Fredholm theory, singular integrals and Tb theorems
In 2010-11 I gave a graduate course/seminar series on these topics
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
T(1) and T(b) theorems for L2-boundedness of SIOs. Below
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)
We show how to solve the classical Dirichlet boundary value problem
the Laplace operator by inverting the double layer potential on the
boundary of the domain. This is our main example of an integral
throughout the course. See . We show how to bound
integral operators on Lp through Schur estimates. The
remainder of of Part I is set in Banach space, where we use standard
functional analysis. See however .
We study compact operators in Banach spaces. Emphasis is put on
"total boundedness" in understanding compact sets. In Hilbert
we study Hilbert-Schmidt operators, where their kernel theorem plays
role analogous to the Arzelà-Ascoli theorem in spaces of
continuous functions, as a tool for proving compactness of
See [4, Appendix A.6].
We study the main difficulty when working in general Banach spaces:
complements and projections to/onto subspaces. As our goal is to
develop the theory for semi-Fredholm operators in Banach spaces,
preliminaries are necessary. Proposition 3.10 is the main result:
semi-Fredholmness is equivalent to lower bounds modulo compact
We prove the closed range theorem for Banach space operators, which
surprisingly more subtle than the Hilbert space version. References
[1, IV Thm. 5.13], which also deals with unbounded operators, and
Section 4]. We show in Corollary 4.12 that the theorem for unbounded
operators follows from the special case of bounded operators, unlike
 where the theorem is proved directly for unbounded
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
semi-Fredholm operators in Banach spaces. To obtain the optimal
we use the Borsuk-Ulam theorem. See [1, IV Lem. 2.3]. For
the method of continuity thm 5.19 is the main result of this
This section contains some complementary result for part I. First a
useful abstract regularity result from , which shows the
of the index not only when perturbing the operator, but also when
perturbing the space. Proposition 6.3 then give full information
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
of Fredholm operators. See [1, VII Thm 1.9].
We introduce the notion of singular integral operators on Rn. We do not make
standing regularity assumption of the kernel, as is usually done in
literature. Five motivating examples are introduced, including the
Cauchy integral on a Lipschitz curve, the Calderón
and the double layer potential on Lipschitz surfaces. To get
with SIOs we first briefly study classical convolution SIOs. The
result of this section is Thm 7.18, which is taken from [2, II 3.2].
We demonstrate in Ex. 8.5, taken from [10, section 7.2], that the
principal value for antisymmetric non-convolution SIOs need not
L2. Proposition 8.2 however shows that the principal
always exists in a slighly weaker sense. The weighted Sobolev test
functions Hts 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 L2-boundedness
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.
We first define dyadic cubes in Rn,
and modify the main branch so that the tree of cubes is connected.
is convenient to have connected, but is not really essential though.
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
and BMO. We prove the fundamental Carleson's theorem. See [13, II
[2, part 1] and . Next we introduce the notion of "perfect
SIOs", which will serve as a "discrete model" for SIOs. The
comes from . We show the necessity of the
We construct the Haar decomposition of L2(Rn). Some novelty is
we work not with individual Haar basis functions but instead with
dimensional subspaces (a vector-valued approach so to speak). We
how a pointwise multiplication operator
splits into three operators under Haar decomposition: a multiplier,
paraproduct and an adjoint paraproduct. The BMO/Carleson bound of
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].
We formulate the T(1) theorem för Hölder regular SIOs,
boundedness of the Calderón commutators from it, and then
the theorem. Inspiration comes from [11, Part II] and .
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 . 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
This final section contains some complementary results for Part II.
We prove Lp-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 L2
invertibility using Rellich estimates. See . Finally we compute
exact spectrum on plane cones using the Fourier/Mellin transform.
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