Narrow Results

Recently, both the bilinear decompositions $h^1({ℝ}^n)\times \mathrm{bmo}({ℝ}^n)\subset L^1({ℝ}^n)+h_{\ast }^{\mathrm{\Phi }}({ℝ}^n)$ and $h^1({ℝ}^n)\times \mathrm{bmo}({ℝ}^n)\subset L^1({ℝ}^n)+h^{log}({ℝ}^n)$ were established. In this paper, the authors prove in some sense that the former is sharp, while the latter is not. To this end, the authors first introduce the local Orlicz-slice Hardy space which contains $h_{\ast }^{\mathrm{\Phi }}({ℝ}^n)$, a variant of the local Orlicz Hardy space, introduced by Bonami and Feuto as a special case, and obtain its dual space by establishing its characterizations via atoms, finite atoms, and various maximal functions, which are new even for $h_{\ast }^{\mathrm{\Phi }}({ℝ}^n)$. The relationship $h_{\ast }^{\mathrm{\Phi }}({ℝ}^n)⊊h^{log}({ℝ}^n)$ is also clarified.

In this paper, the authors first introduce a class of Orlicz-slice spaces which generalize the slice spaces recently studied by Auscher et al. Based on these Orlicz-slice spaces, the authors then introduce a new kind of Hardy-type spaces, the Orlicz-slice Hardy spaces, via the radial maximal functions. This new scale of Orlicz-slice Hardy spaces contains the variant of the Orlicz–Hardy space of Bonami and Feuto as well as the Hardy-amalgam space of de Paul Ablé and Feuto as special cases. Their characterizations via the atom, the molecule, various maximal functions, the Poisson integral and the Littlewood–Paley functions are also obtained. As an application of these characterizations, the authors establish their finite atomic characterizations, which further induce a description of their dual spaces and a criterion on the boundedness of sublinear operators from these Orlicz-slice Hardy spaces into a quasi-Banach space. Then, applying this criterion, the authors obtain the boundedness of $\delta$-type Calderón–Zygmund operators on these Orlicz-slice Hardy spaces. All these results are new even for slice Hardy spaces and, moreover, for Hardy-amalgam spaces, the Littlewood–Paley function characterizations, the dual spaces and the boundedness of $\delta$-type Calderón–Zygmund operators on these Hardy-type spaces are also new.