Function Spaces and Potential Theory

We begin with a review of some of the basic results that will be needed in subsequent chapters. Not all of these results are prerequisites for understanding the book, and the reader is advised to refer to them as the need arises. They are generally stated without proof, although we attempt to provide good references where the proofs can be found. Also, we take this opportunity to introduce some of our notational conventions.

As a motivation for what follows we here give a rather heuristic definition of classical Newton capacity. We begin with a positive charge distribution (a Radon measure) μ on a body K, by which we simply mean a compact subset of R³. The total charge on K is μ(K). A unit test charge placed at a point x in R³ \ K then, according to the laws of physics, experiences a force on it which is equal to −∇U ^μ (x), where (with properly chosen units) the Newton potential of μ at x. The potential U ^μ (x)is interpreted as the amount of work necessary to move a unit charge from ∞ to x, and the total energy stored in the system is given by one half of the energy integral:

Here we interrupt the development of the general theory in order to gain a deeper understanding of some of the aspects of the spaces L ^α,p . In Section 3.1 we give some simple pointwise estimates of potentials in terms of maximal functions. These are going to be used in several of the following chapters. We apply them here to obtain elementary proofs of certain integral inequalities, among which are the Sobolev inequalities of Theorem 1.2.4. In Section 3.2 we pursue a more special subject; we give a sharp exponential integral estimate in the “borderline case” αp = N. Sections 3.3 and 3.5 are devoted to the question under which circumstances a function T “operates” on functions f in L ^α,p in the sense that the composite function T o f also belongs to L ^α,p . This is in part motivated by the desire to prove the equivalence of capacities formulated in Section 2.7. Another consequence is a one-sided approximation theorem, given in Section 3.4, which has turned out to be useful in the theory of nonlinear partial differential equations. Finally, in Section 3.6 we prove an important inequality of B. Muckenhoupt and R. L. Wheeden, comparing Riesz and Bessel potentials with the fractional maximal function M _α f, which will have a role to play later.

In this chapter we shall apply the general theory developed in Chapter 2 to the function spaces known as Besov, and Lizorkin-Triebel spaces. In Section 4.1 we define the Besov spaces B ^p,q _α , and present their theory in a way that suits our purposes. In Section 4.2 the Lizorkin—Triebel spaces F ^p,q _α are defined, and then most of the section is devoted to a proof of the fact that this scale of spaces contains the spaces L ^{α p} , in fact F ^p,2 _α =L ^α,p for 1 < p < ∞. This result will be proved by means of a multiplier theorem of S. G. Mikhlin, whose proof is also included. In Section 4.3 we continue the presentation of these spaces in a way parallel to Section 4.1. This involves proving a rather deep theorem of J. Peetre. Now the stage is set for our application of the general nonlinear potential theory, which takes its beginning in Section 4.4. The short Section 4.5 is devoted to an important inequality of Th. H. Wolff. In Section 4.6, which is independent of most of the preceding theory in this chapter, we give a representation of the Besov, and Lizorkin—Triebel spaces by means of “smooth atoms”. In Section 4.7 we apply this representation to formulate an “atomic” nonlinear potential theory, which among other things gives a new way of viewing the Wolff inequality. Finally, in Section 4.8 we use the atomic representation to give a characterization of L ^{α, p} by means of a local approximation property. This result implies Strichartz’ theorem, Theorem 3.5.6, whose proof was previously postponed.

Many problems have definitive solutions in terms of capacities, but the latter have the drawback that their geometrical meaning is not transparent. For this reason we devote most of this chapter to comparing the (α, p)-capacities C _{α, p} for 1 < p < ∞ and 0 < αp ≤ N to the more geometric quantities known as Hausdorff measures. As we now know, C ^{α, p} is associated not only to the Sobolev spaces B _α ^{p, p} and Bessel potential spaces L ^{α, p} , but also to the Besov spaces and the Lizorkin—Triebel spaces F _α ^{p, q} , 1 < q < ∞.

A statement such as “f belongs to an L ^p space” can be understood in different ways. The strict interpretation is that f is an equivalence class of functions, the equivalence relation being equality almost everywhere. But one can also think of some representative of this equivalence class, perhaps defined at all points outside a set of measure zero. In particular, if a continuous function is identified with an element in L ^p , the identification means that one representative of the corresponding equivalence class is singled out, and this distinguished element is usually thought of as belonging to L ^p .

In Chapter 6 we investigated the continuity properties of distinguished representatives of the equivalence classes constituting elements of function spaces. In the present chapter we shall study the integrability properties of these representatives.

The term “Poincaré type inequality” is used, somewhat loosely, to describe a class of inequalities that generalize the classical Poincaré inequality, valid for f ∈ W ₀ ^1,p (Ω) in a bounded open Ω ⊂ R ^N . What the inequalities have in common is that an integral norm of a function is estimated in terms of integrals of its derivatives, and some information about the vanishing or the average of the function. Some such knowledge is clearly necessary, since estimates of this kind are false for non-zero constants.

This chapter, and part of the next, are devoted to a result which can be viewed either as an approximation theorem, or as a theorem characterizing the kernel of a trace operator for arbitrary sets. In the present chapter we treat the case of Sobolev spaces W ^m,p (R ^N ) for integer m and 1 < p < ∞. The main result, Theorem 9.1.3, and a number of corollaries are stated and discussed at some length in Section 9.1. The proof, which uses much of the nonlinear potential theory developed previously in the book, occupies the rest of the chapter. The contents are outlined at the end of Section 9.1.

In this chapter we apply the powerful “smooth atomic” method of representing elements in function spaces that was exposed in Chapter 4. In Section 10.1 we give the generalization of Theorem 9.1.3 that was announced in the introduction to Chapter 9. In Section 10.2 similar methods are used to extend H. Whitney’s classic characterization of closed ideals of differentiable functions.

We conclude the book by applying some of the results of earlier chapters to certain approximation problems in L ^P -norm for analytic and harmonic functions. By duality these problems can be reformulated as “stability problems” in Sobolev spaces, which can be given complete solutions in terms of capacities. The main results depend on Theorem 9.1.3 (or Theorem 10.1.1), but mostly only on the easy case when the space is W^{1, P}. The more complicated case of spaces involving higher derivatives is used only in the last section, Section 11.5, but even this section can very well be read before studying the proof of the theorem.