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索伯列夫在許多數學領域中都有重大的基礎性貢獻。[[索伯列夫空間]]s can be defined by some growth conditions on the [[Fourier transform]]s; they and their embedding theorems are an important subject in [[泛函分析]]. [[廣義函數論]] (後被稱為[[分布]]) 在1935年首次由索伯列夫引進，廣義函數的概念後來由[[洛朗·施瓦茨]]繼續深入研究. Sobolev abstracted the classical notion of [[derivative|differentiation]] so expanding the ranges of applications of the technique of Newton and Leibniz. The theory of [[Distribution (mathematics)|distribution]] is considered now as the calculus of the modern epoch.{{By whom|date=March 2010}}

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