Some nonlinear differential inequalities and an application to Hölder continuous almost complex structures

www.elsevier.com/locate/anihpc

Some nonlinear differential inequalities and an application to Hölder continuous almost complex structures

Adam Coffman

, Yifei Pan

aDepartment of Mathematical Sciences, Indiana University – Purdue University Fort Wayne, 2101 E. Coliseum Blvd., Fort Wayne, IN 46805-1499, USA

bSchool of Sciences, Nanchang University, Nanchang 330022, PR China Received 22 July 2009; received in revised form 2 October 2009; accepted 14 October 2009

Available online 2 February 2011

Abstract

We consider some second order quasilinear partial differential inequalities for real-valued functions on the unit ball and find conditions under which there is a lower bound for the supremum of nonnegative solutions that do not vanish at the origin. As a consequence, for complex-valued functionsf (z)satisfying∂f/∂z¯= |f|^α, 0< α <1, andf (0)=0, there is also a lower bound for sup|f|on the unit disk. For eachα, we construct a manifold with anα-Hölder continuous almost complex structure where the Kobayashi–Royden pseudonorm is not upper semicontinuous.

©2011 Elsevier Masson SAS. All rights reserved.

MSC:primary 35R45; secondary 32F45, 32Q60, 32Q65, 35B05 Keywords:Differential inequality; Almost complex manifold

1. Introduction

We begin with an analysis of a second order quasilinear partial differential inequality for real-valued functions of nreal variables,

u−B|u|^ε0, (1)

whereB >0 andε∈ [0,1)are constants. In Section 2, we use a Comparison Principle argument to show that (1) has

“no small solutions,” in the sense that there is a uniform lower boundM >0 for the supremum of solutionsuwhich are nonnegative on the unit ball and nonzero at the origin.

We also consider a generalization of (1):

uu−B|u|^1+ε−C| ∇u|²0, (2)

and find conditions under which there is a similar property of no small solutions, in Theorem 2.4.

* Corresponding author.

E-mail addresses:CoffmanA@ipfw.edu (A. Coffman), Pan@ipfw.edu (Y. Pan).

0294-1449/$ – see front matter ©2011 Elsevier Masson SAS. All rights reserved.

doi:10.1016/j.anihpc.2011.02.001

As an application of the results on the inequality (1), we show failure of upper semicontinuity of the Kobayashi–

Royden pseudonorm for a family of 4-dimensional manifolds with almost complex structures of regularity C^0,α, 0< α <1. This generalizes theα= ¹₂ example of [5]; it is known [6] that the Kobayashi–Royden pseudonorm is upper semicontinuous for almost complex structures with regularityC^1,α.

Our construction of the almost complex manifolds in Section 4 is very similar to that of [5]; we give the details for the convenience of the reader, and to show how the argument breaks down asα→1⁻, due to a shrinking radius of the domain. We also take the opportunity in Section 3 to state some lemmas which allow for a more quantitative description than that of [5].

One of the steps in [5] is a Maximum Principle argument applied to a complex-valued functionh(z)satisfying the equation∂h/∂z¯= |h|^1/2, to get the property of no small solutions. The main difference between our paper and [5] is the use of a Comparison Principle in Section 2 instead of the Maximum Principle, and we arrive at this result:

Theorem 1.1.For anyα∈(0,1), supposeh(z)is a continuous complex-valued function on the closed unit disk, and on the set{z: |z|<1, h(z)=0},hhas continuous partial derivatives and satisfies

∂h

∂z¯ = |h|^α. (3)

Ifh(0)=0thensup|h|> S_α, where the constantS_α>0is defined by:

S_α=

2(1−α) 2−α

1/(1−α)

. (4)

2. Some differential inequalities

LetD_Rdenote the open ball inRⁿcentered at0 with radiusR >0, and letD_Rdenote the closed ball.

Lemma 2.1.Given constantsB >0and0ε <1, let M=

B(1−ε)² 2(2ε+n(1−ε))

¹

1−ε

>0.

Suppose the functionu:D1→Rsatisfies:

• uis continuous onD1,

• u(x) 0forx∈D1,

• on the open setω= {x∈D₁: u(x) =0},u∈C²(ω),

• forx∈ω:

u(x) −B u(x) ε

0. (5)

Ifu(0)=0, thensup_x∈D1u(x) > M . Proof. Define a comparison function

v(x) =M|x|^1−ε2 ,

sov∈C²(Rⁿ)since 0ε <1. By construction ofM, it can be checked thatvis a solution of this nonlinear Poisson equation on the domainRⁿ:

v(x) −Bv(x) ^ε≡0.

Suppose, toward a contradiction, thatu(x) Mfor allx∈D₁. For a pointx₀on the boundary ofω⊆Rⁿ, either

|x₀| =1, in which case by continuity,u(x₀)M=v(x₀), or 0<|x₀|<1 andu(x₀)=0, so u(x₀)v(x₀). Since uvon the boundary ofω, the Comparison Principle [4, Theorem 10.1] applies to the subsolutionuand the solution von the domainω. The relevant hypothesis for the Comparison Principle in this case is that the second term expression

of (5),−BX^ε, is weakly decreasing, which usesB >0 andε0. (To satisfy this technical condition for allX∈R, we define a functionc:R→Rbyc(X)= −BX^ε forX0, andc(X)=0 forX0. Thencis weakly decreasing inX,vsatisfiesv(x) +c(v(x)) ≡0 andusatisfiesu(x) +c(u(x)) 0.)

The conclusion of the Comparison Principle is thatuv onω, however0∈ωandu(0) > v(0), a contradic- tion. 2

Of course, the constant functionu≡0 satisfies the inequality (5), and so does the radial comparison functionv, so the initial conditionu(0)=0 is necessary.

Example 2.2.In then=1 case,M=(^B(1₂₍₁⁻₊^ε)_ε)²)^1−ε1 . For pointsc₁,c₂∈R,c₁< c₂, define a function

u(x)=

⎧⎪

⎨

⎪⎩

M(x−c2)^1−2ε ifxc2,

0 ifc₁xc₂,

M(c₁−x)¹−²ε ifxc₁.

Thenu∈C²(R), and it is nonnegative and satisfies u=B|u|^ε (the case of equality in then=1 version of (5)).

Forc₁<0< c₂, this gives an infinite collection of solutions of the ODEu=B|u|^ε which are identically zero in a neighborhood of 0, so the ODE does not have a unique continuation property. Forc₁>0 orc₂<0, the functionu satisfiesu(0)=0 and the other hypotheses of Lemma 2.1, and its supremum on(−1,1)exceedsM even though it can be identically zero on an interval not containing 0.

Example 2.3.In the casen=2,B=1,ε=0, (5) becomes the linear inequalityu1 and the number M=¹₄ agrees with Lemma 2 of [5], which was proved there using a Maximum Principle argument.

By applying Lemma 2.1 to the Laplacian of a power ofu, we get the following generalization.

Theorem 2.4.Given constantsB >0,C∈R, andε <1, let M=

⎧⎨

⎩

(_2(2(ε−^B(1_C)⁻₊^ε)_n(1²_−ε)))¹−¹ε ifCε, (^B(1_2n^−ε))^1−1ε ifCε.

Suppose the functionu:D₁→Rsatisfies:

• uis continuous onD₁,

• u(x) 0forx∈D₁,

• on the open setω= {x∈D₁: u(x) =0},u∈C²(ω),

• forx∈ω:

u(x)u( x) Bu(x) ^1+ε+C∇u(x) ².

Ifu(0)=0, thensup_x∈D1u(x) > M.

Proof. Letμ=min{ε, C}, soμε <1, and on the setω, u(x)u( x) Bu(x) ^1+ε+μ∇u(x) ².

Consider the functionu^1−μonD1, sou^1−μ∈C⁰(D1)∩C²(ω), and on the setω,

u^1−μ

=(1−μ)u^−μ−1

uu−μ| ∇u|² (1−μ)u^−μ−1Bu^1+ε

=(1−μ)B

u^1−μ(ε−μ)/(1−μ)

.

Since(1−μ)B >0, andμε <1⇒0^ε₁⁻₋^μ_μ<1, Lemma 2.1 applies tou^1−μ. If(u(0))^1−μ=0, then supu^1−μ>

(1−μ)B(1−^ε₁⁻₋^μ_μ)² 2(2^ε₁⁻₋^μ_μ+n(1−^ε₁⁻₋^μ_μ))

¹

1−ε−μ

1−μ ⇒ supu >

B(1−ε)² 2(2(ε−μ)+n(1−ε))

¹

1−ε

. 2

Functions satisfying a differential inequality of the form (1) or (2) also satisfy a Strong Maximum Principle; the only condition isB >0.

Theorem 2.5. Given any open set Ω⊆Rⁿ, and any constantsB >0,C, ε∈R, suppose the functionu:Ω →R satisfies:

• uis continuous onΩ,

• on the setω= {x∈Ω: u(x) > 0},u∈C²(ω),

• on the setω,usatisfies

uu−B|u|^1+ε−C| ∇u|²0.

Ifu(x₀) >0for somex₀∈Ω, thenudoes not attain a maximum value onΩ.

Proof. Note that the constant functionu≡0 is the only locally constant solution of the inequality forB >0. IfB=0 then obviously any constant function would be a solution.

Given a functionusatisfying the hypotheses,ωis a nonempty open subset ofΩ. Suppose, toward a contradiction, that there is somex₁∈Ω withu(x) u(x₁)for allx∈Ω. In particular,u(x₁)u(x₀) >0, sox₁∈ω. Letω₁be the connected component ofωcontainingx₁.

Forx∈ω₁,usatisfies the linear, uniformly elliptic inequality u(x) +

−B

u(x)ε−1 u(x) +

−C∇u(x) u(x)

· ∇u(x) 0,

where the coefficients (defined in terms of the given u) are locally bounded functions of x, and (−B(u(x)) ^ε−1) is negative for all x∈ω. It follows from the Strong Maximum Principle [4, Theorem 3.5] that since u attains a maximum value at x₁, then u is constant onω₁. Since the only constant solution is 0, it follows that u(x₁)=0, a contradiction. 2

The next lemma shows how an inequality like (5) withn=2 can arise from a first order PDE for a complex-valued function.

Lemma 2.6.Consider constantsα,γ∈Rwith0< α <1. Letω⊆Cbe an open set, and supposeh:ω→Csatisfies:

• h∈C¹(ω),

• h(z)=0for allz∈ω,

• ^∂h_∂¯z= |h|^αonω.

Then, the following inequality is satisfied onω:

|h|^(1−α)γ

4(1−α)γ −(2−α)²

|h|^{(1−α)(γ−2)}. (6)

Remark.The special caseα=¹₂,γ=³₂is Lemma 1 of [5]; its proof there is a long calculation in polar coordinates, which can be generalized to some other values ofαby an analogous argument. However, usingz,z¯coordinates allows for a shorter calculation.

Proof of Lemma 2.6. We first want to show that h is smooth on ω, applying the regularity and bootstrapping technique of PDE to the equation ∂h/∂z¯= |h|^α. We recall the following fact (for a more general statement, see

Theorem 15.6.2 of [1]): for a nonnegative integer , and 0< β <1, ifϕ∈C_loc ^,β(ω)andg:ω→Chas first deriva- tives inL²_loc(ω)and is a solution of∂g/∂z¯=ϕ, theng∈C_loc^+1,β(ω). In our case,ϕ= |h|^α∈C¹(ω)⊆C_loc^0,β(ω)(since h∈C¹(ω)and is nonvanishing), andg=hhas continuous first derivatives, so we can conclude thatg=h∈C_loc^1,β(ω).

Repeating gives thath∈C_loc^2,β(ω), etc.

Since the conclusion is a local statement, it is enough to expressωas a union of open subsetsω_k and establish the conclusion on each subset. For eachz_k∈ω, there is a sufficiently small diskω_k containingz_k, where real exponenti- ation ofh(z)is well defined onω_k, by choosing a single-valued branch of log to defineh^r=exp(rlog(h)).

The condition ^∂h_∂z¯= |h|^α can be re-written h_z¯=(h)¯ _z= |h|^α=h^α/2h¯^α/2.

This leads to

h_zz¯=(h_z¯)_z=

h^α/2h¯^α/2

z

=α 2

h^(α/2)−1h¯^α/2h_z+h^αh¯^α−1

= (h)¯ _zz¯

,

which is used in a line of the next step. For an arbitrary exponentm∈R, |h|^m

z¯z=

h^m/2h¯^m/2

z¯z

= ∂

∂z m

2h^m2−1h_¯zh¯^m2 +h^m2 m

2h¯^m2−1(h)¯ _z¯

=m 2

∂

∂z

h^m2−1+α2h¯^m2+α2 +h^m2h¯^m2−1(h)¯ _¯z

=m 2

m 2 +α

2 −1

h^m2+α2−2h_zh¯^m2+α2 +h^m2+α2−1 m

2 +α 2

h¯^m2+α2−1(¯h)_z

+m

2h^m2−1h_zh¯^m2−1(¯h)_z¯+h^m2 m

2 −1

h¯^m2−2(h)¯ _z(h)¯ _¯z+h^m2h¯^m2−1(h)¯ _z¯z

.

=m 2

Re

(m+α−2)|h|^m+α−4h¯²h_z +

m 2 +α

|h|^m+2α−2+m

2|h|^m−2|h_z|²

.

With the aim of applying Lemma 2.1 to the function|h|^m, we consider the expression (8), with real constantsB,ε, andm=0. In line (9), we assign

ε= 1

m(m+2α−2) (7)

to be able to combine like terms, and in line (10), we chooseB=4m−(2−α)²to complete the square.

|h|^m

−B

|h|^mε

(8)

=4

|h|^m

zz¯−B|h|^mε

=2m

Re

(m+α−2)|h|^m+α−4h¯²h_z +

m 2 +α

|h|^m+2α−2+m

2|h|^m−2|h_z|²

−B|h|^mε

=

m(m+2α)−B

|h|^m+2α−2 (9)

+Re

2m(m+α−2)|h|^m+α−4h¯²h_z

+m²|h|^m−2|h_z|² |h|^m−2

m²+2αm−B

|h|^2α−2|m||m+α−2||h|^α|hz| +m²|hz|²

= |h|^m−2

|m+α−2||h|^α− |m||h_z|2

0. (10)

Considering the form of (7), it is convenient to choosem=(1−α)γ for some constantγ=0. The claim of the lemma follows; theγ=0 case can be checked separately. 2

The parameter γ can be chosen arbitrarily large; to apply Lemma 2.1 to get the “no small solutions” result of Theorem 1.1, we need the RHS coefficient in (6) to be positive, so γ >⁽²₄₍₁⁻₋^α)_α)², and also the RHS exponent(1− α)(γ −2)to be nonnegative, so γ2. In contrast, theα=¹₂,γ=³₂ case appearing in Lemma 1 of [5] has RHS exponent−¹₄. The approach of Theorem 2 of [5] is to use the negative exponent together with the result of Example 2.3 to show that assuminghhas a small solution leads to a contradiction. As claimed, their method can be generalized to apply to other nonpositive exponents, but ₄₍₁⁽²⁻₋^α)_α)² < γ 2 holds only forα <2(√

2−1)≈0.8284.

Proof of Theorem 1.1. Given a continuoush:D₁→Csatisfying the hypotheses of Theorem 1.1, on the setω= {z∈D₁: h(z)=0},h∈C¹(ω), and the conclusion of Lemma 2.6 can be re-written:

|h|^(1−α)γ

4(1−α)γ −(2−α)²

|h|^(1−α)γ1−²γ. (11) The hypotheses of Lemma 2.1 are satisfied withn=2,u(x, y)= |h(x+iy)|^(1−α)γ, andu(0)=0, when the RHS of (11) has a positive coefficient (soγ >⁽²₄₍₁⁻₋^α)_α)²) and the quantityε=1−_γ² is in[0,1)(forγ 2). The conclusion of Lemma 2.1 is:

sup

z∈D₁

h(z)^(1−α)γ> M= 1

4·

4(1−α)γ−(2−α)²

· 2

γ 2γ /2

⇒ sup

z∈D₁

h(z)>

4(1−α)γ −(2−α)² γ²

¹

2(1−α)

.

We can optimize this lower bound, using elementary calculus to show that the maximum value of ^{4(1−α)γ−(2−α)2}

γ²

is achieved at the critical pointγ = ⁽²₂₍₁⁻₋^α)_α)² >max{2,⁽²₄₍₁⁻₋^α)_α)²}, and the lower bound for the sup isS_α as appearing in (4). 2

Note thatS_α is decreasing for 0< α <1, withS_1/2=⁴₉,S_2/3=¹₈, andS_α→0 asα→1⁻. This theorem is used in the proof of Theorem 4.3.

Example 2.7.As noted by [5], a 1-dimensional analogue of Eq. (3) in Theorem 1.1 is the well-known (for example, [2, §I.9]) ODEu(x)=B|u(x)|^α for 0< α <1 andB >0, which can be solved explicitly. By an elementary separa- tion of variables calculation, the solution on an interval whereu=0 is|u(x)| =(±(1−α)(Bx+C))^1−α1 . The general solution on the domainRis, forc₁< c₂,

u(x)=

⎧⎪

⎨

⎪⎩

(1−α)¹−¹α(B(x−c₂))¹−¹α ifxc₂,

0 ifc1xc2,

−(1−α)^1−α1 (B(c₁−x))^1−α1 ifxc₁.

Sou∈C¹(R), and ifu(0)=0, then sup₋1<x<1|u(x)|> ((1−α)B)^1−α1 . 3. Lemmas for holomorphic maps

We continue with the D_R notation for the open disk in the complex plane centered at the origin. The following quantitative lemmas on inverses of holomorphic functionsD_R→Care used in a step of the proof of Theorem 4.3 where we put a mapD_r→C²into a normal form, (14).

Lemma 3.1.(See [3, Exercise I.1].) Supposef :D1→D1is holomorphic, withf (0)=0,|f(0)| =δ >0. For any η∈(0, δ), lets=(₁^δ₋⁻_ηδ^η)η;then the restricted functionf:Dη→D1takes on each valuew∈Ds exactly once. 2

The hypotheses implyδ1 by the Schwarz Lemma.

Lemma 3.2. For a holomorphic map Z₁:D_r →D₂ with Z₁(0)=0, Z₁(0)=1, if r > ⁴

√2

3 then there exists a continuous functionφ:D1→Drwhich is holomorphic onD1and which satisfies(Z1◦φ)(z)=zfor allz∈D1. Remark.It follows from the Schwarz Lemma thatr2, and it follows from the fact thatφis an inverse ofZ₁that φ(0)=0 andφ(0)=1.

Proof of Lemma 3.2. Define a new holomorphic functionf :D₁→D₁by f (z)=1

2·Z₁(r·z),

sof (0)=0,f(0)=₂^r, and Lemma 3.1 applies withδ=^r₂. If we chooseη=^3r₈, thens=₆₄₋^3r_12r² 2, and the assumption r >⁴

√2

3 impliess >¹₂. It follows from Lemma 3.1 that there exists a functionψ:D_s→D_ηsuch that(f◦ψ )(z)=z for allz∈D_1/2⊆D_s; this inverse functionψis holomorphic onD_1/2. The claimed functionφ:D₁→D_rη⊆D_r is defined byφ(z)=r·ψ (¹₂·z), so forz∈D₁,

Z₁ φ(z)

=Z₁

r·ψ 1

2·z

=2·f

ψ 1

2·z

=2·1

2 ·z=z. 2 4. J-holomorphic disks

For S > 0, consider the bidisk Ω_S =D₂ ×D_S ⊆C², as an open subset of R⁴, with coordinates x = (x1, y1, x2, y2)=(z1, z2)and the trivial tangent bundleT ΩS⊆TR⁴. Consider an almost complex structureJ on ΩS given by a complex structure operator onT_xΩSof the following form:

J (x) =

⎛

⎜⎝

0 −1 0 0

1 0 0 0

0 λ 0 −1

λ 0 1 0

⎞

⎟⎠, (12)

whereλ:Ω_S→Ris any function.

A differentiable map Z:D_r →Ω_S is aJ-holomorphic disk ifdZ◦J_std=J◦dZ, whereJ_std is the standard complex structure onD_r⊆C. Letz=x+iybe the coordinate onD_r. ForJ of the form (12), ifZ(z)is defined by complex-valued component functions,

Z:D_r→Ω_S:Z(z)=

Z₁(z), Z₂(z)

, (13)

then theJ-holomorphic property implies thatZ1:D_r→D2is holomorphic in the standard way.

Example 4.1.If the functionλ(z1, z2)satisfiesλ(z1,0)=0 for allz1∈D2, then the mapZ:D2→ΩS:Z(z)=(z,0) is aJ-holomorphic disk.

Definition 4.2.The Kobayashi–Royden pseudonorm onΩ_Sis a functionT Ω_S→R:(x, v) → (x, v)K, defined on tangent vectorsv∈T_xΩ_Sto be the number

glb 1

r: ∃aJ-holomorphicZ:D_r→Ω_S, Z(0)= x, dZ(0) ∂

∂x

= v

.

Under the assumption thatλ∈C^0,α(ΩS), 0< α <1, it is shown by [6] and [7] that there is a nonempty set ofJ- holomorphic disks throughxwith tangent vectorvas in the definition, so the pseudonorm is a well-defined function.

Further, each such disk satisfiesZ∈C¹(D_r).

At this point we pickα∈(0,1)and setλ(z1, z2)= −2|z2|^α. LetS=S_α>0 be the constant defined by formula (4) from Theorem 1.1. Then,(Ω_S, J )is an almost complex manifold with the following property:

Theorem 4.3.If0=b∈D_Sthen(0, b), (1,0)K ³

4√ 2. Remark. Since ³

4√

2 ≈0.53, and (0,0), (1,0)K ¹₂ by Example 4.1, the theorem shows that the Kobayashi–

Royden pseudonorm is not upper semicontinuous onT Ω_S.

Proof. Consider a J-holomorphic map Z :D_r →Ω_S of the form (13), and suppose Z(0)=(0, b)∈Ω_S and dZ(0)(_∂x^∂ )=(1,0). Then the holomorphic functionZ₁:D_r→D₂satisfiesZ₁(0)=0,Z₁(0)=1, andZ₂∈C¹(D_r) satisfiesZ2(0)=b.

Suppose, toward a contradiction, that there exists such a mapZwithb=0 andr >⁴

√2

3 . Then Lemma 3.2 applies toZ₁: there is a re-parametrizationφwhich putsZinto the following normal form:

(Z◦φ):D₁→Ω_S, z→

Z₁ φ(z)

, Z₂ φ(z)

= z, f (z)

, (14)

wheref =Z₂◦φ:D₁→D_S satisfiesf ∈C⁰(D₁)∩C¹(D₁). From the fact thatZ◦φisJ-holomorphic onD₁, it follows from the form (12) ofJ that iff (z)=u(x, y)+iv(x, y), thenf satisfies this system of nonlinear Cauchy–

Riemann equations onD1: du

dy = −dv

dx and du dx +λ

z, f (z)

=dv

dy (15)

with the initial conditionsf (0)=b,u_x(0)=u_y(0)=v_x(0)=0 andv_y(0)=λ(0, b)= −2|b|^α. The system of equa- tions implies

∂f

∂z¯ =1 2

∂

∂x(u+iv)+i ∂

∂y(u+iv)

=1 2

ux−vy+i(vx+uy)

= −1 2λ

z, f (z)

= |f|^α. (16)

So, Theorem 1.1 applies, withf =h. The conclusion is that sup

z∈D₁

f (z)> S_α,

but this contradicts|f (z)|< S=S_α. 2

The previously mentioned existence theory for J-holomorphic disks shows there are interesting solutions of Eq. (16), and therefore also the inequality (11).

Example 4.4. For 0< α <1, (Ω_S, J ), λ(z₁, z₂)= −2|z₂|^α as above, a map Z:D_r →Ω_S of the form Z(z)= (z, f (z)) isJ-holomorphic iff (x, y)=u(x, y)+iv(x, y)is a solution of (15). Again generalizing theα=¹₂ case of [5], examples of such solutions can be constructed (for smallr) by assumingv≡0 andudepends only onx, so (15) becomes the ODEu(x)−2|u(x)|^α=0. This is the equation from Example 2.7; we can conclude thatJ-holomorphic disks inΩ_Sdo not have a unique continuation property.

Acknowledgement

The authors acknowledge the insightful remarks of the referee, whose suggestions led to a shorter proof of Lemma 2.6 and an improvement of Theorem 2.4.

References

[1] K. Astala, T. Iwaniec, G. Martin, Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane, Princeton Math. Ser., vol. 48, Princeton University Press, Princeton, 2009, MR 2472875 (2010j:30040), Zbl 1182.30001.

[2] G. Birkhoff, G.-C. Rota, Ordinary Differential Equations, Ginn & Co., 1962, MR 0138810 (25 #2253), Zbl 0102.29901.

[3] J. Garnett, Bounded Analytic Functions, rev. first ed., Grad. Texts in Math., vol. 236, Springer, 2007, MR 2261424 (2007e:30049), Zbl 1106.30001.

[4] D. Gilbarg, N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer CIM, 2001, MR 1814364 (2001k:35004), Zbl 1042.35002.

[5] S. Ivashkovich, S. Pinchuk, J.-P. Rosay, Upper semi-continuity of the Kobayashi–Royden pseudo-norm, a counterexample for Hölderian almost complex structures, Ark. Mat. (2) 43 (2005) 395–401, MR 2173959 (2006g:32038), Zbl 1091.32009.

[6] S. Ivashkovich, J.-P. Rosay, Schwarz-type lemmas for solutions of∂-inequalities and complete hyperbolicity of almost complex manifolds, Ann. Inst. Fourier (Grenoble) 54 (7) (2004) 2387–2435, MR 2139698 (2006a:32032), Zbl 1072.32007.

[7] A. Nijenhuis, W. Woolf, Some integration problems in almost-complex and complex manifolds, Ann. of Math. (2) 77 (1963) 424–489, MR 0149505 (26 #6992), Zbl 0115.16103.