# 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 L2-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 . 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 .

### 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  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 , 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 Rn. 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 L2. Proposition 8.2 however shows that the principal value 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 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 Rn, 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 . Next we introduce the notion of "perfect dyadic SIOs", which will serve as a "discrete model" for SIOs. The inspiration comes from . We show the necessity of the T(1) condition.

### Section 10

We construct the Haar decomposition of L2(Rn). Some novelty is that we work not with individual Haar basis functions but instead with the 2n-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 .

### 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 . 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 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 its exact spectrum on plane cones using the Fourier/Mellin transform. See .

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