Lorentz space

In mathematical analysis, Lorentz spaces, introduced by George G. Lorentz in the 1950s, are generalisations of the more familiar ${\displaystyle L^{p}}$ spaces.

The Lorentz spaces are denoted by ${\displaystyle L^{p,q}}$. Like the ${\displaystyle L^{p}}$ spaces, they are characterized by a norm (technically a quasinorm) that encodes information about the "size" of a function, just as the ${\displaystyle L^{p}}$ norm does. The two basic qualitative notions of "size" of a function are: how tall is the graph of the function, and how spread out is it. The Lorentz norms provide tighter control over both qualities than the ${\displaystyle L^{p}}$ norms, by exponentially rescaling the measure in both the range (${\displaystyle p}$) and the domain (${\displaystyle q}$). The Lorentz norms, like the ${\displaystyle L^{p}}$ norms, are invariant under arbitrary rearrangements of the values of a function.