Narrow Results

Let $p\in (\frac{n}{n+1},1]$ and $f\in h^p({ℝ}^n)$ be the local Hardy space in the sense of D. Goldberg. In this paper, the authors establish two bilinear decompositions of the product spaces of $h^p({ℝ}^n)$ and their dual spaces. More precisely, the authors prove that $h^1({ℝ}^n)\times \mathrm{\text{bmo}}({ℝ}^n)=L^1({ℝ}^n)+h_{\ast }^{\mathrm{\Phi }}({ℝ}^n)$ and, for any $p\in (\frac{n}{n+1},1)$, $h^p({ℝ}^n)\times {\mathrm{\Lambda }}_{\alpha }({ℝ}^n)=L^1({ℝ}^n)+h^p({ℝ}^n)$, where $\mathrm{\text{bmo}}({ℝ}^n)$ denotes the local BMO space, ${\mathrm{\Lambda }}_{\alpha }({ℝ}^n)$, for any $p\in (\frac{n}{n+1},1)$ and $\alpha :=n(\frac{1}{p}-1)$, the inhomogeneous Lipschitz space and $h_{\ast }^{\mathrm{\Phi }}({ℝ}^n)$ a variant of the local Orlicz–Hardy space related to the Orlicz function $\mathrm{\Phi }(t):=\frac{t}{\mathrm{\text{log}}(e+t)}$ for any $t\in [0,\infty )$ which was introduced by Bonami and Feuto. As an application, the authors establish a div-curl lemma at the endpoint case.