Sobolev Embedding Theorem

The Sobolev Embedding Theorem gives conditions under which a function with a fixed number of weak derivatives, themselves integrable to some power, must also belong to a better-behaved function space, such as a space of continuous or Holder-continuous functions, depending on the relation between the number of derivatives, the integrability power, and the dimension of the domain. Named for Sergei Sobolev, it is a foundational result of the modern theory of partial differential equations, converting rough integrability information into concrete regularity.