The Schwartz class is a vector space of rapidly decaying smooth functions defined in all R d https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429507069/dafc78e3-1d52-4d0c-a74c-44d704eba110/content/inline-math13_1.jpg"/> . As we will see in the coming sections, the Schwartz class is a complete metric space (it fits in the general construction of Fréchet spaces), which contains D ( R d ) https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429507069/dafc78e3-1d52-4d0c-a74c-44d704eba110/content/inline-math13_2.jpg"/> as a dense subset. Because of this density, the dual space of the Schwartz class will be identifiable to a subset of the space of distributions: they will be called tempered distributions, because, in a way, they are allowed to ‘grow’ at infinity in a moderate way that can be tackled by the quick decay of the test functions. Perhaps the main use of the Schwartz class (and, by extension, of its dual space) is due to the fact that in it the Fourier transform is an isomorphism. We will proceed with the introduction of this material (which brings together harmonic analysis and PDE theory) very slowly, following the program:
Study the Schwartz class and the Fourier transform in it, absent of any topological (functional analysis) structure.
Extend the Fourier transform to an isometric isomorphism in L 2 ( R d ) https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429507069/dafc78e3-1d52-4d0c-a74c-44d704eba110/content/inline-math13_3.jpg"/> and recognize functions in H 1 ( R d ) https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429507069/dafc78e3-1d52-4d0c-a74c-44d704eba110/content/inline-math13_4.jpg"/> and H 2 ( R d ) https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429507069/dafc78e3-1d52-4d0c-a74c-44d704eba110/content/inline-math13_5.jpg"/> through their Fourier transforms. At this point we will spend a little time in examining the concept of H 2 regularity of the Laplacian.
Study the metrizable topology of the Schwartz class (as an example of Fréchet spaces, which we will introduce as well) and of its dual space, whose elements are called tempered distributions.
296Define a class of Sobolev spaces tagged in a real parameter, and show that for positive integer values of the parameter we recover the spaces H m ( R d ) https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429507069/dafc78e3-1d52-4d0c-a74c-44d704eba110/content/inline-math13_6.jpg"/> .
Digress on two interesting topics which help explain why we called the trace space H 1 / 2 ( Γ ) https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429507069/dafc78e3-1d52-4d0c-a74c-44d704eba110/content/inline-math13_7.jpg"/> and how this space exists ‘independently’ of the trace operator.
Finally, look into additional interior (away from the boundary) regularity of solutions to the Laplace equation.
Warning. In this chapter all functions will be complex-valued and we will not warn about it anymore. In particular, the L p and H m spaces that we consider now are those whose elements are complex-valued functions. We will give a different name to the newly defined Sobolev spaces, but then prove that they are the same.