A Characterization of the Lorentz space $L(p,r)$ in terms of Orlicz type classes

TL;DR

This paper characterizes Lorentz spaces $L(p,r)$ using Orlicz type classes, linking Stein's differentiability characterization with Calderon's earlier work on real functions.

Contribution

It provides a new description of Lorentz spaces in terms of Orlicz classes, connecting two different characterizations of differentiability.

Findings

Lorentz space $L(p,r)$ characterized via Orlicz classes

Stein's differentiability characterization equivalent to Calderon's

New insights into function space relationships

Abstract

We describe the Lorentz space $L(p, r), 0 < r < p, p > 1$, in terms of Orlicz type classes of functions L . As a consequence of this result it follows that Stein's characterization of the real functions on $R^n$ that are differentiable at almost all the points in $R^n$, is equivalent to the earlier characterization of those functions given by A. P. Calderon.