The space JNp: Nontriviality and duality

We study a function space JN(p) based on a condition introduced by John and Nirenberg as a variant of BMO. It is known that L-p subset of JNp not subset of L-p,L-infinity, but otherwise the structure of JN(p) is largely a mystery. Our first main result is the construction of a function that belongs to JN(p) but not L-P, showing that the two spaces are not the same. Nevertheless, we prove that for monotone functions, the classes JN(p) and L-P do coincide. Our second main result describes JN(p) as the dual of a new Hardy kind of space HKp'. (C) 2018 Elsevier Inc. All rights reserved.