A Fourier multiplier theorem on anisotropic Hardy spaces associated with ball quasi-Banach function spaces

Maximal functions, Hardy spaces and Fourier multiplier theorems on unbounded Vilenkin groups

In this paper, a class of anisotropic Herz-type Hardy spaces associated with a non-isotropic dilation on ℝn are introduced, and the central atomic and molecular decomposition characterizations of those spaces are established. As some applications of the decomposition theory, the authors study the interpolation problem and the boundedness of the central δ-Calderon-Zygmund operators on the anisotropic Herz-type Hardy spaces.

We characterize the relatively compact subsets of the order continuous part $$E_a$$ of a quasi-Banach function space E showing that the strong connection between compactness, uniform absolute continuity, uniform integrability, almost order boundedness and L-weak compactness that appears in the classical setting of Lebesgue spaces remains almost invariant in this new context under mild assumptions. We also present a de la Vallee–Poussin type theorem in this context that allows us to locate each compact subset of $$E_a$$ as a compact subset of a smaller quasi-Banach Orlicz space $$E^\varPhi .$$ Our results generalize the previous known results for the Banach function spaces $$L^1(m)$$ and $$L^1_w(m)$$ associated to a vector measure m and moreover they can also be applied to the quasi-Banach function space $$L^1\left( \Vert m \Vert \right) $$ associated to the semivariation of m.

Let $ \vec{p}\in(0, \infty)^n $ and $ A $ be a general expansive matrix on $ \mathbb{R}^n $. In this article, via the non-tangential grand maximal function, the authors first introduce the anisotropic mixed-norm Hardy spaces $ H_A^{\vec{p}}(\mathbb{R}^n) $ associated with $ A $ and then establish their radial or non-tangential maximal function characterizations. Moreover, the authors characterize $ H_A^{\vec{p}}(\mathbb{R}^n) $, respectively, by means of atoms, finite atoms, Lusin area functions, Littlewood–Paley $ g $-functions or $ g_{\lambda}^\ast $-functions via first establishing an anisotropic Fefferman–Stein vector-valued inequality on the mixed-norm Lebesgue space $ L^{\vec{p}}(\mathbb{R}^n) $. In addition, the authors also obtain the duality between $ H_A^{\vec{p}}(\mathbb{R}^n) $ and the anisotropic mixed-norm Campanato spaces. As applications, the authors establish a criterion on the boundedness of sublinear operators from $ H_A^{\vec{p}}(\mathbb{R}^n) $ into a quasi-Banach space. Applying this criterion, the authors then obtain the boundedness of anisotropic convolutional $ \delta $-type and non-convolutional $ \beta $-order Calderón–Zygmund operators from $ H_A^{\vec{p}}(\mathbb{R}^n) $ to itself [or to $ L^{\vec{p}}(\mathbb{R}^n) $]. As a corollary, the boundedness of anisotropic convolutional $ \delta $-type Calderón–Zygmund operators on the mixed-norm Lebesgue space $ L^{\vec{p}}(\mathbb{R}^n) $ with $ \vec{p}\in(1, \infty)^n $ is also presented.

In this paper, the authors establish the anisotropic weak Hardy spaces associated with very general discrete groups of dilations. Moreover, the atomic decomposition theorem of the anisotropic weak Hardy spaces is also given. As some applications of the above results, the authors prove some interpolation theorems and obtain the boundedness of the singular integral operators on these Hardy spaces.

By using Fourier multiplier theorems we characterize the existence and uniqueness of periodic solutions for a class of second-order differential equations with infinite delay. We also establish maximal regularity results for the equations in various space

In this paper we propose a general coorbit space theory suitable to define coorbits of quasi-Banach spaces using an abstract continuous frame, indexed by a locally compact Hausdorff space, and an associated generalized voice transform. The proposed theory realizes a further step in the development of a universal abstract theory towards various function spaces and their atomic decompositions which has been initiated by Feichtinger and Grochenig in the late 1980s. We combine the recent approaches in Rauhut and Ullrich (J Funct Anal 260(11):3299–3362, 2011) and Rauhut (Stud Math 180(3):237–253, 2007) to identify, in particular, various inhomogeneous (quasi-Banach) spaces of Besov–Lizorkin–Triebel type. To prove the potential of our new theory we apply it to spaces with variable smoothness and integrability which have attracted significant interest in the last 10 years. From the abstract discretization machinery we obtain atomic decompositions as well as wavelet frame isomorphisms for these spaces.

In this paper we explore some basic properties of quasi-Banach function spaces which are important in applications. Namely, we show that they possess a generalised version of Riesz–Fischer property, that embeddings between them are always continuous, and that the dilation operator is bounded on them. We also provide a characterisation of separability for quasi-Banach function spaces over the Euclidean space. Furthermore, we extend the classical Riesz–Fischer theorem to the context of quasinormed spaces and, as a consequence, obtain an alternative proof of completeness of quasi-Banach function spaces.

We obtain a differential characterization for the anisotropic Hardy space by identifying it with a parabolic Hardy space associated with a general continuous group. This allows to be defined using a parabolic differential equation of Calderón and Torchinsky. We also provide a classification of dilations corresponding to equivalent anisotropic Hardy spaces with respect to linear transformations.

We extend the mapping properties for the fractional integral operators, the convolution operators, the Fourier integral operators and the oscillatory integral operators to rearrangementinvariant quasi-Banach function spaces.We also generalize the Fourier restriction theorem and the Sobolev embedding theorem to rearrangement-invariant quasi-Banach function spaces.We obtain the above results by introducing two families of rearrangement-invariant quasi-Banach function spaces.Furthermore, these two families of rearrangement-invariant quasi-Banach function spaces also give us some embedding and interpolation results of Triebel-Lizorkin type spaces and Hardy type spaces built on rearrangement-invariant quasi-Banach function spaces.

In this paper, we investigate the boundedness of composition operators defined on a quasi-Banach space continuously included in the space of smooth functions on a manifold. We prove that the boundedness of a composition operator strongly limits the behavior of the original map, and it provides an effective method to investigate the properties of composition operators using the theory of dynamical system. Consequently, we prove that only affine maps can induce bounded composition operators on any quasi-Banach space continuously included in the space of entire functions of one variable if the function space contains a nonconstant function. We also prove that any polynomial automorphisms except affine transforms cannot induce bounded composition operators on a quasi-Banach space composed of entire functions in the two-dimensional complex affine space under several mild conditions.

Let A be a symmetric expansive matrix and Hp (ℝ n) be the anisotropic Hardy space associated with A. For a function m in L∞ (ℝ n), an appropriately chosen function η in C ∞c (ℝ n) and j ∈ ℤ define mj(ξ) = m(Ajξ)η(ξ). The authors show that if 0 < p < 1 and \(\hat m_j \) belongs to the anisotropic non-homogeneous Herz space K 1/p−1,p 1 (ℝ n), then m is a Fourier multiplier from Hp (ℝ n) to Lp (ℝ n). For p = 1, a similar result is obtained if the space K 0,1 1 (ℝ n) is replaced by a slightly smaller space K(ω). Moreover, the authors show that if 0 < p ≤ 1 and if the sequence {(mj)∨} belongs to certain mixednorm space, depending on p, then m is also a Fourier multiplier from Hp (ℝ n) to Lp (ℝ n).

In this article, the authors establish the characterizations of a class of anisotropic Herz-type Hardy spaces with two variable exponents associated with a non-isotropic dilation on ℝ n {{\mathbb{R}}}^{n} in terms of molecular decompositions. Using the molecular decompositions, the authors obtain the boundedness of the central δ-Calderón-Zygmund operators on the anisotropic Herz-type Hardy space with two variable exponents.

Let X be a ball quasi-Banach function space satisfying some mild assumptions and H X (R n ) the Hardy space associated with X. In this article, the authors introduce both the Hardy space H X (R n+1 + ) of harmonic functions and the Hardy space H X (R n+1 + ) of harmonic vectors, associated with X, and then establish the isomorphisms among , where H X,2 (R n+1 + ) and H X,2 (R n+1 + ) are, respectively, certain subspaces of H X (R n+1 + ) and H X (R n+1 + ). Using these isomorphisms, the authors establish the first order Riesz transform characterization of H X (R n ). The higher order Riesz transform characterization of H X (R n ) is also obtained. The results obtained in this article have a wide range of generality and can be applied to the classical Hardy space, the weighted Hardy space, the Herz-Hardy space, the Lorentz-Hardy space, the variable Hardy space, the mixed-norm Hardy space, the local generalized Herz-Hardy space, and the mixed-norm Herz-Hardy space.

Sharp bilinear decomposition for products of both anisotropic Hardy spaces and their dual spaces with its applications to endpoint boundedness of commutators