A surprising formula for Sobolev norms

Significance The Sobolev spaces, introduced in the 1930s, have become ubiquitous in analysis and applied mathematics. They involve Lp norms of the gradient of a function u . We present an alternative point of view where derivatives are replaced by appropriate finite differences and the Lebesgue space Lp is replaced by the slightly larger Marcinkiewicz space Mp (aka weak Lp space)—a popular tool in harmonic analysis. Surprisingly, these spaces coincide with the standard Sobolev spaces, a fact which sheds additional light onto these classical objects and should have numerous applications. In particular, it rectifies some well-known irregularities occurring in the theory of fractional Sobolev spaces. The proof relies on original calculus inequalities which might be useful in other situations.