Schauder estimates for equations with cone metrics, II

This is the continuation of our paper \cite{GS}, to study the linear theory for equations with conical singularities. We derive interior Schauder estimates for linear elliptic and parabolic equations with a background K\"ahler metric of conical singularities along a divisor of simple normal crossings. As an application, we prove the short-time existence of the conical K\"ahler-Ricci flow with conical singularities along a divisor with simple normal crossings.


Introduction
Regularity of solutions of Complex Monge-Ampère equations is a central problem in complex geometry. Complex Monge-Ampère equations with singular and degenerate data can be applied to study compactness and moduli problems of canonical Kähler metrics in Kähler geometry. In [43], Yau has already considered special cases of complex singular Monge-Ampère equations as generalization of his solution to the Calabi conjecture. Conical singularities along complex hypersurfaces of a Kähler manifold are among the mildest singularities in Kähler geometry and it has been extensively studied, especially in the case of Riemann surfaces [41,28]. The study of such Kähler metrics with conical singularities has many geometric applications, for example, the Chern number inequality in various settings [39,38]. Recently, Donaldson [14] initiated the program of studying analytic and geometric properties of Kähler metrics with conical singularities along a smooth complex hypersurface on a Kähler manifold. This is an essential step to the solution of the Yau-Tian-Donaldson conjecture relating existence of Kähler-Einstein metrics and algebraic Kstability on Fano manifolds [5,6,7,40]. In [14], the Schauder estimate for linear Laplace equations with conical background metric is established using classical potential theory. This is crucial for the openness of the continuity method to find desirable (conical) Kähler-Einstein metric. Donaldson's Schauder estimate is generalized to the parabolic case [8] with similar classical approach. There is also an alternative approach for the conical Schauder estimates using microlocal analysis [23]. There are also various global and local estimates and regularity derived in the conical setting [1,15,9,11,13,12,18,15,24,29,31,44,45].
The Schauder estimates play an important role in the linear PDE theory. Apart from the classical potential theory, various proofs have been established by different analytic techniques. In fact, the blow-up or perturbation techniques developed in [36,42] (also see [33,34,2,3]) are much more flexible and sharper than the classical method. The authors combined the perturbation method in [20] and geometric gradient estimates to establish sharp Schauder estimates for Laplace equations and heat equations on C n with a background flat Kähler metric of conical singularities along the smooth hyperplane {z 1 = 0} and derived explicit and optimal dependence on conical parameters.
In algebraic geometry, one often has to consider pairs (X, D) with X being an algebraic variety of complex dimension n and the boundary divisor D as a complex hypersurface of X. After possible log resolution, one can always assume the divisor D is a union of smooth hypersurfaces with simple normal crossings. The suitable category of Kähler metrics associated to (X, D) is the family of Kähler metrics on X with conical singularities along D. In order to study canonical Kähler metrics on pairs and related moduli problems, we are obliged to study regularity and asymptotics for complex Monge-Ampère equation with prescribed conical singularities of normal crossings. However, the linear theory is still missing and has been open for a while. The goal of this paper is to extend our result [20] and establish the sharp Schauder estimates for linear equations with background Kähler metric of conical singularities along divisors of simple normal crossings. We can apply and extend many techniques developed in [20], however, new estimates and techniques have to be developed because in case of conical singularities along a single smooth divisor, the difficult estimate in the conical direction can sometimes be bypassed and reduced to estimates in the regular directions, while such treatment does not work in the case of simple normal crossings. One is forced to treat regions near high codimensional singularities directly with new and more delicate estimate beyond the scope of [20].
The standard local models for such conical Kähler metrics can be described as below.
(1) We remark that the number β max on the RHS of (1.5) can be replaced by max{β j , β k }.
(2) In Theorem 1.1 and Theorem 1.3 below, we assume β ∈ (1/2, 1) p just for exposition purposes and cleanness of the statement. When some of angles β j lie in (0, 1/2], the pointwise Hölder estimates in Theorem 1.1 are adjusted as follows: if β j ∈ (0, 1/2] in (1.4), we replace the RHS by the RHS of (1.3). In (1.5), if both β j and β k ∈ (0, 1/2], we also replace the RHS by that of (1.3); if at least one of the β j , β k is bigger than 1/2, (1.5) remains unchanged. The inequalities in Theorem 1.3 can be adjusted similarly. The proofs of these estimates are contained in the proof of the case when β j ∈ (1/2, 1) by using the corresponding estimates in (2.2).
An immediate corollary of Theorem 1.1 is a precise form of Schauder estimates for equation (1.2).
For general non-flat C α,α/2 β -conical Kähler metrics g, we consider the linear parabolic equation (1.10) We then have the following parabolic Schauder estimates as an analogue of Theorem 1.2.
Theorem 1.5. Given α ∈ (0, α ′ ), there exists T = T (n,ω, f, α ′ , α) > 0 such that (1.11) admits a unique solution ϕ ∈ C 2+α, 2+α An immediate corollary of Theorem 1.5 is the short time existence for the conical Kähler-Ricci flow defined as below where Ric(ω) is the unique extension of the Ricci curvature of ω from X \ D to X and [D j ] denotes the current of integration over the component D j . In addition we assume ω 0 is a C 0,α ′ β (X, D)-conical Kähler metric such that where s j , h j are holomorphic sections and hermitian metrics of the line bundle associated to D j , respectively, and Ω is a smooth volume form.
Furthermore, ω is smooth in X\D × (0, T ] and the (normalized) Ricci potentials of ω, log ω n The short time existence of the conical Kähler-Ricci flow with singularities along a smooth divisor is derived in [8] by adapting the elliptic potential techniques of Donaldson [14]. Corollary 1.3 treats the general case of conical singularities with simple normal crossings. There have been many results in the analytic aspects of the conical Ricci flow [8,9,15,16,24,32,43]. In [30], the conical Ricci flow on Riemann surfaces is completely classified with jumping conical structure in the limit. Such phenomena is also expected in higher dimension, but it requires much deeper and delicate technical advances both in analysis and geometry.

Preliminaries
We explain the notations and give some preliminary tools which will be used later in this section.

Notations
To distinguish the elliptic from parabolic norms, we will use the ordinary C to denote the norms in the elliptic case and the script C to denote the norms in the parabolic case.
2.1.1. Elliptic case. We will denote d β (x, y) to be the distance of two points x, y ∈ C n under the metric g β . B β (x, r) will be the metric ball under the metric induced by g β with radius r and center x. It is well-known that (C n \S, g β ) is geodesically convex, i.e. any two points x, y ∈ C n \S can be joined by a g β -minimal geodesic γ which is disjoint with S. Definition 2.1. We define the g β -Hölder norm of functions u ∈ C 0 (B β (0, r)) for some α ∈ (0, 1) as We denote the subspace of all continuous functions u such that u C 0,α β < ∞ as C 0,α β (B β (0, r)).
Definition 2.3. Suppose σ ∈ R is a given real number and u is a C 2,α β -function in Ω. We denote d x = d β (x, ∂Ω) for any x ∈ Ω. We define the weighted (semi)norms where T denotes the following operators of second order: When σ = 0, we denote the norms above as [·] * , · * for notation simplicity.
2.1.2. Parabolic case. We denote Q β = Q β (0, 1) = B β (0, 1) × (0, 1] to be parabolic cylinder and to be the parabolic boundary of the cylinder Q β . We write S P = S × [0, 1] as the singular set and Q # β = Q β \S P the complement of S P . For any two space-time points Q i = (p i , t i ), we define their parabolic distance d P,β (Q 1 , Q 2 ) as Definition 2.4. We define the g β -Hölder norm of functions u ∈ C 0 (Q β ) for some α ∈ (0, 1) as norm of a function u on Q β is defined as: where T denotes all the second order operators in (2.1) and the first order operator ∂ ∂t .
For a given set Ω ⊂ Q β we define the following weighted (semi)norms.

Compact Kähler manifolds.
Let (X, D) be a compact Kähler manifold with a divisor D = j D j with simple normal crossings, i.e. on an open coordinates chart (U, z j ) of any x ∈ D, D ∩ U is given by {z 1 · · · z p = 0}, and D j ∩ U = {z j = 0} for any component D j of D. We fix a finite cover {U a , z a,j } of D.
Definition 2.7. A (singular) Kähler metric ω is called a conical metric with cone angle 2πβ along D, if locally on any coordinates chart U a , ω is equivalent to ω β under the the coordinates {z a,j }, where ω β is the standard cone metric (1.1) with cone angle 2πβ j along {z a,j = 0}, and on X\ ∪ a U a ω is a smooth Kähler metric in the usual sense.
A conical metric ω is in C 0,α β (X, D) if for each a, ω is C 0,α β (U a ) and on X\ ∪ a U a ω is smooth in the usual sense. Similarly we can define the C α,α/2 β -conical Kähler metrics on X × [0, 1]. Definition 2.8. A continuous function u ∈ C 0 (X) is said to be in C 0,α β (X, D) if locally on each U a , u is in C 0,α β (U a ) and on X\ ∪ a U a it is C 0,α -continuous in the usual sense. We define the C 0,α β (X, D)-norm of u as The C 0,α β (X, D)-norm depends on the choice of finite covers, and another cover yields a different but equivalent norm. The space C 0,α β (X, D) is clearly independent of the choice of finite covers. The other spaces and norms like , etc, can be defined similarly.

A useful lemma
We will frequently use the following elementary estimates from [20].
satisfies the equation for some F ∈ L ∞ (B C (0, r)), then we have the pointwise estimate that for any z ∈ B C (0, 9r/10)\{0} where the L ∞ -norms are taken in B C (0, r) and C > 0 is a uniform constant.
Finally we remark that the idea of the proof of the estimates in Theorems 1.1 and 1.3 is the same for general 2 ≤ p ≤ n. To explain the argument clearer we prove the theorems assuming p = 2, i.e. the cone metric of ω β is singular along the two components S 1 and S 2 .

Elliptic estimates
In this section, we will prove Theorems 1.1 and 1.2, the Schauder estimates for the Laplace equation (1.2). To begin with, we first observe the simple C 0 -estimate based on maximum principle.
In Subsections 3.1.2 -3.1.5, for notation convenience, we will omit the subscript ǫ in g ǫ , u ǫ , in the proofs of lemmas.
Lemma 3.2 ([10]). Let u ǫ ∈ C 2 (B(p, R)) be a positive ∆ gǫ -harmonic function. There exists a uniform constant C = C(n) > 0 such that (the metric balls are taken under the metric g ǫ ) sup x∈B(p,3R/4) As we mentioned above, we will omit the ǫ in the subscript of u ǫ and g ǫ . The proof of the lemma is standard ( [10]). For completeness and to motivate the proof of Lemmas 3.3 and 3.4 below, we provide a sketched proof. Define f = log u, and it can be calculated that Then by Bochner formula, we have Let φ : [0, 1] → [0, 1] be a standard cut-off function such that φ| [0,3/4] = 1 and φ [5/6,1] = 0 and between 0, 1 otherwise. let r(x) = d gǫ (p, x) be the distance function to p under the metric g = g ǫ .
By abusing notation we also write φ(x) = φ r(x) R . It can be calculated by Laplacian comparison and the Bochner formula (3.7) that at the (positive) maximum point p max of H := φ 2 |∇f | 2 that Laplacian estimate in singular directions. We will prove the estimates of ∆ j,ǫ u ǫ := (|z j | 2 + ǫ) 1−β j ∂uǫ ∂z j ∂z j for a ∆ gǫ -harmonic function u ǫ .
Under the same assumptions as in Lemma 3.2, along the "bad" directions z 1 , z 2 , ∆ ǫ,1 u ǫ and ∆ ǫ,2 u ǫ satisfy the estimates As in the proof of Cheng-Yau gradient estimates, we will work on the function f = f ǫ = log u and we only need to prove the estimate for ∆ 1,ǫ u ǫ . We write ∆ 1,ǫ f := (|z 1 | 2 + ǫ) 1−β 1 ∂ 2 f ∂z 1 ∂z 1 . As above, we will omit the subscript ǫ in ∆ 1,ǫ f . We first observe that (3.10) (3.10) can be checked from the definitions by the property that g ǫ is a product-metric. Indeed On the other hand, note that by (3.6) ∆ gǫ f = ∆ g f = −|∇f | 2 . By choosing a normal frame {e 1 , . . . , e n } at some point x such that dg(x) = 0 and ∆ 1 f = f 11 , we calculate Then it follows that  We want to estimate the upper bound of G. If the maximum value of G = ϕ 2 (−∆ 1 f ) is negative, we are done. So we assume the maximum of G on B(p, R) is positive, which is achieved at some point p max ∈ B(p, 2R/3). Hence at p max , we have (−∆ 1 f ) > 0. By Laplacian comparison that ∆r ≤ 2n−1 r , we get at p max , Thus at p max , the last term on RHS of (3.12) is Substituting this into (3.12), it follows that at p max , ∆G ≤ 0 and (3.14) Therefore at p max ∈ B(p, 2R/3), it holds that (3.8) and the fact that ϕ, ϕ ′ , ϕ ′′ are all uniformly bounded, we can get at p max Then for any x ∈ B(p, R/2), where ϕ = 1, we have Moreover, recall that f = log u and −∆ 1 f = − ∆ 1 u u + |∇ 1 f | 2 , therefore it follows that sup This in particular implies that On the other hand, by considering the functionû = max B(p,R) u − u, which is still a positive g ǫ -harmonic function ∆ gû = ∆ gǫû = 0. Applying (3.15) to the functionû, we get sup x∈B(p,R/2) which yields that sup x∈B(p,R/2) 3.1.4. Mixed derivatives estimates. In this subsection, we will estimate the following mixed derivatives where as before f = log u and u is a positive harmonic function of ∆ gǫ . Here for simplicity, we omit the subscript ǫ in u ǫ , f ǫ and g ǫ . Observing that since g ǫ = g is a product metric with the non-zero components g kk depending only on z k , it follows that the curvature tensor vanishes unless i = j = k = l = 1 or 2, and also R iīiī ≥ 0 for all i = 1, . . . , n.
We fix some notations: we will write f 12 = ∇ 1 ∇ 2 f (in fact this is just the ordinary derivative of f w.r.t. g, since g is a product metric), |f 12 | 2 g = |∇ 1 ∇ 2 f | 2 g , etc. Let us first recall that the equation (3.11) implies Next we calculate ∆|∇ 1 ∇ 2 f | 2 . For notation convenience we will write f 12 = f12g 11 g 22 , and hence |∇ 1 ∇ 2 f | 2 = f 12 f 12 . We calculate (3.21) The first term on the RHS of (3.21) is (by Ricci identities and switching the indices) =g kk f 12 f kk12 + g 11 g 11 f 12 f 21 R 1111 + g 22 g 22 f 12 f 12 R 2222 , (3.22) and the last term on the RHS of (3.21) is the conjugate of the first term, hence we get ∆|∇ 1 ∇ 2 f | 2 = 2Re f 12 (∆f ) 12 + 2f 12 f 12 g 11 g 11 R 1111 + g 22 g 22 R 2222 + g kk f 12k f 12 ,k + g kk f 12k f 12 ,k . (3.23) Recall from (3.6) we have ∆f = −|∇f | 2 , hence the first term on RHS of (3.23) is Combining (3.24) and (3.23), we get On the other hand we have by Kato's inequality Combining (3.25) and (3.26) it follows that Combining (3.20), (3.27) and applying Cauchy-Schwarz inequality, we have (3.28) Note that the sum on the RHS of (3.27) is (recall under our notation so we get the following equation (3.29) Apply maximum principle to Q and if the max Q ≤ 0, we are done. So we may assume that max Q > 0 and is attained at p max , thus at p max , Q 1 > 0, ∆Q ≤ 0, ∇Q 1 = −2η −1 Q 1 ∇η and where we choose η such that |η ′ |, |η ′′ | ≤ 10, for example. Therefore at p max ∈ B(p, R/2) we have

Thus it follows that
On the other hand from (3.32) Therefore we obtain that sup By exactly the same argument we can also get similar estimates for |∇ 1 ∇2u| and |∇ 1 ∇ k u|+|∇ 1 ∇ku| for k = 1. Hence we have proved the following lemma: for all i, j = 1, 2, · · · , n.
Proposition 3.1. The Dirichlet boundary value problem (3.1) admits a unique solution u ∈ C 2 (B β \S)∩ C 0 (B β ) for any ϕ ∈ C 0 (∂B β ). Moreover, u satisfies the estimates in Lemmas 3.2, 3.3 and 3.4 with u ǫ replaced by u and the metric balls replaced by those under the metric g β , which we will refer as "derivatives estimates" throughout this section.
Proof. Given the estimates of u ǫ as in lemmas 3.2, 3.3 and 3.4, we can derive the uniform local C 2,α estimates of u ǫ on any compact subsets of B β (0, 1)\S.
The C 0 estimates of u ǫ follow immediately from the maximum principle (see Lemma 3.1). Take any compact subsets K ⋐ K ′ ⋐ B β (0, 1). By Lemma 3.3, we have 35) and the third-order estimates (3.36) Moreover, applying the gradient estimate to the ∆ gǫ -harmonic function ∆ ǫ,1 u ǫ , we get (3.37) From (3.34), (3.35) and (3.36), we see that the functions u ǫ have uniform C 3 estimates in the "tangential directions" on any compact subset of B β (0, 1). Moreover, for any fixed small constant δ > 0, let T δ (S) be the tubular neighborhood of S. We consider the equation which is strictly elliptic (with ellipticity depending only on δ > 0). Hence by standard elliptic Schauder theory, we also have C 2,α -estimates of u ǫ in the "transversal directions" (i.e. normal to S) and the mixed directions, on the compact subset K\T δ (S). By taking δ → 0, K → B β , and a diagonal argument, up to a subsequence u ǫ converge in C 2,α loc (B β \S) to a function u ∈ C 2,α (B β \S). Clearly, u satisfies the equation ∆ β u = 0 on B β \S, and the estimates (3.34), (3.35) and (3.36) hold for u outside S, which implies that u can be continuously extended through S and defines a continuous function in B β (0, 1). It remains to check the boundary value of u.
Claim: u = ϕ on ∂B β (0, 1) It remains to show the limit function u of u ǫ satisfies the boundary condition u = ϕ on ∂B β (0, 1), which will be proved by constructing suitable barriers as we did in [20].
We consider the case when z 1 (q) = 0 and z 2 (q) = 0. The boundary ∂B β (0, 1) is smooth near q, hence satisfies the exterior sphere condition. We choose an exterior Euclidean ball B C n (q, r q ) which is tangential with ∂B β (0, 1) (only) at q, i.e. under the Euclidean distance q is the unique closest point toq on ∂B β (0, 1). So the function G The function Ψ q (z) = A(d β (z, 0) 2 − 1) + G(z) is ∆ gǫ -subharmonic for A >> 1 and Ψ q (q) = 0, Ψ q (z) < 0 for ∀z ∈ ∂B β (0, 1)\{q}. We are in the same situation as the Case 1, so by the same argument as above, we can show the continuity of u at such boundary point q.
In case z 1 (q) = 0 but z 2 (q) = 0. The boundary ∂B β (0, 1) is not smooth at q and we cannot apply the exterior sphere condition to construct the barrier. Instead we will use the geometry of the metric ball B β (0, 1). Consider the standard cone metric g β 1 = β 2 1), and their boundaries are tangential at the points with vanishing z 2 -coordinate. Thus q ∈ ∂B β (0, 1) ∩ ∂B g β 1 (0, 1) and ∂B g β 1 (0, 1) is smooth at q so there exists an exterior sphere for ∂B g β 1 (0, 1) at q. We define similar function G(z) as in the last paragraph, and by the strict inclusion of the metric balls B β (0, 1) ⊂ B g β 1 (0, 1), it follows that G(q) = 0 and G(z) < 0 for all z ∈ ∂B β (0, 1)\{q}. The remaining argument is the same as before.
For later application, we prove the existence of solution for a more general RHS of the Laplace equation with the standard background metric. Note that this result is not needed in the proof of Theorem 1.1.
Since Ric(g ǫ ) ≥ 0 we have the following Sobolev inequality ( [25]): there exists a constant C = C(n) > 0 such that for any h ∈ C 1 0 (B gǫ (p, r)), it holds that It can be checked by straightforward calculations that Vol gǫ B gǫ (p, 1) ≥ c 0 (n) > 0 for some constant c 0 independent of ǫ. Then Bishop volume comparison yields that for any r ∈ (0, 1), Thus the Sobolev inequality (3.41) is reduced to With (3.42) at hand, we can apply the same proof of the standard De Giorgi-Nash-Moser theory (see the proof of Corollary 4.18 of [22]) to derive the uniform Hölder continuity of v ǫ at p, i.e. there exists a constant C = C(n, β, R 0 ) > 0 such that where in the first inequality we use the relation (3.40). Letting ǫ → 0 we see the continuity of v at p.
• v = ϕ on ∂B β (0, 1). The proof is almost identical to that of Proposition 3.1. For example, the The remaining are the same as in Proposition 3.1. The Case 2 can be dealt with similarly.
for the same constant C in (3.42). That is, Sobolev inequality also holds for the conical metric ω β .

Tangential and Laplacian estimates
In this section, we will prove the Hölder continuity of ∆ k u for k = 1, 2 and (D ′ ) 2 u for the solution u to (1.2). The arguments of [20] can be adopted here. We recall that we assume β 1 , β 2 ∈ ( 1 2 , 1). We fix some notations first.
For a given point p ∈ S, we denote r p = d g β (p, S), the g β -distance of p to the singular set S. For notation simplicity we will fix τ = 1/2 and an integer k p ∈ Z + to be the smallest integer such that τ k < r p , and k i,p ∈ Z + the smallest integer k such that τ k < d β (p, S i ), for i = 1, 2. So k p = max{k 1,p , k 2,p } We denote p 1 ∈ S 1 and p 2 ∈ S 2 the projections of p to S 1 , S 2 , respectively.
For j = 1, 2 we will write ∆ j u := |z j | 2(1−β j ) ∂ 2 u ∂z j ∂z j . We will consider a family of conical Laplace equations with different choices of k ∈ Z + .
(i) If k ≥ k p , the geodesic balls B β (p, τ k ) are disjoint with S and have smooth boundaries. g β is smooth on such balls. By standard theory we can solve the Dirichlet problem for By similar argument as in the proof of Proposition 3.1, such u k exists.
We remark that we may take f (p) = 0 by consideringũ = u − f (p) 2(n−2) |s − s(p)| 2 . If the estimate holds forũ, it also holds for u. So from now on we assume f (p) = 0.  .46). There exists a constant C = C(n) > 0 such that for all k ∈ Z + , the following estimates hold We will also denote λB k (p) to the concentric ball withB k (p) but the radius scaled by λ ∈ (0, 1). This lemma follows straightforwardly from Lemma 3.1 and the definition of ω(r). So we omit the proof. By triangle inequality, we get the following estimates Since u k −u k+1 are g β -harmonic functions on 1 2B k , applying the gradient and Laplacian estimates (3.5) and (3.9) for harmonic functions, we get: Lemma 3.6. There exists a constant C(n) > 0 such that for all k ∈ Z + it holds that

50)
and where we recall that D ′ denotes the first order operators ∂ ∂s i for i = 5, . . . , 2n. The following lemma can be proved by looking at the Taylor expansion of u k at p for k >> 1 as in Lemma 2.8 of [20].
Lemma 3.7. The following limits hold:
Combining Lemmas 3.6 and 3.7, we obtain the following estimates on the 2nd-order (tangential) derivatives.
Proof. From triangle inequality we have for any given z ∈ B β (0, 1/2)\S The estimates for ∆ i u can be proved similarly.
For any other given point q ∈ B β (0, 1/2)\S, we can solve Dirichlet boundary problems as u k with the metric balls centered at q, and we obtain a family of functions v k such that whereB k (q) are metrics balls centered at q given bỹ Similar estimates as in Lemmas 3.5, 3.6 and 3.6 also hold for v k within the ballsB k (q).
We are now ready to state the main result in this subsection on the continuity of second order derivatives.
There exists a constant C = C(n) > 0 such that if u solves the conical Laplace equation (1.2), then the following holds for i = 1, 2: Proof. We only prove the estimate for (D ′ ) 2 u, and the one for ∆ i u can be dealt with in the same way. We may assume r p = min(r p , r q ). We fix an integer ℓ such that τ ℓ is comparable to d, more precisely, we take We calculate by triangle inequality We will estimate I 1 -I 4 one by one.
• I 1 and I 4 : by (3.51) and (3.52) we have and similar estimate holds for I 4 as well • I 3 : by the choice of ℓ, it is not hard to see that 2 3B ℓ (q) ⊂B ℓ (p). In particular u ℓ and v ℓ are both defined on 2 3B ℓ (q) and satisfy the equations respectively on this ball. From (3.47) for u ℓ and similar estimate for v ℓ we get whereq is the center of the ballB ℓ (q). U is g β -harmonic in 2 3B ℓ (q) and satisfies the estimate: • I 2 : this is a little more complicated than the previous estimates. We define h k = u k−1 −u k for k ≤ ℓ. h k is g β -harmonic onB k (p) and by (3.47 On the other hand, the function (D ′ ) 2 h k is also g β -harmonic on 2 3B k (p) so the gradient estimate implies that Integrating this along the minimal g β -geodesic γ connecting p and q and noting that γ avoids S since (C n \S, g β ) is strictly geodesically convex, we get (3.57) Observe that p, q ∈B 2 (p) and the function (D ′ ) 2 u 2 is g β -harmonic onB 2 (p). From (3.47) and derivative estimates we have Again by gradient estimate we have ). Integrating along the minimal geodesic γ we arrive at Combining (3.57), we obtain that Combing this with the estimates for I 1 , I 2 , I 3 , I 4 , we get Proposition 3.4 now follows from this and the fact that ω(r) is monotonically increasing.

Mixed normal-tangential estimates along the directions S
Throughout this section, we fix two points p, q ∈ B β (0, 1/2)\S and assume r p ≤ r q . Recall that we introduce the weighted "polar coordinates" (r i , θ i ) for (z 1 , z 2 ) as Let u k (resp. v k ) be the solutions to equations (3.44), (3.45) and (3.46) onB k (p) (resp.B k (q)). Recall u k − u k+1 satisfies (3.49) and apply gradient estimates to the g β -harmonic function u k − u k+1 , we get the bound of ∇ g β (u k − u k+1 ) L ∞ ( 1 3B k (p)) , which in particular implies that for i = 1 or 2 Similarly D ′ u k −D ′ u k+1 is also g β -harmonic on 1 2B k (p) and apply gradient estimates to this function we get for i = 1, 2 The following lemma can be proved by the same way as in Lemma 2.10 of [20] since p ∈ S, so we omit the proof.
Lemma 3.8. The following limits hold: for i = 1 or 2 and Similar formulas also hold for v k at the point q.
We are going to estimate the quantities We will estimate the case for i = 1 and J, since the other cases are completely the same. By triangle inequality we have Lemma 3.9. There exists a constant C(n) > 0 such that J 1 , J 3 and J 4 satisfy Proof. The estimates for J 1 and J 4 can be proved similarly as in proving those of I 1 and I 4 as in Section 3.2, using (3.60) and (3.61). J 3 can be estimated similar to that of I 3 as in Section 3.2, using (3.60). So we omit the details.
To estimate J 2 , as in Section 3.
Lemma 3.10. There exists a constant C = C(n, β) > 0 such that for any z ∈ 1 4B k (p)\S, the following pointwise estimate holds for all k ≤ min(ℓ, k p ) Proof. We define a function F as The Laplacian estimates (3.9) and derivative estimates applied to the g β -harmonic function The intersection of B β (x, τ k ) with complex plane C passing through x and orthogonal to the hyperplane S 1 lies in a metric ball of radius τ k under the standard cone metricĝ β 1 on C. We view the equation (3.63) as defined on the ballB := B C (x, (τ k ) 1/β 1 ) ⊂ C. The estimate (2.2) applied to the function D ′ h k gives rise to Therefore on B C (x, (τ k ) 1/β 1 /2)\{x} the following holds Remark 3.3. By similar arguments we can get the following estimates as well for any k ≤ min(ℓ, Lemma 3.11. There exists a constant C = C(n, β) > 0 such that for all k ≤ min(k p , ℓ) and z ∈ 1 4B k (p)\S the following pointwise estimates hold Proof. Applying the gradient estimate to the g β -harmonic function D ′ h k , we get . The function ∂ θ 1 D ′ h k is also a continuous g β -harmonic function so the derivatives estimates implies on 1 3B k (p)\S where F 1 is defined below We apply similar arguments as in the proof of Lemma 3.10. For any x ∈ S 1 ∩ 1 4B k (p), we view the equation (3.70) as defined on the C-ball B C (x, (τ k ) 1/β 1 ), and by the estimate (2.2) we have on Equivalently, this means that on B C (x, (τ k ) 1/β 1 /2)\{x} Again by the inclusion (3.66), we get (3.68). The estimate (3.69) follows from Lemma 3.10, (3.68), (3.64) and the equation (from (3.63)) below There exists a constant C(n, β) > 0 for k ≤ min(k 2,p , ℓ), the following pointwise estimates hold for any z (3.71) Proof. By the Laplacian estimate in (3.9) for the harmonic function D ′ h k on 1 2B k (p), we have From (3.73) and the Laplacian estimates (3.9), we see that sup 1 2.4B k (p) |F 2 | ≤ Cτ −3k ω(τ k ). By similar argument by considering x ∈ 1 3B k (p)∩S 1 , we obtain from (3.74) that onB : This implies that for any z (3.75) Now taking ∂ ∂r 1 on both sides of ∆ β D ′ h k = 0, we get ) ω(τ k ). By similar argument for any y ∈ 1 3.2B k (p) ∩ S 2 , we apply the estimate (2.2) to ∂D ′ h k ∂r 1 , we get on A 1 := B C (y, (τ k ) 1/β 2 /2)\{y}, the punctured ball in the complex plane C of (Euclidean) radius (τ k ) 1/β 2 /2 and orthogonal to S 2 passing through y, Varying y ∈ 1 3.2B k (p) ∩ S 2 we get for any z ∈ 1 4B k \S, that the following pointwise estimate holds and Proof. We will consider the different cases r p = min(r p , r q ) ≤ 2d and r p = min(r p , r q ) > 2d.
◮ r p ≤ 2d. In this case, it is clear by the choice of ℓ that r p ≈ τ kp ≤ 2d ≤ τ ℓ+2 , so k p ≥ ℓ + 2. From our assumption when solving (3.45), r p = d β (p, S 1 ), i.e. r 1 (p) = r p ≤ 2d. By triangle inequality we have r 1 (q) ≤ 3d. We also remark that for k ≤ ℓ, τ k ≥ τ ℓ > 8d, in particular, the geodesics considered below all lie inside the balls 1 4B k (p), and the estimates in Lemma 3.10 -Lemma 3.12 holds for points on these geodesics.
◮ r p > 2d but ℓ ≥ k p + 1. When k ≤ k p , the estimate (3.78) follows in the same way as the case above. So it suffices to consider the case when k p +1 ≤ k ≤ ℓ. In this case the ballsB k (p) = B β (p, τ k ) and it can be seen by triangle inequality that the geodesic γ ⊂ 1 3B k (p)\S. Since the metric ballŝ B k (p) are disjoint with S we can use the smooth coordinates w 1 = z β 1 1 and w 2 = z β 2 2 as before, and everything becomes smooth under these coordinates inB k (p).
The estimate (3.79) can be shown by the same argument, so we skip the details.
Iteratively applying (3.78) for k ≤ ℓ, we get where the inequality can be proved by the same argument as in proving (3.78). Combining the estimates for J 1 , J 2 , J 3 , J 4 we finish the proof of (1.4).
We remark that in solving (3.45), we assume r 1 (p) ≤ r 2 (p), we need also to deal with the following case, whose proof is more or less parallel to that of Lemma 3.13, so we just point out the differences and sketch the proof.
• k 2,p + 1 ≤ k ≤ ℓ. The ballsB k (p) are disjoint with S 2 , so we can introduce the complex coordinate w 2 = z β 2 2 on these balls as before. Let t 1 , t 2 be the real and imaginary parts of w 2 , respectively. The derivatives estimates imply that where ∂ 2 w 2 denotes the full second order derivatives in the {t 1 , t 2 }-directions. And ∂ ∂r 1 In this case we know that r 1 (p) ≈ τ kp ≥ 2τ k ≥ τ ℓ > 8d, so along γ Integrating along γ we get • k ≤ k 2,p . This case is completely the same as in the proof of (3.78), by replacing r 1 by r 2 , β 1 by β 2 . So we omit the details.

Mixed normal directions
In this section, we will deal with the Hölder continuity of the following four mixed derivatives: which by our previous notation correspond to N 1 N 2 u. Since the proof for each of them is more or less the same, we will only prove the Hölder continuity for ∂ 2 u ∂r 1 ∂r 2 . The following holds at p and q by the same reasoning of Lemma 3.7 lim k→∞ ∂ 2 u k ∂r 1 ∂r 2 (p) = ∂ 2 u ∂r 1 ∂r 2 (p), lim k→∞ ∂ 2 v k ∂r 1 ∂r 2 (q) = ∂ 2 u ∂r 1 ∂r 2 (q).
By triangle inequality Lemma 3.15. We have the following estimate Proof. We consider the different cases that k ≥ k p + 1 and ℓ ≤ k ≤ k p .
• k ≥ k p + 1. In this case the ballsB k (p) are disjoint with S and we can introduce the smooth coordinates w 1 = z β 1 1 and w 2 = z β 2 2 . Under the coordinates {w 1 , w 2 , z 3 , . . . , z n }, g β becomes the standard Euclidean metric g C n and the metric ballsB k (p) become the standard Euclidean ball with the same radius and center p. Since the g β -harmonic functions u k − u k+1 satisfy (3.49), by standard gradient estimates for Euclidean harmonic functions we get where we use D w i to denote either ∂ ∂w i or ∂ ∂w i for simplicity. From (3.88) and similar formula for ∂ ∂r 1 , we get • If ℓ ≥ k 2,p + 1 and ℓ ≤ k p = k 1,p . For all ℓ ≤ k, the ballsB k (p) are disjoint with S 2 and center at p 1 . We can still use w 2 = z β 2 2 as the smooth coordinate. The cone metric g β becomes smooth in w 2 -variable and we can apply the standard gradient estimate to the g β -harmonic function D w 2 (u k − u k−1 ) to get Again by (3.88), we get • If ℓ ≤ k 2,p , the case when k ≥ k 2,p +1 can be dealt with similarly as above. In the case ℓ ≤ k ≤ k 2,p , r 2 (p) ≈ τ k 2,p ≤ τ k ≤ τ ℓ ≈ 8d. Now the ballsB k (p) are centered at p 1,2 ∈ S 1 ∩ S 2 . We can proceed as in the proof of Lemma 3.12 with the harmonic functions u k − u k−1 replacing D ′ h k as in that lemma to prove that for any z ∈ 1 3B k (p)\S ) ω(τ k ).
In particular, the estimate in each case holds at p and from r 1 (p) ≤ r 2 (p) ≤ τ k , we obtain Therefore by triangle inequality The estimate for L 4 can be dealt with similarly by studying the derivatives of v k at q.
Lemma 3.18. For any k ≤ ℓ and any point z ∈ 1 3B k (p)\S the following estimates hold Proof. This follows from almost the same argument as in the proof of Lemma 3.17, by studying the harmonic functions h k and D ′ h k instead of ∂h k ∂θ 1 .
Lemma 3.19. The following estimate holds for any k ≤ ℓ and any z ∈ 1 3B k (p)\S Proof. By the Laplacian estimates (3.9) we have Applying again the Laplacian estimate (3.9) to the g β -harmonic function ∆ 1 h k , We consider the equation From the estimates above, we see F 7 L ∞ ( 1 1.8B k (p)) ≤ Cτ −2k ω(τ k ). • When k 2,p + 1 ≤ k ≤ min(ℓ, k p ), we directly apply gradient estimate to ∆ 1 h k to get • When k ≤ k 2,p , the ballsB k (p) are centered at p 1,2 , and we can apply the usual C-ball type estimate to get that for any z Recall the following equation holds Lemma 3.20. There exists a constant C = C(n, β) > 0 such that for all k ≤ ℓ and z ∈ 1 3B k (p)\S Proof. It follows from the Laplacian estimate (3.9) that And by (3.9) again we have We look at the equation and F 8 satisfies sup 1 1.4B k (p) |F 8 | ≤ Cτ −2k ω(τ k ). By the estimate (2.2) as we did before it follows that for any z ∈ 1 2B k (p)\S (remember here k ≤ min(ℓ, k p )) Taking ∂ ∂r 1 on both sides of the equation ∆ β ∂h k ∂θ 2 = 0, we get (3.103) Here |F 9 (z)| ≤ Cr 1 (z) for any z ∈ 1 2B k (p)\S. Therefore we get by the usual C-ball argument that • If k ≤ k 2,p , then for any z ∈ 1 3B k (p)\S ∂ ∂r 2 Lemma 3.21. The following estimate holds for any k ≤ ℓ and any z ∈ 1 3B k (p)\S ∂ ∂r 2 Proof. We first observe the following equation It can be shown that for any z ∈ 1 2B k (p)\S by the C-ball argument that ∂ ∂r 1 From Lemma 3.18, we have for any z ∈ 1 2B k (p)\S From Lemma 3.20, we have for any z ∈ 1 2B k (p)\S Therefore for any z ∈ 1 3B k (p)\S we have It remains to estimate L 2 . For simplicity, we denote h k := −u k + u k−1 as before, where we take k ≤ ℓ. We will denote β max = max(β 1 , β 2 ). Lemma 3.22. Let d = d β (p, q). There exists a constant C(n, β) > 0 such that for all k ≤ ℓ Proof. ◮ Case 1: First we assume that r p ≤ 2d so r q ≤ 3d and ℓ + 2 ≤ k p , in particular, the ballŝ B k (p) are centered at either p 1 ∈ S 1 or 0, depending on whether k ≥ k 2,p + 1 or k ≤ k 2,p . As in the proof of Lemma 3.13, let γ : [0, d] → B β (0, 1)\S be the g β -geodesic connecting p and q. The two points q ′ and p ′ are defined as in (3.80), γ 1 , γ 2 , γ 3 the g β -geodesics as defined in that lemma. We calculate by triangle inequality that Integrating along γ 3 on which the coordinates (r 1 ; r 2 , θ 2 ; z 3 , . . . , z n ) are the same as p, we get by (3.93) Integrating along γ 2 along which the coordinates (θ 1 ; r 2 , θ 2 ; z 3 , . . . , z n ) are the same as p ′ or q ′ , we get by (3.99) that To deal with the term L ′ 3 , we consider different cases of k, either ℓ ≥ k ≥ k 2,p + 1 or k ≤ k 2,p .
In this case ℓ ≤ k p , Lemmas 3.17 -3.21 hold for all k ≤ ℓ and r 1 (p) ≈ τ kp ≤ τ ℓ , so r 1 (γ(t)) ≤ d + r 1 (p) ≤ 9 8 τ ℓ ≤ 9 8 τ k . We calculate the gradient of ∂ 2 h k ∂r 1 ∂2 along the geodesic γ as follows (1). When k 2,p + 1 ≤ k ≤ ℓ we have along γ Then by Lemmas 3.17 -3.21 we have along γ that Integrating this inequality along γ we get (2). When k ≤ k 2,p , we have along γ Then by Lemmas 3.17 -3.21 we have along γ that Integrating this inequality along γ we again get ◮ Case 3: here we assume min(r p , r q ) = r p ≥ 2d, but ℓ ≥ k p + 1. The case when k ≤ k p can be dealt with by the same argument as in Case 2, so we omit it and only consider the cases when k p + 1 ≤ k ≤ ℓ, so here r 2 (p) ≥ r 1 (p) ≥ τ k ≥ τ ℓ > 8d so r 1 (γ(t)) ≥ 7 8 τ k and r 2 (γ(t)) ≥ 7 8 τ k for any point γ(t) in the geodesic γ. By triangle inequality it follows that γ ⊂ 1 3B k (p) = B β (p, τ k /3). As before, we can introduce smooth coordinates w 1 = z β 1 1 and w 2 = z β 2 2 , and g β becomes the standard smooth Euclidean metric g C n under these coordinates. Moreover, h k are the usual Euclidean harmonic function ∆ g C n h k = 0 onB k (p). By the standard derivative estimates we have From the equation From this we see that Integrating along γ we see that Combining the estimates in all three cases, we finish the proof of Lemma 3.22.

By Lemma 3.22 it follows that
To finish the proof, it suffices to estimate the first term on the RHS of the above equation. Recall we assume u 2 is a g β -harmonic function defined on the ballB 2 (p), which is centered at p 1,2 ∈ S 1 ∩S 2 and has radius 2τ 2 . u 2 satsifies the L ∞ estimate by maximum principle: there exists some C = C(n) > 0 such that Recall that the proofs of the estimates in Lemmas 3.17 -3.21 in the case when k ≤ k 2,p work for any g β -harmonic functions defined on suitable balls, and we can repeat the arguments there replacing the L ∞ -estimates of h k that h k L ∞ ≤ Cτ 2k ω(τ k ), by the L ∞ -estimate of u 2 as in (3.107), to get similar estimates as in those lemmas, which we will not repeat here. Given these estimates, we can repeat the proof of Lemma 3.22 to prove the following estimates This inequality, combined with (3.106) give the final estimate of the term L 2 , that By Lemma 3.15, Lemma 3.16 and the estimate (3.108) for L 2 , we are ready to prove the following estimate (see the equation (1.5)) Proposition 3.5. For the given p, q ∈ B β (0, 1/2)\S, there is a constant C = C(n, β) > 0 such that Proof. From Lemma 3.15, Lemma 3.16 and the estimate (3.108) for L 2 , we have where the last inequality follows from the fact that ω(r) is monotonically increasing.
Finally, we remark that the estimates for the other operators in (3.89) follows similarly, so we omit the detailed proof and just state that the estimates are the same as the estimates for ∂ 2 u ∂r 1 ∂r 2 as in Proposition 3.5.

Non-flat conical Kähler metrics
In this section, we will consider the Schauder estimates for general conical Kähler metrics on B β (0, 2) ⊂ C n with cone angle 2πβ along the simple normal crossing hyper-surface S. Let ω be such a metric. By definition, there exists a constant C ≥ 1 such that where ω β is the standard flat conical metric as before. Since ω is closed and B β (0, 2) is simply connected, we can write ω = √ −1∂∂φ for some strictly pluri-subharmonic function φ. By elliptic regularity, φ is Holder continuous under the Euclidean metric on B β (0, 2).
Our interest is to study the Laplacian equation where f ∈ C 0,α β (B β (0, 1)) and u ∈ C 2,α β . We will prove the following scaling-invariant interior Schauder estimates, and the proof follows closely from that of Theorem 6.6 in [19]. So we mainly focus on the differences. Proposition 3.6. There exists a constant C = C(n, β, g * .

(3.111)
Proof. Given any points Let µ ∈ (0, 1/4) be a small number to be determined later. Denote d = µd x 0 and B := B β (x 0 , d), . . , z n }, under which g β becomes the Euclidean one and the components of g become C α in the usual sense. The equation (3.110) has C α leading coefficients and we can apply Theorem 6.6 in [19] to conclude that (the following inequality is understood in the new coordinates) Recall T denotes the second order operators appearing in (2.1). Let D denote the ordinary first order operators in {w 1 , w 2 , z 3 , . . . , z n }. Then we calculate Then we get d 2+α (3.113) Letx 0 ∈ S be the nearest point of x 0 to S. We consider the ballŝ B := B β (x 0 , 2d) which is contained in B β (0, 1) by triangle inequality. As in [14], we introduce a (non-holomorphic) basis of T * 1,0 (C n \S) as ǫ j := dr j + √ −1β j r j dθ j , dz k j=1,2; k=3,...,n , and the dual basis of T 1,0 (C n \S): We can write the (1, 1)-form ω in the basis {ǫ j ∧ǭ k , ǫ j ∧ dz k , dz k ∧ǭ j , dz j ∧ dz k } as ω = g ǫ jǭk ǫ j ∧ǭ k + g ǫ jk ǫ j ∧ dz k + g kǭ j dz k ∧ǭ j + g jk dz j ∧ dz k , (3.115) We remark that all the second order derivatives of φ appearing in (3.115) are linear combination of |z j | 2−2β j ∂ 2 ∂z j ∂zj N j N k (j = k), N j D ′ and (D ′ ) 2 , which are studied in Theorem 1.1. The standard metric ω β becomes the identity matrix under the basis above for (1, 1)-forms. If ω is C 0,α β , all the coefficients in the expression of ω in (3.114) are C 0,α β continuous and the cross terms g ǫ jǭk with j = k, g ǫ jk tend to zero when approaching the corresponding singular sets S j or S k . Moreover, the limit of g jk dz j ∧ dz k as tending to S 1 ∩ . . . ∩ S p defines a Kähler metric on it. Rescaling or rotating the coordinates if necessary we may assume atx 0 ∈ S, g ǫ jǭj (x 0 ) = 1, g jk (x 0 ) = δ jk and the cross terms vanish atx 0 . Let ω β be the standard cone metric under these new coordinates nearx 0 , and we can write the equation (3.110) as for some Hermtian matrix η(z) = (η jk ) n j,k=1 , η jk = g jk (z) − g jk β . It is not hard to see the term η(z).i∂∂u can be written as 2 j,k=1 (3.116) and g with the upper indices denotes the inverse matrix of g. We consider the equivalent form of the equation (3.110) onB, Observe that x 0 , y 0 ∈ B β (x 0 , 3d/2) we can apply the scaled inequality (1.7) of Theorem 1.1 to conclude that , thus d 2+α . (3.117) Combining (3.113), (3.117) and (3.118) we get d 2+α

Proposition 3.6 then follows from interpolation inequalities.
Remark 3.4. It follows easily from the proof of Proposition 3.6 that the estimate (3.111) also holds for metric balls B β (p, R) ⊂ B β (0, 1) whose center p may not lie at S.
Proof. Given the estimates in Proposition 3.6, the proof is identical to that of Lemma 6.20 in [19]. So we omit the details.
Proof. Consider the function Hence ∆ g (N w ± u) ≤ 0 and from the definition of w we also have w| ∂B β ≡ 0, by maximum principle we obtain that |u( x , hence the lemma is proved. Given any function f ∈ C 0,α β (B β ), the Dirichlet problem ∆ g u = f in B β and u ≡ 0 on ∂B β admits a unique solution u ∈ C 2,α β (B β ) ∩ C 0 (B β ). Proof. The proof of this proposition is almost identical to that of Theorem 6.22 in [19]. For completeness, we provide the detailed argument. Fix a σ ∈ (0, 1). We define a family of operators ∆ t = t∆ g + (1 − t)∆ g β and it is straightforward to see that ∆ t is associated to some cone metric which also satisfies (3.109). We study the Dirichlet problem Equation ( * 0 ) admits a unique solution u 0 ∈ C 2,α β (B β ) ∩ C 0 (B β ) by Proposition 3.2. By Theorem 5.2 in [19], in order to apply the continuity method to solve ( * 1 ), it suffices to show ∆ −1 t defines a bounded linear operator between some Banach spaces. More precisely, define By definition any u ∈ B 1 is continuous on B β and u = 0 on ∂B β . By Lemmas 3.23 and 3.24, we have for some constant C independent of t ∈ [0, 1]. Thus ( * 1 ) admits a solution u ∈ B 1 .
Corollary 3.2. For any given ϕ ∈ C 0 (∂B β ) and f ∈ C 0,α β (B β ), the Dirichlet problem ∆ g u = f, in B β , and u = ϕ, on ∂B β , (3.120) Proof. We may extend ϕ continuously to B β and assume ϕ ∈ C 0 (B β ). Take a sequence of functions ϕ k ∈ C 2,α β (B β ) ∩ C 0 (B β ) which converges uniformly to ϕ on B β . The Dirichlet problem ∆ g v k = f − ∆ g ϕ k in B β and v k = 0 on ∂B β admits a unique solution v k ∈ C 2,α β (B β ) ∩ C 0 (B β ). Thus the function u k := v k + ϕ k ∈ C 2,α β satisfies ∆ g u k = f in B β and u k = ϕ k on ∂B β . u k is uniformly bounded in C 0 (B β ) by maximum principle. Corollary 3.1 gives uniformly C 2,α β (K)-bound on any compact subset K ⋐ B β . Letting k → ∞ and K → B β , by a diagonal argument and up to a subsequence u k → u ∈ C 2,α β (B β ). On the other hand, from ∆ g (u k − u l ) = 0 we see that {u k } is a Cauchy sequence in C 0 (B β ) thus u k converges uniformly to u on B β . Hence u ∈ C 0 (B β ) and satisfies the equation (3.120).
Corollary 3.4. Let X be a compact Kähler manifold and D = j D j be a divisor with simple normal crossings. Let g be a conical Kähler metric with cone angle 2πβ along D. Suppose u ∈ H 1 (g) is a weak solution to the equation ∆ g u = f in the sense that X ∇u, ∇ϕ ω n g = − X f ϕω n g , ∀ϕ ∈ C 1 (X) for some f ∈ C 0,α β (X). Then u ∈ C 2,α β (X) ∩ C 0 (X) and there exists a constant C = C(n, β, g, α) such that Proof. We can choose finite covers of D, {B a }, {B ′ a } with B ′ a ⋐ B a and centers at D. By assumption u is a weak solution to the equation ∆ g u = f in each B a , then by Corollary 3.3 we conclude that u ∈ C 2,α β (B a ) for each B a . On X\S, the metric g is smooth so standard elliptic theory implies that u ∈ C 2,α loc (X\S). Since {B a } covers D, u ∈ C 2,α β (X). We can apply Corollary 3.1 to obtain that for some constant C > 0 On X\ ∪ a {B ′ a } the metric g is smooth, the usual Schauder estimates apply. We finish the proof of the Corollary by the definition of C 2,α β (X) (c.f. Definition 2.8).
Remark 3.5. Let (X, D, g) be as in Corollary 3.4. It is easy to see by variational method weak solutions to ∆ g u = f always exist for any f ∈ L 2 (X, ω n g ) satisfying X f ω n g = 0.

Parabolic estimates
In this section, we will study the heat equation with background metric ω β and prove the Schauder estimates for such solution u ∈ C 0 (Q β ) ∩ C 2,1 (Q # β ) to the equation for a function f ∈ C 0 (Q β ) with some better regularity.
Proof. We will only prove the estimate for |∇ 1 ∇ 2 u ǫ |. The others are similar, so we omit the proof. By similar calculations as in deriving (3.27), we have and similar to (3.20) (4.10) Combining (4.10), (4.9) and Cauchy-Schwarz inequality, we get We define a similar cut-off function η as ϕ in the proof of Lemma 4.2, such that η = 1 on B gǫ (0, R/2) and vanishes outside B gǫ (0, 3R/5). We denote We can argue similarly as the F in the proof of Lemma 4.2 that at the maximum point (p 0 , t 0 ) of G, for which we assume G(p 0 , t 0 ) > 0 so it follows that G(p 0 , t 0 ) ≤ C 1 + t 0 R 2 . Therefore by definition of G, it holds that on B gǫ (0, R/2) × (0, R 2 ) thus by Lemmas 4.1 and 4.2, we conclude that on B gǫ (0, R/2) × (0, R 2 ) Existence of u to (4.2). We will show the limit function of u ǫ as ǫ → 0 solves (4.2).
Proof. Let u ǫ be the C 2,1 -solution to the equation (4.4). The C 0 -norm of u ǫ follows from maximum principle (4.3).

Sketched proof of Theorem 1.3
With Proposition 4.1, we can prove the Schauder estimates for the solution u ∈ C 0 (Q β )∩C 2,1 (Q # β ) to the equation (4.1) for a Dini-continuous function f , by making use of almost the same arguments as in the proof of Theorem 1.1. So we will not provide the full details, and only point out the main differences. For any given points Q p = (p, t p ), Q q = (q, t q ) ∈ (B β (0, 1/2)\S) × (t, 1). To define the approximating functions u k as in (3.44), we define u k in this case as the solution to the heat equation and u k = u on ∂ P B k (p) × (t p −t · τ 2k , t p ] , whereB k (p) is defined in (3.48). We can now apply the estimates in Proposition 4.1 to the functions u k or u k − u k−1 , instead of the ones in Lemmas 3.3 and 3.4 as we did in Sections 3.2, 3.3 and 3.4 to prove the Schauder estimates for u. Thus we finish the proof of Theorem 1.3.
Recall T denotes the operators in T and ∂ ∂t , then by (4.16) it follows that (4.17) Letx ∈ S be the projection of x to S andP x = (x, t x ) be the corresponding space-time point. DenoteQ := B β (x, 2d) × (t x − 4d 2 , t x ]. As the Case 1.2 in the proof of Proposition 3.6, we may choose suitable complex coordinates so that g ǫ jǭk (P x ) = δ jk and for j, k ≥ p + 1 g jk (P x ) = δ jk , and the cross terms in the expansion of g in (3.114) vanish atP x . Thus the equation (4.15) can be re-written as for some (1, 1)-form η as in the proof of Proposition 3.6. From the rescaled version of Theorem 1.3 we conclude that . (4.18) • Case 2. d P,β (P x , P y ) ≥ d/2. Then we calculate (recall Q β := B β (0, 1) Combining (4.17), (4.18) and (4.19), we obtain Observe that for any P ∈ Q or ∈Q, d P,β (P, ∂ P Q β ) ≥ (1− 2µ)d Px . Then it follows from definition that f . We calculate where in the last inequality we use the interpolation inequality, by which we also have 8 If µ is chosen small such that µ α (2C 1 [g] * C α, α 2 β (Q β ) + 1) < 1/2, combining the above inequalities we get d 2+α .
Taking supremum over all P x = P y ∈ Q β , we obtain that .

The proposition is proved by invoking the interpolation inequalities.
Remark 4.2. It follows from the proof that the estimates in Proposition 4.2 also hold on Q β (p, R) := B β (p, R) × (0, R 2 ) ⊂ Q β , i.e. the cylinder whose spatial center p may not lie in S.
It is easy to derive the following local Schauder estimate for C Corollary 4.2. Let K ⋐ B β (0, 1) be a compact subset and ε 0 ∈ (0, 1) be a given number. Assumptions as in Proposition 4.2, there exists a constant C = C(n, β, α, g, K, ε 0 ) > 0 such that .
Lemma 4.4. Let σ ∈ (0, 1) be a given number and u ∈ C 2+α, 2+α Then there is a constant C = C(n, α, β, g, σ) > 0 such that Proof. The lemma follows from definitions of the norms and the estimates in Proposition 4.2.
Proof. The uniqueness follows from maximum principle, so it suffices to show the existence. We will use the continuity method. Define a continuous family of linear operators: for s ∈ [0, 1], It can been seen that L s = ∂ ∂t − ∆ gs for some conical Kähler metric g s which uniformly equivalent to g β and has uniform C α, α 2 β -estimate. So the interior Schauder estimates holds also for L s . Fix a σ ∈ (0, 1). Define Observe that any u ∈ B 1 is continuous in Q β and vanishes on ∂ P Q β . L s defines a continuous family of linear operators from B 1 to B 2 . By Lemmas 4.4 and 4.5 we have 1], and ∀u ∈ B 1 .
By Corollary 4.1 and Remark 4.1, L 0 is invertible, thus by Theorem 5.2 in [19], L 1 is also invertible. Proof. The proof is identical to that of Corollary 3.2 by an approximation argument. We may assume ϕ ∈ C 0 (Q β ) and choose a sequence of ϕ k ∈ C 2+α, 2+α 2 β (Q β ) which converges uniformly to ϕ on Q β . The equations ∂v k ∂t = ∆ g v k + f − ∆ g ϕ k , v k ≡ 0 on ∂ P Q β admits a unique C 2+α, 2+α . The C 0 -convergence u k → u is uniform on Q β by maximum principle so u = ϕ on ∂ P Q β , as desired.
We recall the definition of weak solutions and refer to Section 7.1 in [17] for the notations.
Definition 4.1. We say a function u on Q β is a weak solution to the equation ∂u ∂t = ∆ g u + f , if (1) u ∈ L 2 0, 1; H 1 (B β ) and ∂u ∂t ∈ L 2 0, 1; H −1 (B β ) ; (2) For any v ∈ H 1 0 (B β ) and t ∈ (0, 1) On can use the classical Galerkin approximations to construct weak solution to the equation ∂u ∂t = ∆ g u + f (see Section 7.1.2 in [17]). If f has better regularity, so does the weak solution u. Proof. Sobolev inequality holds for the metric g so by the proof of the standard De Giorgi-Nash-Moser theory for parabolic equations implies that u is in fact continuous on Q β . Since the metric g is smooth on Q # β , the weak solution u is also a weak solution in Q # β with the smooth background metric, so u ∈ C 2+α, 2+α 2 loc (Q # β ) in the usual sense by the classical Schauder estimates. Thus it suffices to consider points at S. We choose the worst such points 0 ∈ S only, since the case when centers are in other components of S is even simpler. We fix the point P 0 = (0, t 0 ) ∈ Q β with t 0 > 0. Fix an r ∈ (0, √ t 0 ). By Corollary 4.3 the equation with boundary value v = u on ∂ P Q β (P 0 , r) admits a unique solution v ∈ C 2+α, α+2 2 β (Q β (P 0 , r)). Then by maximum principle u = v in Q β (P 0 , r). Thus u ∈ C 2+α, α+2 2 β (Q β (P 0 , r)) too. The argument also works at other space-time points in S P , we see that u ∈ C 2+α, 2+α 2 β (Q β ), as desired.
Proof. The weak equation can be constructed using the Galerkin approximations ( [17]). The uniqueness is an easy consequence of maximum principle. The regularity of u follows from the local results in Lemma 4.6. The estimate follows from maximum principle, a covering argument as in Corollary 3.4 and the local estimates in Corollary 4.2.
The interior estimate in Corollary 4.4 is not good enough to show the existence of solutions to non-linear partial differential equations since the estimate becomes worse as t approaches t = 0. We need some global estimates in the whole time interval t ∈ [0, 1] if the initial u 0 has better regularity.

Schauder estimate near t = 0
In this subsection, we will prove a Schauder estimate in the whole time interval for the solutions to the heat equation when the initial value is 0 or has better regularity. We consider the model case with the background metric g β first, then we generalize the estimate to general non-flat conical Kähler metrics. 4.4.1. The model case. In this subsection, we will assume the background metric is g β . Let u be the solution to the equation and u = ϕ ∈ C 0 on ∂B β × (0, 1], where f ∈ C α,α/2 β (Q β ). In the calculations below, we should have used the smooth approximating solutions u ǫ , where ∂ t u ǫ = ∆ gǫ u ǫ + f and u ǫ = u on ∂ P Q β . But by letting ǫ → 0, the corresponding estimates also hold for u. So for simplicity, below we will work directly on u.
We fix 0 < ρ < R ≤ 1 and denote B R := B β (0, R) and Q R := B R × [0, R 2 ] in this section. Let u be the solution to (4.22). We first have the following Caccioppoli inequalities.
Proof. We fix a cut-off function η such that supp η ⊂ B R and η = 1 on B ρ , |∇η| g β ≤ 2 R−ρ . Multiplying both sides of the equation (4.22) by η 2 u, and applying integration by parts we get (4.23) follows from this inequality by integrating over t ∈ [0, s 2 ] for all s ≤ ρ. To see (4.24), observe that the Bochner formula yields that Multiplying both sides of this inequality by η 2 and applying IBP, we get (4.25) By a standard Moser iteration argument we get the following sub-mean value inequality.
Multiplyingu 2 = ∂u 2 ∂t on both sides of the equation for u 2 and noting thatu 2 = 0 on ∂B R × (0, R 2 ), we get Integrating over t ∈ [0, R 2 ], we obtain Then for ρ < R we have The estimate is proved by an iteration lemma (see Lemma 3.4 in [22]).
Lemma 4.11. Suppose u satisfies the equations (4.22). There exists a constant C = C(n, β, α) > 0 such that for any ρ ∈ (0, R/2) Proof. From Lemma 4.10, we get On the other hand by Hölder inequality The lemma is proved by combining the two inequalities above.
Therefore for any x 0 ∈ B β (x, R/4) It is then elementary to see by triangle inequality that (by adjusting R slightly if necessary) as desired. The estimate foru follows from the equationu = ∆ g u + f . Remark 4.3. By a simple parabolic rescaling of the metric and time, we see from (4.28) that for any 0 < r < R < 1/10 that (4.29) 4.4.2. the non-flat metric case. In this subsection, we will consider the case when the background metrics are general non-flat C α,α/2 β -conical Kähler metrics g = g(z, t). Suppose u ∈ C 2+α, 2+α 2 β (Q β ) satisfies the equation ∂u ∂t = ∆ g u + f, in Q β , u| t=0 = 0, (4.30) and u ∈ C 0 (∂ P Q β ).
Proposition 4.4. There exists a constant C = C(n, β, α, g) > 0 such that Proof. Choose suitable complex coordinates at the origin x = 0, we may assume the components of g in the basis {ǫ j ∧ǭ k , · · · } satisfies g ǫ jǭk (·, 0) = δ jk and g jk (·, 0) = δ jk at the origin 0. As in the proof of Proposition 4.2, we can write the equation (4.30) as where η is given in the proof of Proposition 3.6. By (4.29) we get β (Q R ) + C(ε) u C 0 (Q R ) . By choosing R 0 = R 0 (n, β, α, g) > 0 small enough and suitable ε > 0, for any 0 < r < R < R 0 < 1/10, the combination of the above inequalities yields that By Lemma 4.13 below (setting φ(r) = [T u] C α,α/2 β (Qr) ), we conclude that . This is the desired estimate when the center of the ball is the worst possible. For the other balls B β (x, r) with center x ∈ B β (0, 1/2), we can repeat the above procedures and use the smooth coordinates w j = z β j j in case the ball is disjoint with S j . Finitely many such balls cover B β (0, 1/2) so we get the [T u] The proposition is proved by combining this inequality, the equation for u, interpolation inequalities, and the interior Schauder estimates in Corollary 4.2.
Proof. We setû = u − u 0 andf = f − ∆ g u 0 .û satisfies the conditions in Proposition 4.4, so the corollary follows from Proposition 4.4 applied toû and triangle inequalities. Then the estimate (4.31) follows from Corollary 4.7 and a covering argument as in the proof of Corollary 3.4.

Conical Kähler-Ricci flow
Let X be a compact Kähler manifold and D = j D j be a divisor with simple normal crossings.
Let ω 0 be a fixed C 0,α ′ β (X) conical Kähler metric with cone angle 2πβ along D andω t be a family of C α ′ , α ′ where f ∈ C α ′ ,α ′ /2 β (X × [0, 1]) is a given function. We will use an inverse function theorem argument in [4] which was outlined in [21] to show the short time existence of the flow (5.1).
Proof. The uniqueness of the solution follows from maximum principle. We will break the proof of short-time existence into three steps.
Step 2. We consider the small ball The map Ψ is well-defined and C 1 with the differential DΨ φ at any φ ∈ B is given by ≤ δ/2d P,g 0 (x, t), (y, t ′ ) α .