Distributions, Partial Differential Equations, and Harmonic Analysis

In this chapter the space of distributions is introduced and studied from the perspective of a topological vector space with various other additional features, such as the concept of support, multiplication with a smooth function, distributional derivatives, tensor product, and a partially defined convolution product. Here the nature of distributions with higher order gradients continuous or bounded is also discussed.

The action of the Fourier transform is extended to the setting of tempered distributions, and several distinguished subclasses of tempered distributions are introduced and studied, including homogeneous and principal value distributions. Significant applications to harmonic analysis and partial differential equations are singled out. For example, a general, higher dimensional jump-formula is deduced in this chapter for a certain class of tempered distributions, which includes the classical harmonic Poisson kernel that is later used as the main tool in deriving information about the boundary behavior of layer potential operators associated with various partial differential operators and systems. Also, one witnesses here how singular integral operators of central importance to harmonic analysis, such as the Riesz transforms, naturally arise as an extension to the space of square-integrable functions, of the convolution product of tempered distributions of principal value type with Schwartz functions.

While Lebesgue spaces play a most basic role in analysis, it is highly desirable to consider a scale of spaces which contains provisions for quantifying smoothness (measured in a suitable sense). This is the key feature of the so-called Sobolev spaces, introduced and studied at some length in this chapter in a completely self-contained manner. The starting point is the treatment of global \(L^2\)-based Sobolev spaces of arbitrary smoothness in the entire Euclidean space, using the Fourier transform as the main tool. We then proceed to define Sobolev spaces in arbitrary open sets, both via restriction (which permits the consideration of arbitrary amounts of smoothness) and in an intrinsic fashion (for integer amounts of smoothness, demanding that distributional derivatives up to a certain order are square-integrable in the respective open set). When the underlying set is a bounded Lipschitz domain, both these brands of Sobolev spaces (defined intrinsically and via restriction) coincide for an integer amount of smoothness. A key role in the proof of this result is played by Calderón’s extension operator, mapping functions originally defined in the said Lipschitz domain to the entire Euclidean ambient with preservation of Sobolev class. Finally, the fractional Sobolev space of order 1 / 2 is defined on the boundary of a Lipschitz domain as the space of square-integrable functions satisfying a finiteness condition involving a suitable Gagliardo–Slobodeckij semi-norm. This is then linked to Sobolev spaces in Lipschitz domains via trace and extension results.

In this chapter, solutions to various problems listed at the end of each of the Chapters 1–12 are provided.