A surprising formula for Sobolev norms

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