Real-Variable Theory for New Herz-Type Hardy Spaces Associated with Ball Quasi-Banach Function Spaces

Let $X$ be a ball quasi-Banach function space, $\alpha\in \mathbb{R}$ and $q\in(0,\infty)$. In this paper, the authors first introduce the new Herz-type Hardy spaces $\mathcal{H\dot{K}}_{X}^{\alpha,\,q}({\mathbb {R}}^n)$ and $\mathcal{HK}_{X}^{\alpha,\,q}({\mathbb {R}}^n)$ associated with ball quasi-Banach function space $X$, via the non-tangential grand maximal function. Then, under some mild assumptions on $X$, the authors establish the real-variable theory for $\mathcal{H\dot{K}}_{X}^{\alpha,\,q}({\mathbb {R}}^n)$ and $\mathcal{HK}_{X}^{\alpha,\,q}({\mathbb {R}}^n)$, in terms of maximal function characterizations, atomic and molecular decompositions, and obtain the boundedness of some sublinear operators from $\mathcal{H\dot{K}}_{X}^{\alpha,\,q}({\mathbb {R}}^n)$ to $\mathcal{\dot{K}}_{X}^{\alpha,\,q}({\mathbb {R}}^n)$ and from $\mathcal{HK}_{X}^{\alpha,\,q}({\mathbb {R}}^n)$ to $\mathcal{K}_{X}^{\alpha,\,q}({\mathbb {R}}^n)$. As appliccations, we give two concrete function spaces which are members of Herz-type Hardy spaces associated with ball quasi-Banach function spaces.Comment: 37 pages, submitte

On Homogeneous Decomposition Spaces and Associated Decompositions of Distribution Spaces

A new construction of decomposition smoothness spaces of homogeneous type is considered. The smoothness spaces are based on structured and flexible decompositions of the frequency space $\mathbb{R}^d\backslash\{0\}$. We construct simple adapted tight frames for $L_2(\mathbb{R}^d)$ that can be used to fully characterise the smoothness norm in terms of a sparseness condition imposed on the frame coefficients. Moreover, it is proved that the frames provide a universal decomposition of tempered distributions with convergence in the tempered distributions modulo polynomials. As an application of the general theory, the notion of homogeneous $\alpha$-modulation spaces is introduced.Comment: 27 page