The Space of Tempered Distributions
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.