Trading generalized derivatives for classical ones

Generalized functions have generalized derivatives. This is how we make sense of things like delta “functions” that are not functions, or functions that are not differentiable satisfying a differential equation. More on that here.

A major theme in the modern approach to partial differential equations is to first look for solutions in a space of generalized functions, then (hopefully) prove that the generalized function that solves your equation is a plain old function with derivatives in the classical sense.

You can differentiate generalized functions as many times as you’d like; the derivatives always exist, but these derivatives may not correspond to ordinary functions, i.e. may not be regular distributions [1]. And if the (generalized) derivatives are regular distributions, the functions they correspond to may not have as many (classical) derivatives as you’d like.

Rather than generalized derivatives existing, we’re primarily interested in generalized derivatives being regular and having finite Lebesgue norms. If generalized functions are “nice” in this sense, then we can “trade” them for classical functions, and the Sobolev embedding theorem sets the “exchange rate” for the trade.

There are a lot of variations on the Sobolev embedding theorem, but one version of the theorem, but one version says that if a function f on ℝn has generalized derivatives up to order k that all correspond to Lp functions, then f is equal (almost everywhere) to a function that has r derivatives, with each of the derivatives being Hölder continuous with exponent α if

n < p


r + α < k – n/p.

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[1] A distribution is a continuous linear functional on a space of test functions. If the effect of a distribution on a test function φ is equal to the integral of the product of a function f and φ, then the distribution is called regular, and we leave the distinction between the distribution and the function f implicit.

The Dirac delta function δ acts on a test function φ by returning φ(0). This is not a regular distribution because there is no function you can integrate against φ and always get φ(0) out.

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