ON THE REGULARITY OF SOLUTIONS TO THEBELTRAMI EQUATION IN THE PLANE

by

Bindu K Veetel

B. Sc. in Mathematics, Bangalore University, India

M. Sc. in Mathematics, Bangalore University, India

A dissertation submitted to

The Faculty of

the College of Science of

Northeastern University

in partial fulllment of the requirements

for the Degree of Doctor of Philosophy

April 17, 2014

Dissertation directed by

Robert McOwen

Professor of Mathematics

Acknowledgements

First of all, I would like to thank my advisor, Prof. Robert McOwen,

with whom I had a great learning experience. His encouraging advice

and guidance kept me motivated through the ups and downs of re-

search and thesis work. Thank you for the patience and understanding

throughout the course of this research work.

Thanks to Prof. Maxim Braverman, Prof. Gordana Todorov, Prof.

Jerzy Weyman, Prof. Alex Suciu, Prof. Christopher King and Ms. Ra-

jini Jesudason for all the help and guidance that enriched my learning

and teaching experiences. I am also thankful to all the professors who

helped me better understand various aspects of Mathematics through

the graduate courses offered. Special thanks to Gabriel Cunningham

for all the discussions that made my initial years of research work en-

riching, easier and fruitful. I would also like to thank all the faculty

members, office staff and all my friends for making my journey through

PhD a pleasant one.

I am grateful to my parents and brother for their continuing support

and love throughtout these years. None of my efforts through these

years would have been fruitful but for the relentless support, patience

and encouragement of my family- my husband Saji Nair and my sons

Nandan and Nikhil.

ii

Abstract of Dissertation

We obtain estimates on regularity of solutions to the inhomogeneous

Beltrami equation in the plane namely ∂f − µ ∂f − ν ∂f = h when h

and the coefficients µ and ν are Dini continuous; the coefficients sat-

isfy an ellipticity condition |µ(z)| + |ν(z)| ≤ κ < 1. In the case when

h and the coefficients µ and ν are Holder continuous with exponent

α, 0 < α < 1, it is well known (cf.[3]) that the solutions and their first

order derivatives are Holder continuous. In our case, we find that al-

though the solutions are in C1, their derivatives are less regular than

the coefficients. Nevertheless, we are able to get a priori estimates. One

application of this work is to stability analysis of the inverse problem

of Calderon in two dimensions.

iii

Table of Contents

Acknowledgements ii

Abstract of Dissertation iii

Table of Contents iv

1. Introduction 1

2. Dini Continuous Functions 9

3. Constant coefficients in bounded domain 12

4. The space C(λ) on the torus 26

5. Non constant coefficients in a bounded domain 53

References 57

iv

1. Introduction

In this thesis, we study the regularity of solutions to the inhomogeneous

Beltrami equation in the plane, namely

(1)∂f − µ ∂f − ν ∂f = h,

where ∂ = ∂z =1

2(∂x + i∂y), ∂ = ∂z =

1

2(∂x − i∂y).

We assume the ellipticity condition, |µ(z)| + |ν(z)| ≤ κ < 1 for all

z ∈ Ω ⊂ C where Ω is a bounded domain in C. When the coefficients

µ and ν and the function h are Holder continuous with exponent α ∈

(0, 1) i.e. in the Holder space Cα(Ω), it is well known (cf.[3]) that the

first order derivatives of a solution to the Beltrami equation (1) also

lie in the space Cα(Ω). In this thesis, we consider the coefficients

µ and ν and h to have modulus of continuity ω satisfying the Dini

condition∫ ε

0ω(r)rdr < ∞. The result that we obtain is that the first

order derivatives of a solution of (1) will have modulus of continuity σ

that could be weaker than ω, in particular, need not satisfy the Dini

condition. Nevertheless, we are able to obtain interior estimates of the

form

(2) ‖∂f‖Cσ(D) + ‖∂f‖Cσ(D) ≤ C(‖h‖Cω(U) + ‖f‖C0(U))

where the constant C depends on µ, ν and the domains D,U where D

is compactly contained in U and U is compactly contained in Ω. (In

(2), we have used Cω to denote functions continuous with respect to

the modulus of continuity ω, even though this is inconsistent with the

notation Cα for Holder continuity; cf. Section 3). The techniques we

use are mostly based on Fourier analysis. Littlewood-Paley theory and

1

Fourier multiplier operators on the torus are used to describe and work

with functions that we encounter. Our main result is given as Theorem

5 in section 6.

One of the applications of (2) is to stability of the inverse problem

of Calderon in two dimensions. An inverse problem assumes a direct

problem that is well-posed: a solution exists, is unique and stable under

perturbation of parameters. It uses information about the solutions to

obtain a parameter in the problem.

To describe the Calderon problem, let u be the unique solution to

the Dirichlet problem

(3)∇ · γ∇u = 0 in a bounded domain U ⊂ R2,

u = f on ∂U.

If we let Λγ(f) = γ ∂u∂ν

where ν is the exterior unit normal on ∂U ,

then the map Λγ : H1/2(∂U) → H−1/2(∂U) is called the the Dirich-

let to Neumann map. If γ represents the electrical conductivity of U ,

then Λγ is the current induced at the boundary by applying a voltage

f at the boundary of U . Hence experimental data involving current

and voltage measurements at the boundary determines Λγ. The in-

verse problem of Calderon is the determination of conductivity γ in U

from the boundary measurements, i.e. from Λγ. In particular, solving

the inverse problem requires showing that the Dirichlet to Neumann

map Λγ uniquely determines γ. Stability is a stronger statement than

2

uniqueness; it says that

(4) ‖γ1 − γ2‖L∞ ≤ V (‖Λγ1 − Λγ2‖∗),

where ‖ · ‖∗ denotes the operator norm H1/2(∂U) → H−1/2(∂U) and

V (ρ) is a “stability function” that satisfies V (ρ)→ 0 as ρ→ 0. Hence

stability implies that small changes in the Dirichlet to Neumann map

will only correspond to small changes in the conductivity. This inverse

problem has important real world applications like in medical imaging

to detect tumors in the human body, reconstruction of the interior of

human body using exterior measurements.

Astala and Paivarinta [5] showed that Λγ uniquely determines γ ∈

L∞ by reducing the problem to a Beltrami equation. In fact, they

recover the complex solutions of (3) by letting

(5) uγ = <(fν) + i=(f−ν)

where fν is the complex geometric optics solution of the R-linear Bel-

trami equation,

(6) ∂f = ν ∂f,

and ν is defined in terms of conductivity γ as ν = 1−γ1+γ

. Notice that the

above equation is a special case of (1) where the coefficient µ and the

function h are zero. In [5], it is shown that for each k ∈ C, there exits

a unique solution fν(z, k) of (6) of the form

fν(z, k) = eikzMν(z, k)

3

where

Mν(z, k) = 1 +O(z−1) as |z| → ∞.

In order to control the behaviour of Mν as |z| → ∞, [5] shows that

fν(z, k) = eikφ(z,k) where for each fixed k, φ satisfies a Beltrami equa-

tion.

The work of Barcelo, Faraco and Ruiz [4], stem from that of Astala

and Paivarinta in [5]. They use regularity results for (6) to obtain

stability in the case of Holder continuous conductivity. In particular,

they obtain that when the coefficients µ, ν and the function h are Holder

continuous with exponent α ∈ (0, 1), any solution f in the Sobolev

space W 1,2(Ω) satisfies the estimate

‖∂f‖Cα(D) + ‖∂f‖Cα(D) ≤ C‖f‖C0(U)

where the constant C depends on µ, ν, α and the domains D,U such

that D is compactly contained in U and U is compactly contained in

Ω. This estimate is then used to obtain regularity results for (6). The

interior estimate (2) obtained in this thesis can be used to obtain cor-

responding estimates needed for stability of the complex solutions of

the inverse problem of Calderon in two dimensions in the Dini case.

This will be further discussed in the upcoming paper [14].

The basic structure of our work is as follows: In section 2, we define

Dini continuous functions and Banach spaces associated with these

functions. In section 3, we obtain the regularity of solutions to the

4

inhomogeneous Beltrami equation with constant coefficients, namely

(7) ∂f − µ0 ∂f − ν0 ∂f = h where h is Dini continuous.

This is done first for the C-R equation, the special case of the Beltrami

equation where the coefficients µ and ν are zero, and then translated

to that of the Beltrami equation (7). But these estimates do not easily

generalize to variable coefficients. Hence in section 4, we follow [15] by

introducing a space of functions C(λ) on a torus using Fourier multi-

pliers and a Littlewood-Paley partition of unity; here the function λ is

defined in terms of modulus of continuity ω. These functions need not

be Dini continuous but are well behaved under the Beurling transform.

We first obtain regularity estimates in terms of C(λ) on solutions of

the Beltrami equation with constant coefficients µ0 and ν0 but with

h ∈ C(λ). We then use these estimates to obtain similar estimates on

solutions of the Beltrami equation (1) with variable coefficients in C(λ).

We make use of tools like the Beurling transform, Fourier transform

and paraproduct. In section 5, we translate our results on the torus

to regularity estimates for (1) with Dini continuous coefficients in a

bounded domain Ω.

The following are some terms used in our work:

Modulus of continuity:

It is a continuous, increasing function on [0,∞), usually denoted by ω

such that ω(0) = 0. The definition of modulus of continuity was first

given by H. Lebesgue in 1910 for functions of one real variable. It is a

5

precise way to measure smoothness of a function.

Beltrami Equation:

Beltrami equations play an important part in complex function theory

and theory of differential equations. Various theorems like the mea-

surable Riemann mapping theorem can be proved using the Beltrami

equation. The first global solution in C of the Beltrami equation of the

form

∂f = µ∂f

was given by Venkua [17] for compactly supported µ. The Lp proper-

ties of the operators associated with the Beltrami equation have been

used to prove the measurable Riemann mapping theorem in [3].

Littlewood-Paley partition of unity:

This is a tool in analysis which allows us to decompose a function into

pieces such that the frequency supports of these pieces are almost dis-

joint. On Rn, it can be defined as a collection ψk(ξ) of smooth func-

tions with compact support such that for k > 0, each ψk(ξ) has support

in 2k−1 < |ξ| < 2k+1, ψ0(ξ) has support in |ξ| < 2 and∑∞

k=0 ψk(ξ) = 1.

Fourier Transform:

Let S(R2) denote Schwartz space i.e. the space of smooth functions

such that all derivatives decay rapidly at infinity. The Fourier transform

F : S(R2)→ S(R2) is defined by

(8) (Ff)(ξ) = f(ξ) =1

(2π)2

∫R2

f(z) e−iz.ξ dz for ξ ∈ R2

6

and the inverse Fourier transform F−1g is given by

(9) (F−1g)(z) = g(z) =

∫R2

g(ξ) eiz.ξ dξ.

If S ′(R2) denote the dual of the Schwartz space, i.e. the space of tem-

pered distributions, the Fourier transform can be extended to F :

S ′(R2)→ S ′(R2) by defining

(10) u(φ) = u ˆ(φ)

for φ ∈ S(R2).

If f, g ∈ L1(R2) and one of them has compact support, then

(fg)∨(z) = (f∨ ∗ g∨)(z) =

∫R2

f∨(z − z)g∨(z)dz.

Fourier Multiplier Operator:

Using the Fourier transform, we can define Fourier multiplier operators

M(D) for any bounded, measurable function m(ξ) as

(M(D)f)∧(ξ) = m(ξ)f(ξ).

The function m(ξ) is called its symbol or multiplier. Examples of

Fourier multiplier operators include differential operators. Fourier mul-

tiplier operators are a special case of the pseudodifferential operators

which are extensions of the concept of differential operators and that

of singular integral operators and are used extensively in the field of

partial differential equations and quantum field theory.

Paraproduct:

According to S. Jansen and J. Peetre [11],“The name paraproduct de-

notes an idea rather than a unique definition; several definitions exist

7

and can be used for the same purpose.” The term paraproduct means

beyond product. It can be considered as a bilinear, non-commutative

operator that is used to reconstruct a product such that it generally has

better properties than the usual product of functions. The first version

of a paraproduct appeared in A. P. Calderon’s work on commutators

[7]. Several versions have appeared since then. J.-M. Bony introduced

a paraproduct in his work,“Theory of paradifferential operators” [6],

which came to be known as Bony’s paraproduct. It is defined using a

Littlewood-Paley partition of unity and helps in reconsructing a prod-

uct fg based on the frequency supports of f and g. It separates out

the parts where frequency supports of f and g are disjoint and hence

enables analysis of each part of the product appropriately. The part

where the supports are not disjoint is considered as the error term and

hence plays the role of the best part of the product reconstruction.

Smoothing operator:

It is an operator that maps non smooth functions or even distibutions

to smooth functions. Any Fourier multiplier operator ψ(D) ∈ L1(Rn)

with compact support can be considered as a smoothing operator. The

Fourier multiplier operator ψ0(D) associated with the Littlewood-Paley

partition of unity ψj(ξ) is an example of a smoothing operator used

in this work.

Fredholm operators:

A bounded linear operator T : X → Y between two Banach spaces X

and Y is said to be Fredholm if it has finite dimensional kernel and

8

cokernel. The index of the Fredholm operator T is given by ind(T )=

nul(T ) - def(T ) where nul(T ) is the dimension of kernel of T and def(T )

is the dimension of cokernel of T . The composition of Fredholm op-

erators is also Fredholm with index of the composition equal to the

sum of index of each operator. If K : X → X is a compact operator,

then the operator T = I +K is Fredholm with ind(T )=0. A bounded

linear operator T : X → Y between two Banach spaces X and Y is

said to be semi-Fredholm if the range of T is a closed subspace of Y

and atleast one of kernel T or cokernel T is of finite dimension. It is

well known that if an operator T satisfies an a priori estimate of the

form ‖f‖X ≤ C(‖Tf‖Y + ‖f‖Y ) where X is compactly contained in

Y , then T has a finite dimensional kernel and closed range, i.e. is a

semi-Fredholm operator.

2. Dini Continuous Functions

Consider a function ω with the following properties: ω(0) = 0, ω(r) is

continuous and strictly increasing for 0 ≤ r ≤ 1 and satisfies the Dini

condition

(11)

∫ ε

0

ω(r)

rdr <∞

for some ε > 0. For technical reasons, also assume

(12) 0 < ω(r) ≤ 1

and that

(13) r−βω(r) is decreasing on (0, 1] for some β ∈ (0, 1).

9

When convenient, we extend ω to r > 1 by letting ω(r) = ω(1). This

extension of ω will still satisfy (13) on (0,∞). As a consequence of (13),

it is easy to see that ω(2r) ≤ 2βω(r). Hence there exists constants C1

and C2 such that C1 ω(2r) ≤ ω(r) ≤ C2 ω(2r).

For 0 ≤ r ≤ 1, define a function σ by

(14) σ(r) :=

∫ r

0

ω(s)

sds.

Using (13), it is easy to see that

(15) C1 σ(2r) ≤ σ(r) ≤ C2 σ(2r) for some constants C1, C2

and ω(r) ≤ βσ(r) certainly implies

(16) ω(r) ≤ σ(r).

We shall use ω and σ as modulii of continuity for functions. Let U

be a bounded domain in R2.

Definition 1. For any modulus of continuity ω, let Cω(U) denote those

functions f ∈ C(U) for which |f(x) − f(y)| ≤ C ω(|x − y|) for all

x, y ∈ U . Cω(U) is a Banach space under the norm

(17) ‖f‖Cω(U) := supx∈U|f(x)|+ sup

x,y∈Ux 6=y

|f(x)− f(y)|ω(|x− y|)

.

Definition 2. For any domain Ω, the collection of all functions f

such that f ∈ Cω(U) for all U compactly contained in Ω is denoted by

Cω(Ω).

10

Definition 3. Let C1,ω(U) denote those functions f ∈ C1(U) whose

first order derivatives ∂f/∂z and ∂f/∂z are in Cω(U). C1,ω(U) is a

Banach space under the norm

(18) ‖f‖C1,ω(U) := ‖∂f‖Cω(U) + ‖∂f‖Cω(U) + ‖f‖C0(U).

Definition 4. For any domain Ω, the collection of all functions f

such that f ∈ C1,ω(U) for all U compactly contained in Ω is denoted

by C1,ω(Ω).

When ω satisfies the Dini condition (11), the functions in Cω(Ω) are

called Dini continuous.

Throughout this paper, we shall use ω to denote the modulus of con-

tinuity with the properties (11), (12) and (13).

Using (14), Cσ(U) and C1,σ(U) are Banach spaces defined as above

using σ as the modulus of continuity. In fact, using (16), we see that

Cω(Ω) is contained in Cσ(Ω). When ω(r) = rα with α ∈ (0, 1), we get

the special case of Holder continuous functions. In this case, σ(r) is so

equivalent to rα and hence Cω(Ω) coincides with Cσ(Ω).

In this work, we also consider the collection of all functions f whose

first-order derivatives are square-integrable in every U compactly con-

tained in any domain Ω, which is generally denoted by W 1,2loc (Ω).

11

3. Constant coefficients in bounded domain

Throughout this work, we shall denote an open disk with center z0 and

radius r as Dr(z0) and an open disk with center as the origin and radius

r as Dr. We shall consider Ω to be a bounded domain in R2.

Theorem 1. Let h ∈ Cω(Ω) and µ0 and ν0 be constants satisfying

|µ0|+ |ν0| ≤ κ < 1. Let f ∈ W 1,2loc (Ω) satisfy

(19) ∂f − µ0 ∂f − ν0 ∂f = h.

Then f ∈ C1,σ(Ω) and satisfies

(20) ‖∂f‖Cσ(D) + ‖∂f‖Cσ(D) ≤ C(κ,D,U) (‖h‖Cω(U) + ‖f‖Cω(U))

for any domain D, U such that D is compactly contained in U and U

is compactly contained in Ω.

We first consider the special case of the Beltrami equation where

the coefficients µ and ν are zero. This is the inhomogeneous Cauchy-

Riemann equation of the form

(21) ∂p = q.

If q is integrable on a domain Ω, then the domain potential of q is given

by

(22) p(z) =1

π

∫Ω

q(τ)

(z − τ)dAτ

and is a distribution solution of the equation (21) in Ω since 1πz

is the

fundamental solution for the C-R operator ∂ .

12

An important result that we use in this section is that for any disk

DR(z0),

(23)

∫DR(z0)

1

(z1 − τ)2dAτ = 0 for z1 ∈ DR(z0).

Using polar coordinates, this can easily be seen for the case when z1 =

z0. When z1 6= z0, we can write∫DR(z0)

1

(z1 − τ)2dAτ = lim

ε→0

∫DR(z0)\Dε(z1)

1

(z1 − τ)2dAτ .

Now Green’s theorem can be used to get (23).

We first prove results which are necessary to prove the above theo-

rem.

Lemma 1. Consider a function φ ∈ Cω(D1). Then

(24) ∂z

∫D1

φ(τ)

(z − τ)dAτ = −

∫D1

φ(τ)

(z − τ)2dAτ for z ∈ D1.

Proof: Let

w(z) :=

∫D1

φ(τ)

(z − τ)dAτ and u(z) := −

∫D1

φ(τ)− φ(z)

(z − τ)2dAτ .

Note that using (23), u(z) = −∫D1

φ(τ)(z−τ)2

dAτ .

Consider a function η ∈ C∞(R) such that 0 ≤ η ≤ 1, 0 ≤ η′ ≤ 2 and

η(t) =

0, for t ≤ 1

1, for t ≥ 2

Denote η( |z−τ |ε

) as ηε(z − τ). For ε > 0, define

wε(z) :=

∫D1

ηε(z − τ)φ(τ)

(z − τ)dAτ .

13

Note that wε(z) is smooth as it is the convolution with the smooth

function 1zη( |z|

ε). Now w is the uniform limit of wε since

|wε(z)− w(z)| =∣∣∣∣∫|z−τ |<2ε

1− ηε(z − τ)

(z − τ)φ(τ) dAτ

∣∣∣∣≤∫|z−τ |<2ε

|1− ηε(z − τ)||z − τ |

|φ(τ)| dAτ

≤ C(ε) supτ∈D1

|φ(τ)|

where C(ε) tends to 0 as ε tends to 0. We now prove that u is the

uniform limit of ∂zwε. Consider

(25)

∂zwε(z) =

∫D1

∂z

[ηε(z − τ)

(z − τ)

](φ(τ)− φ(z)) dAτ

+ φ(z)

∫D1

∂z

[ηε(z − τ)

(z − τ)

]dAτ

The second integral can be written as∫D1

∂z

[ηε(z − τ)

(z − τ)

]dAτ = −

∫D1

∂τ

[ηε(z − τ)

(z − τ)

]dAτ

=−i2

∫∂D1

[ηε(z − τ)

(z − τ)

]dτ

=−i2

∫∂D1

1

(z − τ)dτ

since ε can be chosen so that ηε(z − τ) = 1 for τ on ∂D1 for fixed z.

Now using polar coordinates, it can be seen that∫∂D1

1

(z − τ)dτ = 0

Hence

(26)

∫D1

∂z

[ηε(z − τ)

(z − τ)

]dAτ = 0.

14

Using (26) in (25),we get

∂zwε(z) =

∫D1

∂z

[ηε(z − τ)

(z − τ)

](φ(τ)− φ(z)) dAτ

We now have

(27)

|u(z)− ∂zwε(z)| =∣∣∣∣∫|z−τ |<2ε

∂z

[1− ηε(z − τ)

(z − τ)

](φ(τ)− φ(z)) dAτ

∣∣∣∣≤∫|z−τ |<2ε

∣∣∣∣∂z 1− ηε(z − τ)

z − τ

∣∣∣∣ |φ(τ)− φ(z)| dAτ

But∣∣∣∣∂z 1− ηε(z − τ)

z − τ

∣∣∣∣ ≤ ( 1

ε |z − τ |η′(|z − τ |ε

)+|1− ηε(|z − τ |)||z − τ |2

)≤(

2

ε |z − τ |+

1

|z − τ |2

)Substituting the above in (27), we get

|u(z)− ∂zwε(z)|

≤∫|z−τ |<2ε

(2

ε |z − τ |+

1

|z − τ |2

)|φ(τ)− φ(z)| dAτ

≤ ‖φ‖Cω(D1)

∫|z−τ |<2ε

(2

ε |z − τ |+

1

|z − τ |2

)ω(|τ − z|) dAτ

≤ ‖φ‖Cω(D1)(4ω(2ε) + 2π σ(2ε))

which tends to 0 as ε tends to 0. Hence we can conclude that ∂zw(z) =

u(z) which is our required result.

In the case of Holder continuous functions, it is well known (cf.[3]) that

the Holder norm of derivative of the solution p to the inhomogeneous

C-R equation (21) is bounded by the Holder norm of q. We now obtain

15

similar bounds in the case of (21) for Dini continuous functions. This

is where we get to see the role of the Cσ norm.

Lemma 2. For q ∈ Cω(Dε), the domain potential p(z) = 1π

∫Dε

q(τ)(z−τ)

dAτ

lies in C1,σ(Dε/2) and satisfies the estimate

(28) ‖∂p‖Cσ(Dε/2) + ‖∂p‖Cσ(Dε/2) ≤ K(ε)‖q‖Cω(Dε).

Proof: For z ∈ Dε/2, using (24) and (23), we can write

∂p(z) =−1

π

∫Dε

q(τ)

(z − τ)2dAτ =

−1

π

∫Dε

q(τ)− q(z)

(z − τ)2dAτ .

Since ∂p(z) = q(z), we have ‖∂p‖Cσ(Dε/2) ≤ K‖q‖Cω(Dε). Now let us

estimate ‖∂p‖Cσ(Dε/2). For any z1, z2 ∈ Dε/2, we get

|∂p(z1)− ∂p(z2)| = 1

π

∣∣∣∣∫Dε

q(τ)− q(z1)

(z1 − τ)2dAτ −

∫Dε

q(τ)− q(z2)

(z2 − τ)2dAτ

∣∣∣∣Let |z1− z2| = δ < ε, z3 = z1+z2

2so that Dδ(z3) is contained in Dε. The

above equation can now be estimated as

(29)

|∂p(z1)− ∂p(z2)|

≤ 1

π

∣∣∣∣∫Dδ(z3)

q(τ)− q(z1)

(z1 − τ)2dAτ

∣∣∣∣+1

π

∣∣∣∣∫Dδ(z3)

q(τ)− q(z2)

(z2 − τ)2dAτ

∣∣∣∣+

1

π

∣∣∣∣∫Dε−Dδ(z3)

q(τ)− q(z1)

(z1 − τ)2dAτ +

∫Dε−Dδ(z3)

q(τ)− q(z2)

(z2 − τ)2dAτ

∣∣∣∣Now |q(τ)− q(z1| ≤ ‖q‖Cω(Dε) ω(|τ − z1|) gives

(30)

∣∣∣∣∫Dδ(z3)

q(τ)− q(z1)

(z1 − τ)2dAτ

∣∣∣∣ ≤ ‖q‖Cω(Dε)

∫Dδ(z3)

ω(|τ − z1|)|z1 − τ |2

dAτ

≤ ‖q‖Cω(Dε)

∫D2δ(z1)

ω(|τ − z1|)|z1 − τ |2

dAτ

= 2π ‖q‖Cω(Dε) σ(2δ).

16

Using (15), (30) can be written as

(31)

∣∣∣∣∫Dδ(z3)

q(τ)− q(z1)

(z1 − τ)2dAτ

∣∣∣∣ ≤ C ‖q‖Cω(Dε) σ(δ).

Similarly

(32)

∣∣∣∣∫Dδ(z3)

q(τ)− q(z2)

(z2 − τ)2dAτ

∣∣∣∣ ≤ C ‖q‖Cω(Dε) σ(δ).

Now we estimate the last term in (29). First observe that using (23),

we get ∣∣∣∣∫Dε−Dδ(z3)

q(τ)− q(z1)

(z1 − τ)2dAτ −

∫Dε−Dδ(z3)

q(τ)− q(z2)

(z2 − τ)2dAτ

∣∣∣∣=

∣∣∣∣∫Dε−Dδ(z3)

[1

(z1 − τ)2− 1

(z2 − τ)2

](q(τ)− q(z2)) dAτ

∣∣∣∣Now by the mean value theorem, there exists zτ between z1 and z2 such

that

(33)1

(z1 − τ)2− 1

(z2 − τ)2= (z1 − z2).

−2

(zτ − τ)3.

Hence we get

(34)

∣∣∣∣∫Dε−Dδ(z3)

[1

(z1 − τ)2− 1

(z2 − τ)2

](q(τ)− q(z2)) dAτ

∣∣∣∣≤ ‖q‖Cω(Dε) |z1 − z2|

∫Dε−Dδ(z3)

∣∣∣∣ 2

(zτ − τ)3

∣∣∣∣ ω(|τ − z2|) dAτ

= 2δ ‖q‖Cω(Dε)

∫Dε−Dδ(z3)

ω(|τ − z2|)|zτ − τ |3

dAτ .

But

|z2 − τ | ≥ ||z2 − z3| − |τ − z3|| ≥ |2ε−3δ

2| ≥ δ/2 =

|z1 − z2|2

.

Using property (13) of ω(r), we get

|z2 − τ |−β ω(|τ − z2|) ≤(|z1 − z2|

2

)−βω(|z1 − z2|).

17

Using these we get

∫Dε−Dδ(z3)

ω(|τ − z2|)|zτ − τ |3

dAτ ≤∫Dε−Dδ(z3)

ω(|z1 − z2| ( |z1−z2|2)−β

|z − τ |3 |z2 − τ |−βdAτ

≤ 2βδ−β ω(δ)

∫Dε−Dδ(z3)

1

|zτ − τ |3 |z2 − τ |−βdAτ .

Using (33), it can be seen that |z3 − τ | ≤ 2|zτ − τ |. Also

|z2 − τ | ≤δ

2+ |z3 − τ | ≤

3|z3 − τ |2

.

Using these, the above integral can be written as

(35)∫Dε−Dδ(z3)

ω(|τ − z2|)|zτ − τ |3

dτ ≤ 2βδ−β ω(δ)

∫Dε−Dδ(z3)

3β 23−β

|z3 − τ |3−βdAτ

≤ 23 3β δ−β ω(δ)

∫|z3−τ |≥δ

1

|z3 − τ |3−βdAτ

= 24 3β δ−β ω(δ) πδ−1+β

1− β

≤ Cδ−1ω(δ).

Now using (35) in (34) we get

(36)

∣∣∣∣∫Dε−Dδ(z3)

[1

(z1 − τ)2− 1

(z2 − τ)2

](q(z2)− q(τ)) dAτ

∣∣∣∣≤ C1‖q‖Cω(Dε)ω(δ).

Using (31),(32), (36) and (16) in (29), we get

|∂p(z1)− ∂p(z2)| ≤ K‖q‖Cω(Dε)σ(δ)

or

‖∂p‖Cσ(Dε/2) ≤ K(ε)‖q‖Cω(Dε).

18

Hence we get our required result (28).

To further understand the regularity properties of solutions to the

Cauchy- Riemann equation and the Beltrami equation, it is convenient

to make use of analytic properties of operators that are defined in

the whole complex plane. For any bounded function φ with compact

support on C, the Cauchy transform is defined by

(37) (Cφ)(z) :=1

π

∫C

φ(τ)

(z − τ)dAτ

and the Beurling transform is the singular integral operator defined

by

(38) (Sφ)(z) :=−1

π

∫C

φ(τ)

(z − τ)2dAτ .

When φ is integrable in a domain Ω ⊂ C with compact support in Ω

and if φ is extended by zero outside Ω, then the domain potential of φ

coincides with the Cauchy transform of φ. The Cauchy transform p =

Cq solves the C-R equation (21) when q is bounded and has compact

support. Note that using (23) in Lemma 1, we can conclude that for

any φ ∈ Cω(U) with compact support in K ⊂ U and extended by zero

outside K,

(39) ∂zCφ(z) = Sφ(z) for z ∈ C.

Also for any φ ∈ C1,ω(U) with compact support in K ⊂ U ,

(40) φ(z) = C(∂φ(z)) for z ∈ C.

19

Proposition 1. For any domain Ω and q ∈ Cω(Ω), every solution p ∈

W 1,2loc (Ω) of the equation (21) lies in C1,σ(Ω) and satisfies the estimate

(41) ‖∂p‖Cσ(Dε/2) + ‖∂p‖Cσ(Dε/2) ≤ C(‖q‖Cω(Dε) + ‖p‖Cω(Dε))

where C = C(ε) and Dε,Dε/2 are concentric disks of radius ε and ε/2

respectively that are compactly contained in Ω.

Proof: We first prove that the solution p lies in C1,σ(Ω). Note that it is

sufficient to prove that p lies in C1,σ(Dε/2). Consider a smooth function

φ with compact support in Dε such that φ ≡ 1 in Dε/2. Let q1 = φ q.

Now since q1 has compact support in Dε then let p1 = Cq1, which is also

the domain potential, so we know by Lemma 2 that p1 ∈ C1,σ(Dε/2).

Hence for z ∈ Dε/2, ∂(p − p1)(z) = (q − q1)(z) = 0. So (p − p1) is

holomorphic in Dε/2 and hence is smooth in Dε/2. Thus p ∈ C1,σ(Dε/2).

Consider the smooth function φ introduced above. Define p = φ p.

Then

(42) ∂p = ∂φ p+ φ q.

Since ∂φ p + φ q ∈ Cω(Dε) with compact support in Dε, using (40),

p = C(∂p) solves (42). For z ∈ Dε/2, p(z) can be written as

p(z) = p(z) = C∂p(τ) =1

π

∫Dε

∂p(τ)

(z − τ)dAτ =

1

π

∫Dε

(∂φ p+ φ q)(τ)

(z − τ)dAτ .

Using Lemma 2, p satisfies the estimate

‖∂p‖Cσ(Dε/2) + ‖∂p‖Cσ(Dε/2) ≤ K‖∂φ p+ φ q‖Cω(Dε).

But p = p in Dε/2. Hence

(43) ‖∂p‖Cσ(Dε/2) + ‖∂p‖Cσ(Dε/2) ≤ K(‖∂φ p‖Cω(Dε) + ‖φ q‖Cω(Dε)).

20

For z1, z2 ∈ Dε,

|(∂φ p)(z1)− (∂φ p)(z2)|ω(|z1 − z2|)

=|∂φ(z1)(p(z1)− p(z2)) + p(z2)(∂φ(z1)− ∂φ(z2)|

ω(|z1 − z2|)

≤ |∂φ(z1)| |p(z1)− p(z2)|ω(|z1 − z2|)

+ |p(z2)| |∂φ(z1)− ∂φ(z2)|ω(|z1 − z2|)

.

We now get

(44)‖∂φ p‖Cω(Dε) ≤ ‖∂φ‖C0(Dε)‖p‖Cω(Dε) + ‖p‖C0(Dε)‖∂φ‖Cω(Dε)

≤ C1‖p‖Cω(Dε)

where C1 = C1(ε). Similarly we can see that ‖φ q‖Cω(Dε) ≤ C2‖q‖Cω(Dε)

where C2 = C2(ε). Using this and (44) in (43), we get the desired result

(41).

Proof of Theorem 1: As in [3],we first reduce the Beltrami equa-

tion (19) to an inhomogeneous Cauchy-Riemann equation and then

try to obtain the bounds for the solution of (19) using the solution of

the inhomogeneous Cauchy-Riemann equation. For any constants a, b,

define

(45) p(z) := f(ξ) + b f(ξ), where ξ = z + a z

which can be written as

(46) f(ξ) =p(z)− b p(z)

1− |b|2where z =

ξ − a ξ1− |a|2

.

21

Let u(z) := h(ξ) = h(z + a z). Choose constants a, b such that f(ξ)

satisfying (19) can be reduced to the form

(47) ∂p(z) = u(z) + ab u(z).

We choose the pair of values for a and b as

(48) a =−2µ0

1 + |µ0|2 − |ν0|2 + (1 + |µ0|2 − |ν0|2)2 − 4|µ0|2)1/2,

(49) b =−2 ν0

1 + |ν0|2 − |µ0|2 + (1− |ν0|2 + |µ0|2)2 − 4|µ0|2)1/2.

This gives |a|, |b| ≤ κ < 1.

Using Proposition 1, any solution p ∈ W 1,2loc (Ω) of (47) lies in C1,σ(Dε/2)

and satisfies

∥∥∂p∥∥Cσ(Dε/2)

+ ‖∂p‖Cσ(Dε/2) ≤ K(ε) (‖u‖Cω(Dε) + ‖p‖Cω(Dε))

Using the definition of u(z), the above relation can be written as

(50)∥∥∂p∥∥

Cσ(Dε/2)+ ‖∂p‖Cσ(Dε/2) ≤ K1(ε)(‖h‖Cω(Dε) + ‖p‖Cω(Dε)).

Using (85) and Proposition 1, we can see that any solution f ∈ W 1,2loc (Ω)

of (19) lies in C1,σ(Dε/2).

Now we estimate ‖∂f‖Cσ(Dε/2)+‖∂f‖Cσ(Dε/2) in terms of ‖∂p‖Cσ(Dε/2)+

‖∂p‖Cσ(Dε/2). We first note that the map from z to ξ takes unit circles

to ellipses which are contained in circles with double the radius. Since

ξ = z + az, we get

|ξ1 − ξ2| ≤ |z1 − z2|(1 + |a|) ≤ 2|z1 − z2|.

22

This gives

(51) ω(|ξ1 − ξ2|) ≤ ω(2|z1 − z2|) ≤ Cω(|z1 − z2|).

Again since z = ξ−bξ1−|b|2 , we get |z1 − z2| ≤ |ξ1−ξ2|

1−κ . Hence

(52) σ(|z1 − z2|) ≤ σ(|ξ1 − ξ2|

1− κ) ≤ (1− κ)−βσ(|ξ1 − ξ2|)

using the definition of sigma. Now let us obtain a relation between

sigma norm of p and sigma norm of f . Let ξ = φ(z). Hence (85) can

be written as

p(z) = f(φ(z)) + bf(φ(z)).

Now the Cω norm of p can be written as

(53)‖p‖Cω(Dε/2) ≤ ‖f φ‖Cω(Dε/2) + |b|‖f φ‖Cω(Dε/2)

≤ C1(1 + |b|)‖f‖Cω(Dε)

where the last inequality is obtained using (51) and the fact that φ(Dε/2)

is contained in Dε.

Now we get a relation between sigma norm of the derivatives of f

and p. Let the map from ξ to z be denoted by ψ. Now using (46), we

get

fξ(ξ) =(1 + ab) pz(z)− (a+ b) pz(z)

(1− |b|2)(1− |a|2)

=(1 + ab) pzψ(ξ)− (a+ b) pzψ(ξ)

(1− |b|2)(1− |a|2)

where we have used z = ψ(ξ) in the last equality. Using z = ξ−a ξ1−|a|2 ,

we get (1− κ)|z| ≤ |ξ|. Choose δ such that Dδ is contained in φ(Dε/2).

23

Now the sigma norm can be obtained as

‖fξ‖Cσ(Dδ) ≤|1 + ab| ‖pz ψ‖Cσ(Dδ) + |a+ b| ‖pz ψ‖Cσ(Dδ)

(1− |b|2)(1− |a|2)

≤|1 + ab| ‖pz ψ‖Cσ(φ(Dε/2)) + |a+ b| ‖pz ψ‖Cσ(φ(Dε/2))

(1− |b|2)(1− |a|2).

Using (52), the above inequality can be written as

(54)

‖fξ‖Cσ(Dδ) ≤|1 + ab| (1− κ)−β‖pz‖Cσ(Dε/2)

(1− |b|2)(1− |a|2)

+|a+ b|(1− κ)−β ‖pz‖Cσ(Dε/2)

(1− |b|2)(1− |a|2).

Similarly we can get

(55)

‖fξ‖Cσ(Dδ) ≤|a+ b| (1− κ)−β‖pz‖Cσ(Dε/2)

(1− |b|2)(1− |a|2)

+|1 + ab|(1− κ)−β ‖pz‖Cσ(Dε/2)

(1− |b|2)(1− |a|2).

Adding (54) and (55), we get

‖fξ‖Cσ(Dδ) + ‖fξ‖Cσ(Dδ) ≤(1− κ)−β(‖pz‖Cσ(Dε/2) + ‖pz‖Cσ(Dε/2))

(1 + |b|)(1 + |a|)

≤(1− κ)−βK1(ε)(‖h‖Cω(Dε) + ‖p‖Cω(Dε))

(1 + |b|)(1 + |a|).

The last inequality is obtained using (50). Using (53), we now get

‖fξ‖Cσ(Dδ) + ‖fξ‖Cσ(Dδ) ≤K2(ε)(‖h‖Cω(Dε) + ‖f‖Cω(D2ε)

)

(1− κ)β(1 + |b|)(1 + |a|).

Hence

(56) ‖fξ‖Cσ(Dδ) + ‖fξ‖Cσ(Dδ) ≤ C(κ, ε))(‖h‖Cω(D2ε)+ ‖f‖Cω(D2ε)

).

For any D compactly contained in U , we now need to get a similar

inequality with norms in D and U . Consider any point ξ ∈ D. Choose

24

ε such that D2ε(ξ) is contained in U . Now for ξ1, ξ2 ∈ D such that

ξ = (ξ1 + ξ2)/2 and |ξ1 − ξ2| < δ, we get

|∂f(ξ1)− ∂f(ξ2)|σ(|ξ1 − ξ2|)

≤ supη,ζ∈Dδ(ξ)

|∂f(η)− ∂f(ζ)|σ(|η − ζ|)

|∂f(ξ)| ≤ supζ∈Dδ(ξ)

|∂f(ζ)|.

Combining the above two inequalities, we get that for any ξ ∈ D,

|∂f(ξ1)− ∂f(ξ2)|σ(|ξ1 − ξ2|)

+ |∂f(ξ)| ≤ ‖∂f‖Cσ(Dδ(ξ)).

Similarly,

|∂f(ξ1)− ∂f(ξ2)|σ(|ξ1 − ξ2|)

+ |∂f(ξ)| ≤ ‖∂f‖Cσ(Dδ(ξ)).

Adding the above two inequalities and using (56), we get

|∂f(ξ1)− ∂f(ξ2)|σ(|ξ1 − ξ2|)

+ |∂f(ξ)|+ |∂f(ξ1)− ∂f(ξ2)|σ(|ξ1 − ξ2|)

+ |∂f(ξ)|

≤ ‖∂f‖Cσ(Dδ(ξ)) + ‖∂f‖Cσ(Dδ(ξ))

≤ C(κ, ε)(‖h‖Cω(D2ε(ξ))+ ‖f‖Cω(D2ε(ξ))

)

≤ C(κ, ε)(‖h‖Cω(U) + ‖f‖Cω(U)).

Taking the supremum over ξ and dependant ξ1, ξ2, we get

‖∂f‖Cσ(D) + ‖∂f‖Cσ(D) ≤ C(κ,D,U)(‖h‖Cω(U) + ‖f‖Cω(U)).

In case of the Beltrami equation with variable coefficients, to obtain

the estimates on domains D,U compactly contained in Ω, the standard

method for generalising from constant to variable coefficients is the

following: We first freeze the coefficients at a point in Ω and consider

functions with support in a small disk around this point: the disk is

25

chosen such that the difference between the variable coefficients and

value of the coefficient at the center of the disk are sufficiently small.

We now look at the Beltrami equation satisfied by this function and

then obtain estimates similar to (20). In obtaining this estimate, the

standard absorption trick cannot be used since it requires having the

same norms on the right hand side as the left hand side of the inequality.

The norms on the right hand side and left hand side of (20) are not

the same. We have the C1,σ norm of the solution bounded by the Cω

norm of the solution. Hence the above mentioned absorption cannot

be done in this case.

To overcome this problem, we now look at functions that are less

regular than the functions in the Cω space but are more regular than

the functions in the Cσ space. The idea is that when the coefficients

of the Beltrami equation are functions with this kind of regularity, we

can obtain a result where the norm of the derivative of solution to the

Beltrami equation is bounded by the same norm of the solution. This

space of functions is defined using Fourier multipliers.

We first work with functions on a torus, obtain the required estimates

to solutions of the Beltrami equation and then use this to get estimates

on any bounded domain. We work with functions on a torus rather

than the whole complex plane since this gives us boundedness of the

Beurling transform which will be a key tool used in the next section.

4. The space C(λ) on the torus

In [15], the function space C(λ)(Rn) with

(57) λ(j) = ω(2−j)

26

was defined for a general modulus of continuity ω using a Littlewood-

Paley decomposition ψj as the collection of all f ∈ Rn such that

‖(ψj(D)f)‖L∞(Rn) ≤ Cλ(j).

where ψj(D) is the Fourier multiplier associated with the Littlewood-

Paley partition of unity ψj(ξ). C(λ)(Rn) is a Banach space under the

norm

‖f‖C(λ)(Rn) := supj

‖(ψj(D)f)‖L∞(Rn)

λ(j).

The classical results in Fourier analysis on Rn were used in [15] to anal-

yse these functions and their properties. We can use these results to

study functions on a torus by considering them as periodic functions

on R2.

Let T2 = (R/2πZ)2 denote the 2-dimensional torus. We identify T2

with [−π, π)× [−π, π) ⊂ R2 and functions on T2 with functions on R2

that are 2π -periodic in each of the coordinate directions.

For functions on a torus, the toroidal Fourier transform is defined

from C∞(T2) to S(Z2) where S(Z2) denotes the space of rapidly de-

caying functions on Z2. However, in our work we will be using the

Fourier transform on R2. But since periodic functions on R2 are not

L1, while taking the Fourier transform, we consider them as tempered

distributuions, for which the Fourier transform is given by (10).

In order to define the C(λ) space , we first need a smooth Littlewood-

Paley decomposition which is obtained as follows: Choose a compactly

27

supported function Ψ0 such that

Ψ0(ξ) =

1 for |ξ| ≤ 1

0 for |ξ| ≥ 2

for ξ ∈ R2. For k ≥ 1, set

(58) Ψk(ξ) = Ψ0(2−kξ).

Let

(59) ψ0(ξ) = Ψ0(ξ) and ψk(ξ) = Ψk(ξ)−Ψk−1(ξ) for k ≥ 1

so that

(60) Ψk(ξ) = ψ0(ξ) + ...+ ψk(ξ) and∞∑k=0

ψk(ξ) = 1.

The collection ψk(ξ) forms a Littlewood-Paley partition of unity.

Note that for each k, Ψk(ξ) is supported on a disk |ξ| < 2k+1. For

k > 0, each ψk(ξ) is supported on the annulus 2k−1 < |ξ| < 2k+1 and

ψ0(ξ) is supported on |ξ| < 2.

Since ψj(ξ) has compact support, we consider ψj(D) : S ′(R2) →

C∞(R2) as a Fourier multiplier operator associated with the function

ψj(ξ). Here since we want to define ψj(D) on periodic functions on R2,

we again consider the functions as tempered distributions. It can also

be written as a convolution given by

(ψj(D)f)(z) =

∫R2

ψj∨(z)f(z − z)dz.

Note that a Fourier multiplier operator preserves periodicity, as can

be seen from the following argument: Let f be a function on R2 with

28

period 2π and P (D) be a Fourier multiplier operator with symbol p(ξ).

Then we have

(P (D)f)(z) =

∫R2

p∨(z)f(z − z)dz

=

∫R2

p∨(z)f(z − z + 2π)dz

=

∫R2

p∨(z)f((z + 2π)− z)dz

= (P (D)f)(z + 2π).

Hence ψj(D) can be considered as an operator on the torus.

We are now ready to define the function space C(λ)(T2). We assume

that ω is a Dini modulus of continuity, i.e. satisfies the Dini condition

(11).

Definition 5. C(λ)(T2) is the collection of all periodic functions f ∈ R2

such that

(61) ‖(ψj(D)f)‖L∞(R2) ≤ Cλ(j)

and is a Banach space under the norm

(62) ‖f‖C(λ)(T 2) := supj

‖(ψj(D)f)‖L∞(R2)

λ(j).

The space C(λ)(T2) is identified with C(λ)P (R2) where the subscript

P refers to periodic functions on R2.

Definition 6. We denote by C1,(λ)(T2), those functions f ∈ C1(T2)

whose first order derivatives ∂f/∂z and ∂f/∂z are in C(λ)(T 2). C1,(λ)(T2)

is a Banach space under the norm

(63) ‖f‖C1,(λ)(T2) := ‖∂f‖C(λ)(T2) + ‖∂f‖C(λ)(T2) + ‖f‖C0(T 2).

29

The relation between the spaces C(λ), Cω and Cσ that holds on Rn

(as shown in [15]) also holds in the case of a torus T2 i.e.

(64) Cω(T2) ⊂ C(λ)(T2) ⊂ Cσ(T2).

This is true since the spaces Cω(T2), C(λ)(T2) and Cσ(T2) are just the

periodic functions on R2 which belong to the corresponding spaces on

R2.

We observe here that λ(j) has the following properties:

Lemma 3. The positive decreasing sequence λ(j) given by λ(j) =

ω(2−j) satisfies the following properties:

(65)∑j

λ(j) <∞,

(66)∑j≥l

λ2(j) ≤ cλ(l),

and λ(j) is slowly varying, i.e.

(67) λ(j) ≤ Cλ(j + 1).

Proof:

Proof of (65):∫ 1

0

ω(t)

tdt ≥

∞∑j=0

ω(2−j)

2−j(2−j − 2−(j+1)) =

1

2ω(2−j) =

1

2

∞∑j

λ(j).

Using (11), we get∑∞

j=0 λ(j) <∞.

Proof of (66): Since λ(j) is a decreasing sequence, we can write

Σj≥lλ2(j) ≤ λ(l)Σj≥lλ(j).

30

Using (65), we get the final result.

Proof of (67): Using (13), we get

λ(j) = ω(2−j) ≤ 2βω(2−(j+1)) ≤ Cλ(j + 1).

A tool that we will use extensively in this section is the Beurling

transform. It is defined for functions on a torus as a Fourier multiplier

operator by

(68) (Sf)(ξ) = m(ξ)f(ξ) where m(ξ) = ξ/ξ for ξ ∈ R2.

In working with Fourier multiplier operators, convolution of func-

tions will be an operation that we will use frequently in this section.

We shall make use of the following property of convolution of functions:

If f ∈ L1(R2), g ∈ L∞(R2) and one of them has compact support, then

(69) ‖f ∗ g‖L∞(R2) ≤ ‖f‖L1(R2)‖g‖L∞(R2).

We now prove the boundedness of the Beurling transform which plays

a vital role in obtaining our estimates.

Lemma 4. If h ∈ C(λ)(T2), then

(70) ‖Sh||C(λ)(T2) ≤ C‖h‖C(λ)(T2).

Proof: We first observe that the symbol of S has a singularity at

ξ = 0. We handle this singularity by writing S in terms of a smoothing

operator. Using the Littlewood-Paley partition of unity, we get

S =∞∑k=0

ψk(D)S = ψ0(D)S +∞∑k=1

ψk(D)S.

31

Let S :=∑∞

k=1 ψk(D)S. We now have

(71) ‖Sh‖C(λ)(T2) ≤ ‖(ψ0(D)S)h‖C(λ)(T2) + ‖Sh‖C(λ)(T2)

Observe that as the symbol of ψ0(D)S has compact support, ψ0(D)S

is a smoothing operator which takes a periodic function h ∈ C(λ)(R2)

to a smooth periodic function and so is bounded from C(λ)(T2) to

C(λ)(T2), i.e.

(72) ‖ψ0(D)Sh‖C(λ)(T2) ≤ C2‖h‖C(λ)(T2).

Let us now look at the boundedness of S. To estimate ‖Sh‖C(λ)(T 2), we

first consider

‖ψl(D)Sh‖L∞(R2) = ‖ψl(D)∞∑k=1

ψk(D)Sh‖L∞(R2)

=

∥∥∥∥∥F−1

(ψl(ξ)

∞∑k=1

ψk(ξ)ξ

ξFh

)∥∥∥∥∥L∞(R2).

Using the fact that ψl(ξ) is supported only in 2l−1 ≤ |ξ| ≤ 2l+1, we get

‖ψl(D)Sh‖L∞(R2) =

∥∥∥∥∥F−1

(ψl(ξ)

k=l+1∑k=l−1

ψk(ξ)ξ

ξFh

)∥∥∥∥∥L∞(R2)

=

∥∥∥∥∥F−1

(k=l+1∑k=l−1

ψk(ξ)ξ

ξψl(ξ)Fh

)∥∥∥∥∥L∞(R2)

=

∥∥∥∥∥F−1

(k=l+1∑k=l−1

ψk(ξ)ξ

ξ

)∗ F−1(ψl(ξ)Fh)

∥∥∥∥∥L∞(R2)

.

We now use (69) to get

(73)

‖ψl(D)Sh‖L∞(R2)

≤

∥∥∥∥∥F−1

(k=l+1∑k=l−1

ψk(ξ)ξ

ξ

)∥∥∥∥∥L1(R2)

∥∥F−1(ψl(ξ)Fh)∥∥L∞(R2)

.

32

Let us now estimate the term∥∥∥F−1

(∑k=l+1k=l−1 ψk(ξ)

ξξ

)∥∥∥L1(R2)

.

Using the definition of the inverse Fourier transform, (58) and (59),

we get

F−1

(k=l+1∑k=l−1

ψk(ξ)ξ

ξ

)=

∫R2

(Ψ0(2−(l+1)ξ)−Ψ0(2−(l−2)ξ))ξ

ξeiz·ξ dξ

=

∫R2

(Ψ0(2−(l+1)ξ)−Ψ0(2−(l−2)ξ))ξ

ξeiz·ξ

Let Ψ(η) := Ψ0(2−1η) − Ψ0(22η). Observe that Ψ(η) is supported in

2−1 < |η| < 22. The above expression can now be written as

F−1

(k=l+1∑k=l−1

ψk(ξ)ξ

ξ

)≤∫R2

Ψ(2−lξ)ξ

ξeiz·ξ dz

≤∫R2

Ψ(η)η

ηeiz·2

lη22l dη

≤ 22l

∫R2

Ψ(η)η

ηei2

lz·η dη

= 22lF−1(mΨ)(2lz)

where m(η) = ηη.

Now the L1 norm of F−1(∑k=l+1

k=l−1 ψk(ξ)ξξ

)can be written as∥∥∥∥∥F−1

(k=l+1∑k=l−1

ψk(ξ)ξ

ξ

)∥∥∥∥∥L1(R2)

=∥∥∥22lF−1(mΨ)(2lz)

∥∥∥L1(R2)

=

∫R2

22l∣∣∣F−1(mΨ)(2lz)

∣∣∣ dz= 22l

∫R2

∣∣∣F−1(mΨ)(z)∣∣∣ 2−2ldz

=∥∥∥F−1(mΨ)

∥∥∥L1(R2)

≤ C

33

where C is independent of l. This is true due to the fact that as mΨ

is smooth and has compact support, it is in Schwartz class and hence

in L1(R2).

Hence

‖ψl(D)Sh‖L∞(R2) ≤ C1‖ψl(D)h‖L∞(R2)

which gives

(74) ‖Sh‖C(λ)(T2) ≤ C2‖h‖C(λ)(T2).

Using (72) and (74 ) in (71), we get

(75) ‖Sh‖C(λ)(T2) ≤ C‖h‖C(λ)(T2).

Taylor had shown in [15] that C(λ)(R2) is a Banach algebra. The

following lemma uses a slightly different proof to show that C(λ)(T2) is

a Banach algebra.

Lemma 5. Assume f, g ∈ C(λ)(T 2). Then

(76) ‖fg||C(λ)(T2) ≤ C‖f‖C(λ)(T2)‖g‖C(λ)(T2).

Proof: Here we use Bony’s paraproduct decomposition. For the prod-

uct fg, it is given by

fg = Tfg + Tgf +Rfg

where Tf is Bony’s paraproduct defined by

Tfg =∑k≥5

Ψk−5(D)f · ψk+1(D)g

34

and Rfg, which is used to denote the remainder is given by

Rfg =∑j

φj(D)f · ψj(D)g where φj(D)f =∑|j−k|≤3

ψk(D)f

Hence

(77) ‖fg‖C(λ)(T2) ≤ ‖Tfg‖C(λ)(T2) + ‖Tgf‖C(λ)(T2) + ‖Rfg‖C(λ)(T2).

Let us first estimate ‖Rfg‖C(λ)(T 2). We have

‖(ψl(D)Rfg)‖L∞(R2) = ‖ψl(∑j

φj(D)f · ψj(D)g)‖L∞(R2)

= ‖ψl(D)∑j

(∑|j−k|≤3

ψk(D)f) · ψj(D)g‖L∞(R2).

Now let us consider the term in the right hand side of the above equa-

tion. The transform of the jth term of Rfg is

F(∑|j−k|≤3

ψk(D)f · ψj(D)g)(ξ)

= F((ψj−3 + ...+ ψj+3)(D)f.ψj(D)g)(ξ)

= [(ψj−3 + ...+ ψj+3) ∗ f) ∗ (ψj ∗ g)](ξ)

= |ψl(ξ)∑j

[((ψj−3 + ...+ ψj+3) ∗ f) ∗ (ψj ∗ g)](ξ))∨|

The term∑

j(ψj−3+...+ψj−3)(D)f is supported in 2j−4 ≤ |ξ| ≤ 2j+4

and the support of ψj(D)g is in 2j−1 ≤ |ξ| ≤ 2j+1. Hence the support

of transform of the jth term of Rfg is in |ξ| ≤ C2j+4. Now ψl(ξ) is

supported in 2l−1 < |ξ| ≤ 2l+1 . Hence for the annulus to intersect

the disc |ξ| ≤ 2j+4, we need C2j+4 ≥ 2l−1. We can now say that there

35

exists some integer N such that j ≥ l −N . Hence we can write

‖ψl(D)Rfg‖L∞(R2) = ‖(ψl(D)∑j≥l−N

(∑|j−k|≤3

ψk(D)f)ψj(D)g‖L∞(R2)

= ‖F−1(ψl(ξ)) ∗∑j≥l−N

(F∑|j−k|≤3

ψk(D)f ψj(D)g)∨‖L∞(R2).

Now we can use (69) to get

‖ψl(D)Rfg‖L∞(R2)

≤ ‖ψl∨‖L1(R2)

∑j≥l−N

(‖(ψj−3 + ...+ ψj+3)(D)f)‖L∞(R2)‖ψj(D)g‖L∞(R2)

≤ C‖f‖C(λ)(T 2)‖g‖C(λ)(T 2)

∑j≥l−N

(λ(j − 3) + ....+ λ(j + 3))λ(j)

≤ C‖f‖C(λ)(T2)‖g‖C(λ)(T2)

∑j≥l−N

(k1λ(j) + k2λ(j) + k2λ(j) + 4λ(j))λ(j)

≤ C‖f‖C(λ)(T2)‖g‖C(λ)(T2)

∑j≥l−N

C1λ2(j).

The last inequality is obtained using (67) and the property that λ(j) is

a positive decreasing sequence. Now∑

j≥l−N(k1λ(j)+k2λ(j)+k2λ(j)+

4λ(j))λ(j) ≤ C1λ2(j). We now use (66) to get

‖(ψl(D)Rfg‖L∞(R2) ≤ C2‖f‖C(λ)(T2)‖g‖C(λ)(T2)λ(l).

Using the above inequality we get

(78) ‖Rfg‖C(λ)(T2) ≤ K1‖f‖C(λ)(T2)‖g‖C(λ)(T2).

Now let us estimate ‖Tfg‖C(λ)(T2). We have

‖(ψl(D)Tfg)‖L∞(R2) = ‖ψl(D)∑k≥5

Ψk−5(D)f ψk+1(D)g‖L∞(R2)

= ‖ψl(D)∑k≥5

(ψ0 + ....+ ψk−5)(D)f ψk+1(D)g‖L∞(R2)

36

Since ψl is supported in the annulus 2l−1 ≤ |ξ| ≤ 2l+1, we must have

k − 5 ≥ l − 1 or k ≥ l + 4. Let m = min(k − 5, l + 1). Now we have

‖(ψl(D)Tfg)‖L∞(R2)

= ‖ψl(D)∑k≥l+4

m∑j=l−1

ψj(D)f ψk+1(D)g‖L∞(R2)

≤ ‖ψl∨‖L1(T2)

∑k≥l+4

‖ψj(D)f‖L∞(R2)‖ψk+1(D)g‖L∞(R2)

≤ C‖f‖C(λ)(T2)‖g‖C(λ)(T2)

∑k≥l+4

(m∑

j=l−1

λ(j))λ(k + 1)

≤ C‖f‖C(λ)(T2)‖g‖C(λ)(T2)

∑k≥l+4

λ(l − 1) + λ(l)) + λ(l + 1)λ(k + 1)

≤ C‖f‖C(λ)(T2)‖g‖C(λ)(T2)

∑k≥l+4

C1λ(l)λ(k + 1)

≤ C2‖f‖C(λ)(T2)‖g‖C(λ)(T2)λ(l)

The last inequality is obtained by using∑

k≥l+4 λ(k+ 1) ≤ C for some

constant C. We now get

(79) ‖Tfg‖C(λ)(T2) ≤ K2‖f‖C(λ)(T2)‖g‖C(λ)(T2).

Similarly we can get

(80) ‖Tgf‖C(λ)(T2) ≤ K3‖f‖C(λ)(T2)‖g‖C(λ)(T2).

Using (78),(79) and (80) in (77), we get (76).

We now obtain a result on the ∂ operator which will be useful in

analysing the Beltrami equation.

Proposition 2. The operator ∂ : C1,(λ) → C(λ) is Fredholm of index

zero with kernel and cokernel consisting of constants.

37

Proof: Define a parametrix P for ∂ by Ph = (φ(ξ)2iξh(ξ))∨ where φ is

a smooth function satisfying φ(0) = 0 and φ(ξ) = 1 for |ξ| ≥ 1. Using a

similar proof as in lemma 4 for the Beurling transform, it can be shown

that for h ∈ C(λ)(T2),

‖Ph||C1,(λ)(T2) ≤ C‖h‖C(λ)(T2).

Now

∂Ph = (φ(ξ)h(ξ))∨ = ((1− Φ(ξ))h(ξ))∨

where Φ is a smooth function with compact support defined by 1 −

Φ(ξ) = φ(ξ). Hence P ∂ = ∂P = I − K where K = Φ(D). But

Φ(D) : C(λ)(T2) → C∞(T2) is a smoothing operator. Since C∞(T2) is

compactly contained in C(λ)(T2), we get that K is a compact operator

on C(λ)(T2). Hence ∂ : C1,(λ) → C(λ) is invertible modulo compact

operators and is a Fredholm operator of index zero.

Let ∂f = 0 on T2. Hence f is analytic and periodic. This implies

that f is a constant. Hence kernel ∂ consists of constants.

Let ∂f = h. Using integration by parts, we see that∫T2

h =

∫T2

∂f = 0.

This shows that h is orthogonal to the constants. But the cokernel

is one-dimensional (since the map has index zero and one-dimensional

kernel), so the cokernel is exactly the constants.

We now have the necessary tools to to obtain estimates for the con-

stant coefficient Beltrami equation.

38

Theorem 2. Let h ∈ C(λ)(T2) and µ0 and ν0 be constants satisfying

|µ0| + |ν0| ≤ κ < 1. Let L0 denote the operator given by L0f = ∂f −

µ0 ∂f − ν0 ∂f . Then L0 : C1,(λ)(T2) → C(λ)(T2) is Fredholm of index

zero with kernel and cokernel consisting of constants. Moreover, if

f ∈ C1,(λ)(T2) satisfies the equation

(81) ∂f − µ0 ∂f − ν0 ∂f = h,

then we have the estimate

(82) ‖∂f‖C(λ)(T2) + ‖∂f‖C(λ)(T2) ≤ K‖L0f‖C(λ)(T2).

where K is a constant depending on µ0 and ν0.

Proof: We first observe that as in Section 4, for functions on R2,

the equation L0f = h can be reduced to an inhomogeneous Cauchy-

Riemann equation of the form (47), i.e. we have

(83) A−1L0Ap = ∂p.

Here A is a transformation given by

(84) Ap(z) :=p(z)− b p(z)

1− |b|2= f(ξ) where z =

ξ − a ξ1− |a|2

and A−1 is given by

(85) A−1f(ξ) := f(ξ) + b f(ξ) = p(z), where ξ = z + a z.

Note that the constants a and b are chosen as in (48) and (49) such

that |a|, |b| ≤ κ < 1. We first prove that there exists constants c1 and

c2 such that

(86) c1‖Ap‖C(λ)(R2) ≤ ‖p‖C(λ)(R2) ≤ c2‖Ap‖C(λ)(R2).

39

This can be proved as follows: Consider

‖f‖C(λ)(R2) =

∥∥∥∥∥p(z)− b p(z)

1− |b|2

∥∥∥∥∥C(λ)(R2)

≤ 1 + |b|1− |b|2

‖p‖C(λ)(R2)

≤ 1

1− |b|‖p‖C(λ)(R2)

≤ 1

1− κ‖p‖C(λ)(R2)

Hence we obtain

(87) ‖Ap‖C(λ)(R2) ≤1

1− κ‖p‖C(λ)(R2)

or

c1‖Ap‖C(λ)(R2) ≤ ‖p‖C(λ)(R2).

Now consider

‖p‖C(λ)(R2) = ‖f(ξ) + b f(ξ)‖C(λ)(R2)

≤ (1 + |b|)‖f‖C(λ)(R2)

≤ (1 + κ)‖f‖C(λ)(R2).

This gives the estimate

(88) ‖p‖C(λ)(R2) ≤ (1 + κ)‖Ap‖C(λ)(R2) ≤ c2‖Ap‖C(λ)(R2).

Hence we have proved (86).

Similarly we can prove

(89) c3‖A−1f‖C(λ)(R2) ≤ ‖f‖C(λ)(R2) ≤ c4‖A−1f‖C(λ)(R2).

40

We now obtain the estimates (82) on R2. First observe that in

Lemma 4, if we let h = ∂p and use S∂ = ∂, we obtain

‖∂p‖C(λ)(R2) ≤ C‖∂p‖C(λ)(R2).

Using (86), we can now obtain

‖∂p‖C(λ)(R2) + ‖∂p‖C(λ)(R2) ≤ C‖∂p‖C(λ)(R2)

= C‖A−1L0Ap‖C(λ)(R2)

≤ C1‖L0Ap‖C(λ)(R2).

Hence we have

(90) ‖∂p‖C(λ)(R2) + ‖∂p‖C(λ)(R2) ≤ C1‖L0f‖C(λ)(R2).

Now need to get a relation between the C(λ) norms of the first order

derivatives of f and p. Now using (46), we get

fξ(ξ) =(1 + ab) pz(z)− (a+ b) pz(z)

(1− |b|2)(1− |a|2)

Now the C(λ) norm can be obtained as

(91) ‖fξ‖C(λ)(R2) ≤|1 + ab| ‖pz‖C(λ)(R2) + |a+ b| ‖pz‖C(λ)(R2)

(1− |b|2)(1− |a|2)

Similarly we can get

(92) ‖fξ‖C(λ)(R2) ≤|a+ b|‖pz‖C(λ)(R2)) + |1 + ab|‖pz‖C(λ)(R2)

(1− |b|2)(1− |a|2).

Adding (91) and (92), we obtain

‖fξ‖C(λ)(R2) + ‖fξ‖C(λ)(R2) ≤(‖pz‖C(λ)(R2) + ‖pz‖C(λ)(R2)

(1 + |b|)(1 + |a|)

≤C1‖L0f‖C(λ)(R2)

(1 + |b|)(1 + |a|).

41

The last inequality is obtained using (90). Hence

(93) ‖fξ‖C(λ)(R2) + ‖fξ‖C(λ)(R2) ≤ K‖L0f‖C(λ)(R2).

In particular, for periodic functions on R2, we get the estimate

‖∂f‖C(λ)(T2) + ‖∂f‖C(λ)(T2) ≤ K(‖L0f‖C(λ)(R2).

which is our required estimate. Now adding the term ‖f‖C0(T2) on both

sides of the above estimate, we obtain

(94)‖f‖C1,(λ)(T2) ≤ K (‖L0f‖C(λ)(T2) + ‖f‖C0(T2))

≤ K (‖L0f‖C(λ)(T2) + ‖f‖C(λ)(T2)).

Now C1,(λ)(T2) is compactly contained in C(λ)(T2) . Since L0 satisfies

the above estimate, we can conclude that dimension of kernel of L0 is

finite and the range is closed, i.e. L0 is semi-Fredholm. But as we have

seen, A−1L0A = ∂. Consider the homotopy Gt = (1 − t)∂ + tL0 for

0 ≤ t ≤ 1. Since we know from Proposition (2) that ∂ is an operator

with index 0, using Gt, we can conclude that L0 also has index 0. Hence

we have proved that L0 : C1,(λ)(T2) → C(λ)(T2) is Fredholm of index

zero.

We use the estimate (82) to prove that the kernel consists of con-

stants. Let L0f = 0. Then using (82), we obtain

‖∂f‖C(λ)(T2) + ‖∂f‖C(λ)(T2) ≤ 0.

This shows that f has to be a constant and hence the kernel of L0

consists of constants.

42

Now we prove that the cokernel of L0 consists of constants. Consider

L0f = h. Since µ0 and ν0 are constants, we can use integration by parts

to obtain ∫T2

h dz =

∫T2

(∂f − µ0 ∂f − ν0 ∂f) dz = 0.

This shows that h is orthogonal to the constants. But we have al-

ready proved that the map L0 has index zero and that the kernel is

one dimensional. This indicates that the cokernel of L0 should also be

one-dimensional and hence it is exactly the constants.

We shall hereafter use L to denote the Beltrami operator defined by

(95) Lf := ∂f − µ ∂f − ν ∂f.

We need the following two lemmas in the proof of Theorem 3 below:

Lemma 6. Let µ, ν ∈ Cω(T2) satisfy |µ(ξ)| + |ν(ξ)| ≤ κ < 1, for all

ξ ∈ T2. If f ∈ W 1,2(T2) satisfies Lf = 0, then f is a constant.

Proof: If f ∈ W 1,2(T2) satisfies Lf = 0, then

∂f = µ ∂f + ν ∂f.

Taking an inner product with ∂f on both sides, we obtain

(96) (∂f, ∂f) = (µ ∂f, ∂f) + (ν∂f, ∂f).

43

But using the definition of inner product, we have

(∂f, ∂f) =

∫T2

∂f(z) ∂f(z)dz

=1

4

∫T2

(∂x + i∂y)f(x+ iy) (∂x − i∂y)f(x+ iy)dxdy

=1

4

∫T2

|∂xf |2 + |∂yf |2 + i(∂yf∂xf − ∂xf∂yf)dxdy

=1

4‖∇f‖2

L2(T2)

where the term∫T2(∂yf∂xf − ∂xf∂yf)dxdy = 0, using integration by

parts. Hence we can write (96) as

‖∇f‖2L2(T2)

=

∫T2

µ(z)∂f(z) ∂f(z)dz +

∫T2

ν(z)∂f(z) ∂f(z)dz

=

∫T2

µ(z)∂f(z) ∂f(z)dz +

∫T2

ν(z)∂ f(z) ∂f(z)dz

=

∫T2

µ(x+ iy)(∂x − i∂y)f(x+ iy) (∂x − i∂y)f(x+ iy)dxdy

+

∫T2

ν(x+ iy)(∂x + i∂y)f(x+ iy) (∂x − i∂y)f(x+ iy)dxdy

We now obtain

‖∇f‖2L2(T2) ≤ (‖µ‖L∞(T2) + ‖ν‖L∞(T2))‖∇f‖2

L2(T2)

≤ κ‖∇f‖2L2(T2).

The above inequality implies that ∇f=0, i.e. f = 0.

Lemma 7. Let g ∈ Cω(T2) such that ‖g − g0‖Cω(Dε) < C(ε) where

g0 = g at the center of Dε. Then we can find an extension g of g − g0

restricted to Dε such that g has support only in D2ε and

(97) ‖g‖C(λ)(T2) ≤ ‖g − g0‖Cω(D2ε).

44

Proof: We obtain the extension g with the desired properties by re-

flecting the values of g− g0 on D2ε \Dε i.e for each z ∈ D2ε(ξ0), choose

ξ = 2ε z|z|−z on D2ε \Dε so that as |z| → ε, ξ → z and as |z| approaches

the origin, ξ approaches the boundary of D2ε. We define g by

g(ξ) =

(g − g0)(ξ) for ξ ∈ Dε

(g − g0)(z) for ξ ∈ D2ε \ Dε

where ξ is the reflection of z in D2ε(ξ0). We extend g by zero outside

D2ε. First we need to show that this extension is Cω at the boundary

of each of the disks Dε and D2ε.

First we consider the boundary of Dε. Let z0 be a point on the boundary

of Dε. For δ < ε and sufficiently small, consider a small disk Dδ(z0).

We just need to consider points on either side of the boundary. For z,

ξ1 ∈ Dδ(z0) such that z ∈ Dε and ξ1 ∈ D2ε \ Dε, we obtain

‖g‖Cω(Dδ(z0)) = supξ∈Dδ(z0)

|g(ξ)|+ supz,ξ1∈Dδ(z0)

|g(ξ1)− g(z)|ω(|ξ1 − z|)

= supz∈Dδ(z0)

|g(z)|+ supz,ξ1∈Dδ(z0)

|g(z1)− g(z)|ω(|ξ1 − z|)

where z1 is the reflection of ξ1 and z is the reflection of ξ in Dε. As δ

tends to 0, z and ξ1 approach the boundary of Dε and z1 approaches

the boundary of Dε. Now |ξ1−z| = |2ε z1|z1|−z1−z| = |z1(2ε 1|z1|−1)−z|.

But |z1| ≤ ε and |z1| → as ε as δ → 0. Hence 2ε 1|z1| − 1 ≥ 1. We now

obtain |ξ1 − z| ≥ |z1 − z| which gives ω(|ξ1 − z|) ≥ ω(|z1 − z|). Hence

we conclude that

‖g‖Cω(Dδ(z0)) ≤ ‖g‖Cω(Dε)

45

which is finite.

Next we consider the boundary of D2ε. Let z0 be a point on the bound-

ary of Dε. For δ < ε and sufficiently small, consider a small disk Dδ(z0).

Again we only need to consider points on either side of the boundary.

For ξ1, ξ2 ∈ Dδ(z0) such that ξ1 ∈ D2ε and ξ2 lies outside D2ε, we get

‖g‖Cω(Dδ(z0)) = supξ∈Dδ(z0)

|g(ξ)|+ supξ2,ξ1∈Dδ(z0)

|µ(ξ1)− µ(ξ2)|ω(|ξ1 − ξ2|)

= supz∈Dδ(z0)

|g(z)|+ supξ2,ξ1∈Dδ(z0)

|µ(z1)− µ(ξ2)|ω(|ξ1 − ξ2|)

where z and z1 are the reflections of ξ and ξ1 (which has already been

chosen) respectively in Dε. Now g(ξ2) = 0 and hence we obtain

supξ2,ξ1∈Dδ(z0)

|g(ξ1)− g(ξ2)|ω(|ξ1 − ξ2|)

= supξ2,ξ1∈Dδ(z0)

|g(z1)− g0|ω(|ξ1 − ξ2|)

≤ supξ2,ξ1∈Dδ(z0)

‖g‖Cω(Dε(ξ0))ω(|z1 − ξ0|)ω(|ξ1 − ξ2|)

Now |z1 − ξ0| ≤ |ξ1 − ξ2| which gives ω(|z1 − ξ0|) ≤ ω(|ξ1 − ξ2|). Hence

we get ‖g‖Cω(Dδ(z0)) ≤ ‖g‖Cω(Dε(ξ0)) which is finite.

Now we consider the Cω norm of g in D2ε. Using the definition, we

have

‖g‖Cω(D2ε) = supξ∈D2ε

|g(ξ)|+ supz,ξ1∈D2ε

|g(z)− g(ξ1)|ω(|z − ξ1|)

= supz∈Dε|g(z)|+ sup

z,ξ1∈D2ε

|g(z)− g(z1)|ω(|z − ξ1|)

where z and z1 are the reflections of ξ and ξ1 respectively in Dε. Now

|z − ξ1| ≥ |z − z1| which gives ω(|z − ξ1|) ≥ ω(|z − z1|). Hence the

above equation can be written as

‖g‖Cω(D2ε) ≤ supz∈Dε|g(z)|+ sup

z,z1∈Dε

|g(z)− g(z1)|ω(|z − z1|)

= ‖g‖Cω(Dε)

46

which is our required result.

Theorem 3. Let h ∈ C(λ)(T2) and µ, ν ∈ Cω(T2) satisfy |µ(ξ)| +

|ν(ξ)| ≤ κ < 1, for all ξ ∈ T2. Then L : C1,(λ)(T2) → C(λ)(T2)

is Fredholm of index zero with kernel consisting of constants. If f ∈

C1,(λ)(T2) is a solution to the equation

(98) ∂f − µ ∂f − ν ∂f = h

and ‖µ‖Cω(T2) +‖ν‖Cω(T2) < Γ0, then there exists K = K(Γ0) such that

(99) ‖f‖C1,(λ)(T2) ≤ K (‖h‖C(λ)(T2) + ‖f‖C(λ)(T2)).

Proof: We start with the assumption that f ∈ C1,(λ)(T2) and show

that (99) holds. Let ξ0 ∈ T2. Let µ0 = µ(ξ0) and ν0 = ν(ξ0). Choose ε

such that the Cω norms of (µ(z)−µ0) and (ν(ξ)−ν0) are small enough

in a disk Dε(ξ0). Without loss of regularity, we assume here that ξ0 = 0.

Define F := φ f where φ ∈ C0∞(Dε) .

(100) ∂F−µ ∂F−ν ∂F = H where H := φh+∂φ f−µ ∂φ f−ν ∂φ f.

(101)∂F − µ0 ∂F − ν0 ∂F = H0 where

H0 := H + (µ− µ0)Fξ + (ν − ν0)Fξ.

Hence F satisfies the Beltrami equation in T2. Using (82), we can

write

‖∂F‖C(λ)(T2) + ‖∂F‖C(λ)(T2) ≤ K‖L0F‖C(λ)(T2)

47

where L0F := H and K is a constant . This can be written as

(102)‖∂F‖C(λ)(T2) + ‖∂F‖C(λ)(T2)

≤ K‖(L− L0)F‖C(λ)(T2) + ‖LF‖C(λ)(T2).

Now

(103) ‖(L−L0)F‖C(λ)(T2) ≤ ‖(µ−µ0)∂F‖C(λ)(T2)+‖(ν−ν0)∂F‖C(λ)(T2)

Now we need to be able to write the terms ‖(µ− µ0)∂F‖C(λ)(T2) and

‖(ν − ν0)∂F‖C(λ)(T2) in terms of ‖(µ− µ0)‖Cω(Dε) and ‖(ν − ν0)‖Cω(Dε)

respectively. This will enable us to make use of the fact that these

terms are small enough to get absorbed into on the left hand side of

(102).

We first do this for ‖(µ − µ0)∂F‖C(λ)(T2). For this, we first observe

that if we can find an extension µ to T2 of µ − µ0 stricted to Dε such

that ‖µ‖C(λ)(T2) ≤ ‖(µ − µ0)‖Cω(Dε), then since F has support only in

Dε. We have

‖(µ− µ0)∂F‖C(λ)(T2) = ‖µ∂F‖C(λ)(T2) ≤ ‖µ‖C(λ)(T2)‖∂F‖C(λ)(T2)

from which can obtain our desired estimate

(104) ‖(µ− µ0)∂F‖C(λ)(T 2) ≤ ‖µ− µ0‖Cω(Dε)‖∂F‖C(λ)(T2).

For a similar extension ν of ν − ν0, we can obtain

(105) ‖(ν − ν0)∂F‖C(λ)(T2) ≤ ‖ν − ν0‖Cω(Dε)‖∂F‖C(λ)(T2).

Lemma 8 enables us to obtain extensions ν and µ with the desired

properties such that (104) and (105) hold good. Hence we can use

(104) and (105) in (103) to get

‖(L− L0)F‖C(λ)(T 2) ≤ (‖µ− µ0‖Cω(D2ε) + ‖ν − ν0‖Cω(D2ε))‖∂F‖C(λ)(T 2).

48

Substituting the above in (102) gives

‖∂F‖C(λ)(T2) + ‖∂F‖C(λ)(T2)

≤ K(‖µ− µ0‖Cω(D2ε) + ‖ν − ν0‖Cω(D2ε))‖∂F‖C(λ)(T 2) + ‖LF‖C(λ)(T2)).

Now we chose ε small enough so thatK(‖µ−µ0‖Cω(D2ε)+‖ν−ν0‖Cω(D2ε)) <

1/2 and hence the corresponding term can be absorbed into (‖∂F‖C(λ)(T2)+

‖∂F‖C(λ)(T2)). Hence we obtain

(106) ‖∂F‖C(λ)(T2) + ‖∂F‖C(λ)(T2) ≤ K1‖LF‖C(λ)(T2).

Now consider an open cover of T2 by ε disks. Then T2 has a finite

subcover ie

T2 ⊂ ∪Nj=1Dεj(ξj)

where ξj ∈ T2, j = 1, ...N. Consider a partition of unity (φj), j = 1, ..N

subordinate to this subcover. We can now write

‖∂f‖C(λ)(T2) = ‖N∑j=1

φj∂f‖C(λ)(T2)

≤N∑j=1

‖φj∂f‖C(λ)(T2)

≤N∑j=1

(‖∂(φjf)‖C(λ)(T2) + ‖∂φjf‖C(λ)(T2))

≤N∑j=1

(‖∂(φjf)‖C(λ)(T2) + C1‖∂φj‖C(λ)(T 2)‖f‖C(λ)(T 2))

≤ C2

N∑j=1

(‖∂(φjf)‖C(λ)(T 2) + ‖f‖C(λ)(T2))

Similarly we can get

‖∂f‖C(λ)(T2) ≤ C3

N∑j=1

(‖∂(φjf)‖C(λ)(T2) + ‖f‖C(λ)(T2))

49

Hence

‖∂f‖C(λ)(T2) + ‖∂f‖C(λ)(T2)

≤ C5

N∑j=1

(‖∂(φjf)‖C(λ)(T2) + ∂(φjf)‖C(λ)(T2) + ‖f‖C(λ)(T2)).

Now φjf is supported only in Dεj(ξj). Hence (106) can be applied

to φjf to get the estimate

(107)

‖∂f‖C(λ)(T2) + ‖∂f‖C(λ)(T2)

≤ C4

N∑j=1

(‖L(φjf)‖C(λ)(T2) + ‖f‖C(λ)(T2))

We have L(φjf) = φj h+∂φj f−µ ∂φj f−ν ∂φj f . Hence ‖L(φjf)‖C(λ)(T2)

can be estimated as

‖L(φjf)‖C(λ)(T2)

≤ ‖φj h‖C(λ)(T2) + ‖∂φj f‖C(λ)(T2) + ‖µ ∂φj f‖C(λ)(T2) + ‖ν ∂φj f‖C(λ)(T2)

≤ C5(‖h‖C(λ)(T2) + ‖f‖C(λ)(T2) + (‖µ‖C(λ)(T2) + ‖ν‖C(λ)(T2))‖f‖C(λ)(T2)).

Now we have

(108) ‖L(φjf)‖C(λ)(T2) ≤ C6(‖Lf‖C(λ)(T2) + ‖f‖C(λ)(T2))

where C6 = C6(Γ0).

Substituting (108) in (107), we get

(109) ‖∂f‖C(λ)(T2) + ‖∂f‖C(λ)(T2) ≤ C7 (‖h‖C(λ)(T2) + ‖f‖C(λ)(T2)).

We can now obtain

‖∂f‖C(λ)(T2) + ‖∂f‖C(λ)(T2) + ‖f‖C0(T2)

≤ C7 (‖h‖C(λ)(T2) + ‖f‖C(λ)(T2)) + ‖f‖C0(T2)

≤ K (‖h‖C(λ)(T2) + ‖f‖C(λ)(T2))

50

where K = K(Γ0). Hence we obtain the final estimate (99).

We now use the regularity estimates (99) to prove that L is Fredholm

of index zero. Consider a homotopy Lt given by Lt = (1− t)∂ + tL =

∂ + t(−µ∂ − ν∂) where 0 ≤ t ≤ 1. The ellipticity condition is satisfied

in the case of Lt and hence it satisfies the estimate

(110) ‖f‖C1,(λ)(T2) ≤ K (‖Ltf‖C(λ)(T2) + ‖f‖C(λ)(T2)).

Now C1,(λ) is compactly contained in Cλ and Lt satisfies (110). Hence

Lt is semi-Fredholm i.e. dimension of kernel of L is finite and the range

is closed. Now L0=∂. From Proposition (2), ∂ is an operator with index

0 and hence we can conclude that L also has index 0.

We now need to prove that the kernel of L consists of constants. But

Lemma 6 shows that the kernel consists of constants.

Corollary 1. Let h ∈ C(λ)(T2) and µ, ν ∈ Cω(T2) satisfying |µ(ξ)| +

|ν(ξ)| ≤ κ < 1, for all ξ ∈ T2. If f ∈ W 1,2(T2) satisfies Lf = h, then

f ∈ C1,(λ)(T2) .

Proof: Using Theorem 3, let f1 ∈ C1,(λ)(T2) satisfy Lf = h. Hence

L(f − f1) = 0. By Lemma (6), we infer that f − f1 is a constant which

in turn implies that f ∈ C1,(λ)(T2) .

Theorem 4. Let h ∈ Cω(T2) and µ, ν ∈ Cω(T2) satisfy |µ(ξ)| +

|ν(ξ)| ≤ κ < 1 for all ξ ∈ T2. Let f ∈ W 1,2(T2) be a solution to

51

the equation

(111) ∂f − µ ∂f − ν ∂f = h.

Then f ∈ C1,σ(T2). If ‖µ‖Cω(T2) + ‖ν‖Cω(T2) < Γ0, then there exists

K = K(Γ0) such that

(112) ‖f‖C1,σ(T2) ≤ K (‖Lf‖Cω(T2) + ‖f‖C0(T2)).

Proof: Using Corollary 1, we get that if f ∈ W 1,2(T2) is a solution

to (111), then f ∈ C1,(λ)(T2) . Using Theorem 3, we obtain that f

satisfies the estimate

‖f‖C1,(λ)(T2) ≤ K (‖Lf‖C(λ)(T2) + ‖f‖C(λ)(T2)).

Using (64), we obtain that f ∈ C1,σ(T2) and satisfies

(113) ‖f‖C1,σ(T2) ≤ K (‖Lf‖Cω(T2) + ‖f‖Cω(T2)).

We now need to replace the term ‖f‖Cω(T2) with ‖f‖C0(T2) in the above

inequality.

For 1 < q <∞, we have the elliptic estimate for Sobolev spaces

(114) ‖f‖W 1,q(T2) ≤ K1 (‖Lf‖Lq(T2) + ‖f‖Lq(T2))

for coefficients µ and ν that are just bounded and measurable. We also

have

(115) W 1,q(T2) ⊂ Cα(T2) for 1− 2

q> α

and

(116) Cα(T2) ⊂ Cω(T2) ⊂ C0(T2) ⊂ Lq(T2).

52

Using (115) and(116), we obtain

‖f‖Cω(T2) ≤ K2‖f‖Cα(T2) ≤ K3‖f‖W 1,q(T2)

Using (114) and also (116) we now obtain

‖f‖Cω(T2) ≤ K4(‖Lf‖Lq(T2) + ‖f‖Lq(T2))

≤ K5(‖Lf‖C0(T2) + ‖f‖C0(T2))

≤ K6(‖Lf‖Cω(T2) + ‖f‖C0(T2)).

Substituting the above in (112), we now obtain our required estimate

‖f‖C1,σ(T2) ≤ K7(‖Lf‖Cω(T2) + ‖f‖C0(T2)).

We are now ready to obtain similar estimates for the Beltrami equa-

tion in case of a bounded domain.

5. Non constant coefficients in a bounded domain

Let f be defined in a bounded domain Ω. Without loss of generality,

we can assume that Ω ⊂ (−π, π)× (π, π).

We now prove the main theorem of estimates on solutions of Beltrami

equation in Ω.

Theorem 5. Let µ, ν, h ∈ Cω(Ω) satisfying |µ(ξ)|+ |ν(ξ)| ≤ κ < 1 for

all ξ ∈ Ω. Let f ∈ W 1,2`oc (Ω) be a solution to the equation

(117) ∂f − µ ∂f − ν ∂f = h.

Then f ∈ C1,σ(Ω). In fact, for any domains D, U such that D ⊂ U

and U ⊂ Ω, we have f ∈ C1,σ(D) and if ‖µ‖Cω(U) + ‖ν‖Cω(U) < Γ0,

then there exists K = K(κ,Γ0, D, U) such that

(118) ‖f‖C1,σ(D) ≤ K (‖h‖Cω(U) + ‖f‖C0(U)).

53

Proof: We shall use estimates obtained on a torus to get the above

result. Let φ ∈ C0∞(V ) and φ ≡ 1 on D where D ⊂ V and V ⊂ U . We

can consider φf ∈ W 1,2(T2) ⊂ Lq(T2) for 1 ≤ q <∞. Now φf satisfies

a Beltrami equation with L(φf) = φh+∂φ f−µ ∂φ f−ν ∂φ f ∈ Lq(T2).

Now by using (114), we have φf ∈ W 1,q(T2). Using (115), we can take

q large enough to get φf ∈ Cα(T2) for some α > 0 which is contained

in Cω(T2). Now we can use Theorem 4 to infer that φf ∈ C1,σ(T2).

But φf = f in D. Hence f ∈ C1,σ(D). We observe that since D is

any domain compactly contained in Ω, we also obtain that f ∈ C1,σ(Ω).

We have

‖f‖C1,σ(D) = ‖φf‖C1,σ(D) ≤ ‖φf‖C1,σ(V ) = ‖φf‖C1,σ(T2).

Now φf satisfies a Beltrami equation defined on the whole of T 2

with L(φf) ∈ Cω(T2) and L is the Beltrami operator as given by (95).

Hence using (112), we have

(119) ‖φf‖C1,σ(T2) ≤ K (‖L(φf)‖Cω(T2) + ‖φf‖C0(T2)).

Using (119), we get the estimate

(120) ‖f‖C1,σ(D) ≤ K (‖L(φf)‖Cω(V ) + ‖φf‖C0(V )).

We now estimate ‖L(φf)‖Cω(V ).

(121)‖L(φf)‖Cω(V ) ≤ ‖φh‖Cω(V ) + ‖f ∂φ‖Cω(V )

+ ‖µ ∂φ f‖Cω(V )) + ‖ν ∂φ f‖Cω(V ).

Let us estimate each term above. We obtain

(122) ‖φh‖Cω(V ) ≤ ‖φ‖Cω(V ) ‖h‖L∞(V ) + ‖h‖Cω(V ) ‖φ‖L∞(V ).

54

Similarly we find

(123) ‖f ∂φ‖Cω(V ) ≤ c1‖∂φ‖Cω(V ) ‖f‖L∞(V ) + ‖f‖Cω(V ) ‖∂φ‖L∞(V ).

We can also obtain

(124)

‖µ ∂φ f‖Cω(V )) ≤ c2(‖∂φf‖Cω(V ) ‖µ‖L∞(V ) + ‖µ‖Cω(V ) ‖∂φf‖L∞(V ))

≤ c3‖µ‖L∞(V )(‖∂φ‖Cω(V ) ‖f‖L∞(V ) + ‖f‖Cω(V ) ‖∂φ‖L∞(V ))

+ ‖µ‖Cω(V ) ‖∂φ‖L∞(V )‖f‖L∞(V ))

≤ c3‖f‖Cω(V ) ‖µ‖L∞(V )(‖∂φ‖Cω(V ) + ‖∂φ‖L∞(V ))

+ ‖µ‖Cω(V ) ‖∂φ‖L∞(V ).

This now simplifies to

(125) ‖µ ∂φ f‖Cω(V )) ≤ c4(D,U)‖f‖Cω(V ) (‖µ‖L∞(V ) + ‖µ‖Cω(V )).

Similarly we can obtain

(126) ‖ν ∂φ f‖Cω(V ) ≤ c5(D,U)‖f‖Cω(V ) (‖ν‖L∞(V ) + ‖ν‖Cω(V )).

Substituting (122),(123), (124) and (126) in (121) we obtain

(127) ‖L(φf)‖Cω(V ) ≤ C1(D,U) ‖Lf‖Cω(V ) + C2(κ,Γ0, D) ‖f‖Cω(V ).

where C3 = C3(U,D).

Substituting (127) in (120), we get the estimate

(128)‖f‖C1,σ(D) ≤ C3(‖Lf‖Cω(V ) + ‖f‖Cω(V ) + ‖φf‖C0(V ))

≤ C4(‖Lf‖Cω(V ) + ‖f‖Cω(V ) + ‖f‖C0(V ))

where C4 = C4(κ,Γ0, U,D).

55

We now use the similar calculations as for (113) to replace the Cω

norm with the C0 norm on f and improve the estimate above. For

1 < q <∞, we use the elliptic estimate for Sobolev spaces

(129) ‖f‖W 1,q(V ) ≤ K1 (‖Lf‖Lq(U) + ‖f‖Lq(U)).

As in Theorem 5, we also use

(130) W 1,q(V ) ⊂ Cα(V ) for 1− 2

q> α

and

(131) Cα(V ) ⊂ Cω(V ) ⊂ C0(V ) ⊂ Lq(V ).

Using (130) and (131), we obtain

‖f‖Cω(V ) ≤ K2‖f‖Cα(V ) ≤ K3‖f‖W 1,q(V ).

Using (129) and also (131) we now obtain

‖f‖Cω(V ) ≤ K4(‖Lf‖Lq(U) + ‖f‖Lq(U))

≤ K5(‖Lf‖C0(U) + ‖f‖C0(U))

≤ K6(‖Lf‖Cω(U) + ‖f‖C0(U)).

Substituting the above estimate in (128), we get the required estimate.

56

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