( ·− ˜ y ) (cid:13)(cid:13) C ( ˜ B / ( y )) ≤ C Λ ρ ( y ) q ≤ C Λ ρ ( x ) q CHAUDER ESTIMATES ON SMOOTH AND SINGULAR SPACES 21 and (cid:13)(cid:13) u − P ˜ x ( ·− ˜ x ) (cid:13)(cid:13) C ( ˜ B ( x )) ≤ C Λ ρ ( x ) q . Then (39) follows from a combination of the above two inequalities and the factthat ˜ B / ( y ) ⊂ ˜ B ( x ).6.3. Comparison with Donaldson’s C ,αβ space. Donaldson’s definition re-quires both 0 < β < α < min n β − , o , which we assume in this subsectiononly. It is the purpose of this subsection to show that the space U α is equivalentto C ,αβ when the above restriction to β and α applies.In this case, the only monic T -monomials whose order is smaller than 2 + α are(40) 1 , ρ β cos θ, ρ β sin θ, ρ , ξ, ξ . First, let’s recall the definition of C ,αβ space defined by Donaldson. It is definedto be the set of functions satisfying(D1) u is in C α ( ˆ B );(D2) ∂ ρ u , ρ ∂ θ u and ∂ ξ u are in C α ( ˆ B );(D3) ∂ ξ u , ∂ ξ ∂ ρ u , ρ ∂ ξ ∂ θ u and ˜ △ β u are in C α ( ˆ B ), where ˜ △ β = ∂ ρ + ρ ∂ ρ + β ρ ∂ θ is the Laplacian on the cone surface of cone angle 2 πβ .Here C α ( ˆ B ) is the space of H¨older continuous functions with respect to the distance d of X β . Moreover, the C ,αβ norm is the sum of all C α norms mentioned above.We start the comparison with the following lemma, Lemma 6.8. (i) If u is C α ( ˆ B ) , then u is in U α ( ˆ B ) with (41) k u k U α ( ˆ B ) ≤ C k u k C α ( ˆ B ) ; and (ii) if u is in U α ( ˆ B ) , then u is in C α ( ˆ B ) with (42) k u k C α ( ˆ B ) ≤ C k u k U α ( ˆ B ) . Proof. (i) We notice that (H1) is trivial and (H2) is just the definition of H¨oldercontinuous if we choose P x to be the constant u ( x ). For x ∈ S δ ∩ ˆ B and any z ∈ ˜ B / , since P ˜ x is the constant u (˜ x ), we have S x ( u − P ˜ x ( · − ˜ x ))( z ) = ( u − P ˜ x ( · − ˜ x ))(Ψ − x ( z ))= u (Ψ − x ( z )) − u (˜ x ) . For any z ∈ ˜ B / , the distance from Ψ − x ( z ) to ˜ x is at most 2 ρ ( x ) and hence(43) k S x ( u − P ˜ x ( · − ˜ x )) k C ( ˜ B / ) ≤ C k u k C α ( ˆ B ) ρ ( x ) α . For any z , z in ˜ B / , | S x ( u − P ˜ x ( · − ˜ x ))( z ) − S x ( u − P ˜ x ( · − ˜ x ))( z ) | = (cid:12)(cid:12) u (Ψ − x ( z )) − u (Ψ − x ( z )) (cid:12)(cid:12) ≤ k u k C α ( ˆ B ) ( ρ ( x ) d ( z , z )) α . (44)Hence, (H3) and then (41) follow from (43) and (44).(ii) By (H1), it suffices to consider x and x in S δ ∩ ˆ B . Assume that ρ ( x ) ≥ ρ ( x ). If d ( x , x ) ≥ c β ρ ( x )2 : Let ˜ x and ˜ x be the projections of x and x to S respectively. The triangleinequality and (H2) imply that | u ( x ) − u ( x ) | ≤ | u ( x ) − u (˜ x ) | + | u ( x ) − u (˜ x ) | + | u (˜ x ) − u (˜ x ) |≤ k u k U α ( ˆ B ) ( ρ ( x ) α + ρ ( x ) α + d (˜ x , ˜ x ) α ) ≤ C k u k U α ( ˆ B ) d ( x , x ) α . If d ( x , x ) < c β ρ ( x )2 : denote Ψ x ( x ) by z , which is a point in ˜ B / . | u ( x ) − u ( x ) | = | S x ( u )(˜ x ) − S x ( u )( z ) |≤ | S x ( u − P ˜ x ( · − ˜ x ))(˜ x ) − S x ( u − P ˜ x ( · − ˜ x ))( z ) |≤ C k u k U α ( ˆ B ) ρ ( x ) α d (˜ x , z ) α ≤ C k u k U α ( ˆ B ) d ( x , x ) α . Here in the second line above, we used (H3). (cid:3) To conclude this section, we prove Lemma 6.9. If u is in U α ( ˆ B ) , then u is in C ,αβ ( ˆ B ) and k u k C ,αβ ( ˆ B ) ≤ C k u k U α ( ˆ B ) . Remark 6.10. It is natural to ask about the other direction of Lemma 6.9. Whileit is possible to give a direct proof, we omit it because it follows from Lemma 6.8and the Schauder estimate (Theorem 7.1), which is to be proved in the next section.So we conclude that C ,αβ and U α are the same (in the sense above).Proof. The proof consists of several steps. Step 1: By Remark 6.5 and Lemma 6.8, we have u ∈ C α ( ˆ B ). Step 2: All derivatives listed in (D2) and (D3) are bounded. Since the proofsare the same, we prove the claim for ∂ ρ u only. Thanks to (H1), it suffices to consider x ∈ S δ ∩ ˆ B . Let ˜ x be the projection of x onto S , then for the T -polynomial P ˜ x (with order smaller than 2 + α ) in (H2), we have ∂ ρ u ( x ) = ∂ ρ ( u − P ˜ x ( · − ˜ x )) ( x ) + ∂ ρ P ˜ x ( x − ˜ x ) . (45)The second term above is bounded by a multiple of k u k U α ( ˆ B ) , because of (H2)and the fact that the ∂ ρ of each monic T -monomial (listed in (40)) is bounded. Thefirst term is bounded by(46) 1 ρ |∇ S x ( u − P ˜ x ( · − ˜ x )) | (˜ x ) , which is in turn bounded by C k u k U α ( ˆ B ) due to (H3). Step 3: All derivatives of u listed in (D2) and (D3) are in C α .As before, due to (H1), we may assume that x , x ∈ S δ ∩ ˆ B and ρ ( x ) ≥ ρ ( x ). CHAUDER ESTIMATES ON SMOOTH AND SINGULAR SPACES 23 If d ( x , x ) < c β ρ ( x ) / : Let ˜ x be the projection of x onto S and z =Ψ x ( x ) ∈ ˜ B / . | ∂ ρ u ( x ) − ∂ ρ u ( x ) |≤ | ∂ ρ ( u − P ˜ x ( · − ˜ x ))( x ) − ∂ ρ ( u − P ˜ x ( · − ˜ x ))( x ) | + | ∂ ρ P ˜ x ( x − ˜ x ) − ∂ ρ P ˜ x ( x − ˜ x ) |≤ ρ ( x ) |∇ S x ( u − P ˜ x ( · − ˜ x ))(˜ x ) − ∇ S x ( u − P ˜ x ( · − ˜ x ))( z ) | + C k u k U α ( ˆ B ) d ( x , x ) α ≤ ρ ( x ) · C k u k U α ( ˆ B ) ρ ( x ) α d (˜ x , z ) α + C k u k U α ( ˆ B ) d ( x , x ) α ≤ C k u k U α ( ˆ B ) d ( x , x ) α . For the estimate of the second term in the second line above, we check that for eachmonic monomial f in (40), there holds | ∂ ρ f ( x ) − ∂ ρ f ( x ) | ≤ Cd ( x , x ) α . For the first term in the third line above, we used (H3).Again, the same argument works for all other derivatives in (D2) and (D3) inthe case d ( x , x ) < c β ρ ( x )2 . If d ( x , x ) ≥ c β ρ ( x ) / : We study ∂ ρ u , ρ ∂ θ u and ˜ △ β u first. In this case, (H3)implies that(47) | ∂ ρ ( u ( x ) − P ˜ x ( x − ˜ x )) | ≤ ρ ( x ) |∇ S x ( u − P ˜ x ( · − ˜ x )) | (˜ x ) ≤ C k u k U α ( ˆ B ) ρ ( x ) α and(48) | ∂ ρ ( u ( x ) − P ˜ x ( x − ˜ x )) | ≤ ρ ( x ) |∇ S x ( u − P ˜ x ( · − ˜ x )) | (˜ x ) ≤ C k u k U α ( ˆ B ) ρ ( x ) α . By checking the monic monomials in (40) one by one, we find that for either x = x or x = x ,(49) | ∂ ρ P ˜ x ( x − ˜ x ) | ≤ C k u k U α ( ˆ B ) ρ ( x ) α . By (47), (48) and (49) and the fact that ρ ( x ) ≤ ρ ( x ) ≤ c β d ( x , x ), we get | ∂ ρ u ( x ) − ∂ ρ u ( x ) | ≤ C k u k U α ( ˆ B ) d ( x , x ) α . The same argument works for ρ ∂ θ u and ˜ △ β u .Now, let’s turn to the proof for ∂ ξ u , ∂ ρ ∂ ξ u , ρ ∂ θ ∂ ξ u and ∂ ξ u . We prove for ∂ ξ u only and the same proof works for the other three functions. Similar to (45), wehave(50) ∂ ξ u ( x ) = ∂ ξ ( u − P ˜ x ( · − ˜ x ))( x ) + ∂ ξ P ˜ x ( x − ˜ x ) . By (50), we have (cid:12)(cid:12) ∂ ξ u ( x ) − ∂ ξ u ( x ) (cid:12)(cid:12) ≤ (cid:12)(cid:12) ∂ ξ ( u − P ˜ x ( · − ˜ x ))( x ) (cid:12)(cid:12) + (cid:12)(cid:12) ∂ ξ ( u − P ˜ x ( · − ˜ x ))( x ) (cid:12)(cid:12) + (cid:12)(cid:12) ∂ ξ P ˜ x ( x − ˜ x ) − ∂ ξ P ˜ x ( x − ˜ x ) (cid:12)(cid:12) . Using (H3) and the inequality ρ ( x ) ≤ ρ ( x ) ≤ c β d ( x , x ) as before, we can boundthe first two terms by C k u k U α ( ˆ B ) d ( x , x ) α . It remains to estimate the thirdterm above. Remark 6.11. For each f in (40) , the mixed second derivatives ∂ ρ ∂ ξ and ρ ∂ θ ∂ ξ vanish. The proofs are done in these two cases. Again by checking the monic monomials in (40), we notice that ∂ ξ P ˜ x i ( x i − ˜ x i ) = ∂ ξ P ˜ x i (0) , i = 1 , . The proof will be done, if we can show(51) (cid:12)(cid:12) ∂ ξ P ˜ x (0) − ∂ ξ P ˜ x (0) (cid:12)(cid:12) ≤ C | ˜ x − ˜ x | α . To see this, we define a function ˜ u on S ∩ ˆ B by˜ u (˜ x ) = u (˜ x ) = P ˜ x (0) . By (H2) (restricted to S direction), ˜ u satisfies the assumptions in Definition 2.1 inSection 2. Proposition 2.2 then implies that ˜ u is C ,α (on R n − ) in the usual senseand that ( ∂ ξ P ˜ x )(0) = ( ∂ ξ ˜ u )(˜ x ) , from which (51) follows. (cid:3) Schauder estimates for X β We prove here an estimate that by our perspective should be called the Schauderestimate on X β . Recall that D is the set of degrees of all T -monomials and we havedefined generalized H¨older spaces U q only for q / ∈ D . Theorem 7.1. Assume that q > and q, q + 2 / ∈ D . Suppose f ∈ U q ( ˆ B ) and u isa bounded weak solution to △ β u = f. Then u is in U q +2 ( ˆ B ) and k u k U q +2 ( ˆ B ) ≤ C (cid:16) k u k C ( ˆ B ) + k f k U q ( ˆ B ) (cid:17) . This is the main result of this paper. The key step in its proof is an analog ofLemma 3.2. Lemma 7.2. For q > and q + 2 / ∈ D , if f is O ( d q ) on ˆ B r , then there exists some u that is O ( d q +2 ) and satisfies △ β u = f and [ u ] O q +2 , ˆ B r ≤ C [ f ] O q , ˆ B r . The rest of this section consists of two parts. In the first part, we prove Lemma7.2, following the proof of Lemma 3.2 and utilizing many facts about harmonicfunctions on X β proved in Section 5. In the second part, we complete the proof ofTheorem 7.1. CHAUDER ESTIMATES ON SMOOTH AND SINGULAR SPACES 25 Proof of Lemma 7.2. Without loss of generality, we assume r = 1.Setting ˆ A l := ˆ B − l \ ˆ B − l − , for l = 0 , , , · · · and f l = f · χ ˆ A l , we have | f l | ( x ) ≤ Λ f d ( x ) q χ ˆ A l ≤ Λ f − lq on X β , where as before Λ f = [ f ] ˆ O q , ˆ B .Let w l be the solution to the Poisson equation △ β w l = f l on X β in Lemma 4.1,which satisfies that(52) sup X β | w l | ≤ C Λ f − l ( q +2) . Again, w l is harmonic in ˆ B − l − . Let P l be the X β -polynomial in Proposition5.2 applied to w l in ˆ B − l − with q + 2 replacing q . Our plan is to set u l ( x ) = w l ( x ) − P l ( x )and to show that the series u ( x ) = ∞ X l =0 u l ( x )converges and gives the solution needed in Lemma 7.2. Notice that by Proposition5.6, P l is harmonic on the entire X β and u l is harmonic outside ˆ A l .For that purpose, we need an analog of Lemma 3.3. Lemma 7.3. There exists a constant C q depending on n, β and q such that(i) on ˆ B − l − , | u l ( x ) | ≤ C q Λ f (( q +2) ∗ − ( q +2)) l d ( x ) ( q +2) ∗ ; Here ( q + 2) ∗ is the smallest number in D that is larger than q + 2 .(ii) on ˆ A l , | u l ( x ) | ≤ C q Λ f − ( q +2) l or equivalently | u l ( x ) | ≤ C q Λ f d ( x ) q +2 ; (iii) on ˆ B \ ˆ B − l , | u l ( x ) | ≤ C q Λ f − l ( q +2) + X j + kβ + | σ | The following are immediate corollaries of Proposition 5.2.(a) Recall that P l ( x ) is the X β -polynomial in Proposition 5.2 applied to w l , namely, P l ( x ) = X j + kβ + | σ | By the usual Schauder estimate, to prove an estimateof u in U q +2 ( ˆ B ), we do not need to worry about (H1). For (H2), let x ∈ S ∩ ˆ B ,there is a T -polynomial P f (order less than q ) such that f ( y ) = P f ( y − x ) + O ( d ( x, y ) q ) , ∀ y ∈ ˆ B δ ( x ) . By our choice of T (see (P4) in Lemma 6.3), there is some T -polynomial ˜ P u (orderless than q + 2) such that(53) △ β ˜ P u = P f . Notice that ˜ P u is not uniquely determined by P f , since we may add any harmonic T -polynomial to it.If we denote the O ( d ( x, y ) q ) term by e f ( y ), Lemma 7.2 implies the existence of e u ( y ) defined on ˆ B δ ( x ) such that △ β e u ( y ) = e f ( y )and [ e u ] O q +2 , ˆ B δ ( x ) ≤ C [ e f ] O q , ˆ B δ ( x ) . Therefore, u − ˜ P u ( · − x ) − e u is a harmonic function v bounded by C ( k u k C ( ˆ B ) + k f k U q ( ˆ B ) ). Proposition 5.2 implies the existence of some X β -polynomial h u oforder less than q + 2 such that v ( y ) = h u ( y − x ) + O ( d ( x, y ) q +2 ) . Then (H2) is verified by setting P ˜ x = ˜ P u + h u .For (H3), let x ∈ S δ and ˜ x be its projection to S . On one hand, (H2), which isproved above, implies the existence of P ˜ x such that(54) k S x ( u − P ˜ x ( · − ˜ x )) k C ( ˜ B ) ≤ Cρ ( x ) q +2 . On the other hand, by (53) and the definition of P ˜ x , △ β ( S x ( u − P ˜ x ( · − ˜ x ))) = ρ ( x ) S x ( △ β ( u − P ˜ x ( · − ˜ x ))) = ρ ( x ) S x ( f − P f ( · − ˜ x )) . By (H3) for f and the above equation,(55) k△ β S x ( u − P ˜ x ( · − ˜ x )) k C k,α ( ˜ B / ) ≤ Cρ ( x ) q +2 . The usual interior Schauder estimate on ˜ B / , together with (54) and (55), impliesthat k S x ( u − P ˜ x ( · − ˜ x )) k C k +2 ,α ( ˜ B / ) ≤ Cρ ( x ) q +2 . Now the proof of Theorem 7.1 is concluded by Remark 6.7. CHAUDER ESTIMATES ON SMOOTH AND SINGULAR SPACES 27 Appendix A. Proof of Lemma 5.4 The proof of this lemma consists of a bootstrapping argument of a family ofPoisson equations on cone surface, which was used in [17]. More precisely, byLemma 5.3, ∂ σ u is harmonic, which implies that( E σ ) ˜ △ β ( ∂ σξ u ) = −△ ξ ( ∂ σξ u ) . Here ˜ △ β is the Laplacian of the two dimensional cone surface X β , parametrized by( ρ, θ ) and equipped with the cone metric g β = dρ + β ρ dθ . Also by Lemma 5.3, the right hand side of ( E σ ) is bounded by some constantdepending on σ . It then follows that ∂ σξ u (with ξ fixed) is H¨older continuousfunction with respect to the distance of X β . To see this, recall that in termsof the ( u, v ) coordinates, where u = ρ cos θ and v = ρ sin θ , ˜ △ β is a uniformlyelliptic operator with bounded coefficients and hence the H¨older continuity followsfrom De Giorgi’s iteration. (see [17] for detail). This is the starting point of thebootstrapping.A function w is said to have T h -expansion up to order q > w ( ρ, θ ) = X j + kβ 2. The expansion is said to be bounded by Λ if the coefficients A j,k , B j,k and the constant in the definition of O ( ρ q ) are bounded by Λ. Remark A.1. Note that the expansion above is different from the one used in [17] ,where we also included ρ j + kβ cos mθ and ρ j + kβ sin mθ, for k − m ∈ N ∪ { } . This is because in [17] , we dealt with nonlinear equations, while here we are es-sentially working with linear equations. The product of harmonic functions is notnecessarily harmonic and hence there is no need to require the formal series to bemultiplicatively closed. The H¨older continuity of ∂ σξ u means that ∂ σξ u ( ξ ) has an expansion up to someorder q ∈ (0 , 1) uniformly (independent of ξ ) bounded by Λ = Λ( q, σ ). The proofof Lemma 5.4 is now reduced to the following claim. Notice that we prove (22) and(23) simultaneously. Claim. Let u be a bounded solution to˜ △ β u = f on the unit ball centered at the unique singular point of X β . If f has a T h -expansionup to order q bounded by Λ for q = 2 j + kβ for any k, j ∈ N ∪ { } , then u has a T h -expansion up to order q + 2 bounded by a multiple of Λ and C norm of u onthe ball.This is nothing but Lemma 6.9 in [17]. The difference pointed out in Remark A.1does not cause a problem because for the proof, we only require that for each T h -polynomial of order q ′ , there exists a T h -polynomial of order q ′ + 2 that is mappedto the given one by ˜ △ β . As a final remark, we notice that Lemma 6.9 in [17] relies on Lemma 6.10 there,which is the precursor of Lemma 3.2 and Lemma 7.2 in this paper. We find theproof of Lemma 6.10 in [17], which depends on Fourier series, hard to generalizeto higher dimensions. The new proof here of course can be used to prove Lemma6.10. References [1] Luis A. Caffarelli. Interior a priori estimates for solutions of fully nonlinear equations. Ann.of Math. (2) , 130(1):189–213, 1989.[2] Luis A. Caffarelli and Xavier Cabr´e. Fully nonlinear elliptic equations , volume 43 of AmericanMathematical Society Colloquium Publications . American Mathematical Society, Providence,RI, 1995.[3] S. Campanato. Propriet`a di una famiglia di spazi funzionali. Ann. Scuola Norm. Sup. 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| ǫ | . We rewrite (9) Q ǫǫ ( a ǫ ( x + h ) − a ǫ ( x )) = X ǫ (cid:12) σ, | σ |