Sobolev Spaces

This book has given substantial attention to the C k spaces. Such spaces are suitable classes from which to select the defining function for a domain, and the C k spaces are natural in a number of other geometric contexts. However, in the study of partial differential equations and Fourier analysis, the Sobolev spaces are more convenient. The definition of the Sobolev spaces is less near the surface than that of the C k spaces, but theorems about Sobolev spaces are more accessible. In the end, the Sobolev imbedding theorem allows one to pass back and forth between the C k spaces and the Sobolev spaces (however one must pay with a certain lack of precision that is present in the imbedding theorem).