The space $J{N}_p$: nontriviality and duality

Abstract

We study a function space $J{N}_p$ based on a condition introduced by John and Nirenberg as a variant of BMO. It is known that $L^p\subset J{N}_p⊊L^{p,\infty }$, but otherwise the structure of $J{N}_p$ is largely a mystery. Our first main result is the construction of a function that belongs to $J{N}_p$ but not $L^p$, showing that the two spaces are not the same. Nevertheless, we prove that for monotone functions, the classes $J{N}_p$ and $L^p$ do coincide. Our second main result describes $J{N}_p$ as the dual of a new Hardy kind of space $H{K}_{p^{\prime }}$.

corrections, 1 figure added, Section 4 (multidimensional counterexample) is new

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