Lorentz space

Summary

The Lorentz spaces $L^{p,q}(\mu)$ form a two-parameter family of function spaces that refines the scale of Lp spaces. They are defined with the decreasing rearrangement of a function, so they only see the sizes of the level sets of the function. The first index $p$ gives the main size and the second index $q$ gives a fine adjustment. They contain $L^p = L^{p,p}$ and the weak Lp space $L^{p,\infty}$, and they appear in interpolation theory and in sharp forms of Sobolev inequalities.