# Fredholm theory, singular integrals and Tb theorems

In 2010-11 I gave a graduate course/seminar series on these topics
at
Linköping university. There were 7 lectures on Fredholm theory,
with focus on weakly singular integral operators, before Christmas,
followed by 8 lectures on singular integrals (SIOs), with focus on
proving the
T(1) and T(b) theorems for L_{2}-boundedness of SIOs. Below
you
find a
description of the contents, references to the literature, and my hand-written lecture notes:

Part I: Fredholm operator theory

Part II: singular integral operators
(Second version. Some typos corrected 9 March 2011)

## Part I

### Section 1

We show how to solve the classical Dirichlet boundary value problem
for
the Laplace operator by inverting the double layer potential on the
boundary of the domain. This is our main example of an integral
equation
throughout the course. See [3]. We show how to bound
integral operators on L_{p} through Schur estimates. The
remainder of of Part I is set in Banach space, where we use standard
functional analysis. See however [7].

### Section 2

We study compact operators in Banach spaces. Emphasis is put on
using
"total boundedness" in understanding compact sets. In Hilbert
spaces,
we study Hilbert-Schmidt operators, where their kernel theorem plays
a
role analogous to the Arzelà-Ascoli theorem in spaces of
continuous functions, as a tool for proving compactness of
operators.
See [4, Appendix A.6].

### Section 3

We study the main difficulty when working in general Banach spaces:
complements and projections to/onto subspaces. As our goal is to
fully
develop the theory for semi-Fredholm operators in Banach spaces,
these
preliminaries are necessary. Proposition 3.10 is the main result:
upper
semi-Fredholmness is equivalent to lower bounds modulo compact
operators.

### Section 4

We prove the closed range theorem for Banach space operators, which
is
surprisingly more subtle than the Hilbert space version. References
are
[1, IV Thm. 5.13], which also deals with unbounded operators, and
[6,
Section 4]. We show in Corollary 4.12 that the theorem for unbounded
operators follows from the special case of bounded operators, unlike
in
[1] where the theorem is proved directly for unbounded
operators/pairs
of subspaces.

### Section 5

We show the relation between semi-Fredholmness and existence of
inverses modulo compact operators. Then the notion of index is
introduced and the basic
composition theorem is proved along the lines in [5, Thm 1.3.2]. The
main result in section is the fundamental perturbation theorem 5.14
for
semi-Fredholm operators in Banach spaces. To obtain the optimal
result
we use the Borsuk-Ulam theorem. See [1, IV Lem. 2.3]. For
applications,
the method of continuity thm 5.19 is the main result of this
section.

### Section 6

This section contains some complementary result for part I. First a
useful abstract regularity result from [8], which shows the
relevance
of the index not only when perturbing the operator, but also when
perturbing the space. Proposition 6.3 then give full information
about
the Fredholm properties of the double layer potential operator,
including the dimensions of null spaces and cokernels. Finally the
analytic Fredholm theorem 6.8 is proved, which is the main tool for
getting more precise information about the dimension of the
nullspaces
of Fredholm operators. See [1, VII Thm 1.9].

## Part II

### Section 7

We introduce the notion of singular integral operators on R^{n}. We do not make
the
standing regularity assumption of the kernel, as is usually done in
the
literature. Five motivating examples are introduced, including the
Cauchy integral on a Lipschitz curve, the Calderón
commutators
and the double layer potential on Lipschitz surfaces. To get
familiar
with SIOs we first briefly study classical convolution SIOs. The
main
result of this section is Thm 7.18, which is taken from [2, II 3.2].

### Section 8

We demonstrate in Ex. 8.5, taken from [10, section 7.2], that the
principal value for antisymmetric non-convolution SIOs need not
exist
in
L_{2}. Proposition 8.2 however shows that the principal
value
always exists in a slighly weaker sense. The weighted Sobolev test
functions H_{t}^{s} are convenient when working with
Haar bases. This setup for SIOs is somewhat novel. We
define the Haar basis on the real line, and use it to reprove the L_{2}-boundedness
of
the Cauchy integral on the real line in Example 8.16. This serves as
a first illustration of the proof method for the T(1) theorem.

### Section 9

We first define dyadic cubes in R^{n},
and modify the main branch so that the tree of cubes is connected.
This
is convenient to have connected, but is not really essential though.
Standard
cubes will work. This section contains mainly the background from
harmonic analysis needed for the T(1) theorem: the Hardy-Littlewood
maximal function, Carleson measures, non-tangential maximal
functions
and BMO. We prove the fundamental Carleson's theorem. See [13, II
2.2],
[2, part 1] and [15]. Next we introduce the notion of "perfect
dyadic
SIOs", which will serve as a "discrete model" for SIOs. The
inspiration
comes from [12]. We show the necessity of the
T(1) condition.

### Section 10

We construct the Haar decomposition of L_{2}(R^{n}). Some novelty is
that
we work not with individual Haar basis functions but instead with
the 2^{n}-1
dimensional subspaces (a vector-valued approach so to speak). We
show
how a pointwise multiplication operator
splits into three operators under Haar decomposition: a multiplier,
a
paraproduct and an adjoint paraproduct. The BMO/Carleson bound of
the
paraproduct is proved. Mapping properties of these operators are
compared to perfect dyadic SIO, resulting in the main Thm 10.10. The
one-dimensional case is in [12, Lem. 6.1].

### Section 11

We formulate the T(1) theorem för Hölder regular SIOs,
derive
boundedness of the Calderón commutators from it, and then
prove
the theorem. Inspiration comes from [11, Part II] and [14].

### Section 12

We prove a (finite dimensional) vector-valued T(b)-theorem. In the
literature, usually only complex-valued functions are considered for
the T(b)-theorem. Example 12.3 illustrates the usefulness of a more
vector-valued formulation of the T(b)-theorem. We here prove
boundedness of the Cauchy integral and the double layer potential on
Lipschitz curves/surfaces using a multivector-valued SIO: the
Clifford-Cauchy SIO. See [16]. It is
emphasized that a T(b) theorem for a Hölder regular SIO is
equivalent to a T(1) theorem for a non-smooth SIO. Here we see the
usefulness of not making regularity of the SIO kernel a standing
assumption.

### Section 13

This final section contains some complementary results for Part II.
We prove L_{p}-boundedness of Calderón-Zygmund
operators. See [2, part II]. Then we return to the double layer
potential operator (not a compact operator any longer!) on Lipschitz
surfaces and prove L_{2}
invertibility using Rellich estimates. See [17]. Finally we compute
its
exact spectrum on plane cones using the Fourier/Mellin transform.
See
[18].

## References

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properties
of functions.
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from
Lund 1989.)
- W. Rudin: Functional analysis.
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problems on irregular plane domains. J. Differential Equations
34
(1979), no. 3, 361–392.

- Y. Meyer: wavelets and operators.
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multilinear
operators.

- G. David: Singular integrals on surfaces. Springer lecture
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