Sobolev spaces were defined by the Russian mathematician Sergei Lvovich Sobolev (1908-1989) in the 1930s. Denoted by Wm;p (Ω) ; Sobolev spaces are the space of functions whose all m-th order generalized derivatives are in Lp (Ω) space and partial derivatives of these spaces satisfy certain integrability conditions. Notice that generalized derivative refers to the weak derivative which is defined in the previous chapter. In this part we present fundamental properties of Sobolev spaces with several examples. For further reading on Sobolev spaces we cite [29–32].
Keywords: Sobolev space, Banach space, dual space, Sobolev space of real order, Sobolev space of negative order, weighted Lebesgue space, Schwartz space, Plancherel theorem, embedding, weight function.