Sobolev Inequality

A family of embedding inequalities that bound norms of functions in Lebesgue spaces by norms of their weak derivatives in Sobolev spaces, asserting that sufficient derivative integrability implies improved integrability or continuity of the function itself.

An inequality that bounds a norm of a function in a Lebesgue or Hölder space by a Sobolev norm, typically of the form ||u||_{L^q} ≤ C||u||_{W^{k,p}} or variants relating gradient norms to higher integrability or continuity.