Si deve a lui l'introduzione della nozione di [[spazio di Sobolev]], che viene introdotta mediante una richiesta sull'andamento delle [[trasformata di Fourier|trasformate di Fourier]]; gli spazi di Sobolev ed i relativi [[disuguaglianza di Sobolev|teoremi di immersione]] sono di fondamentale importanza per l'[[analisi funzionale]].
Le [[funzione generalizzata|funzioni generalizzate]] (in seguito più comunemente chiamate distribuzioni) furono introdotte per la prima volta da Sobolev nel 1935, e successivamente maggiormente sviluppate da [[Laurent Schwartz]].
[[Generalized function]]s (later known as [[distribution (mathematics)|distribution]]s), were first introduced by Sobolev in 1935 for [[weak solution]]s, and further developed by [[Laurent Schwartz]]. Sobolev abstracted the classical notion of [[derivative|differentiation]] so expanding the ranges of applications of the technique of Newton and Leibniz. The theory of distributions is considered now as the calculus of the modern epoch.
Sobolev graduated from [[Saint Petersburg State University|Leningrad University]] in 1929, where he was a student of [[Vladimir Ivanovich Smirnov (mathematician)|Vladimir Smirnov]]. He worked in Leningrad from 1932, and at the [[Steklov Institute of Mathematics]] in Moscow from 1934. He headed the institute in
evacuation to [[Kazan]] during the [[World War II]]. He was a [[Moscow State University]] professor from 1935 to 1957 and also a deputy director of the [[Kurchatov Institute|Institute for Atomic Energy]] 1943-57 where he participated in the [[A-bomb]]
project of the USSR.
In 1956 Sobolev joined a number of prominent scientists in proposing a large-scale scientific and educational initiative for the Eastern parts of the [[Soviet Union]], which resulted in the creation of the Siberian Division of the [[Russian Academy of Sciences|Academy of Sciences]]. He was the founder and first director of the Institute of Mathematics at [[Akademgorodok]] near [[Novosibirsk]], which was later to bear his name, and played an important role in the establishment and development of [[Novosibirsk State University]].
