quasi-Banach Spaces Research Articles

Noncommutative Calderón–Lozanovskiĭ–Hardy spaces

Let $${\mathcal {M}}$$ be a diffuse von Neumann algebra with a faithful normal semi-finite trace $$\tau $$ , and let E be a symmetric quasi-Banach space. Then for any Orlicz function $$\varphi $$ , we can define the noncommutative Calderon–Lozanovskiĭ spaces $$E_\varphi ({\mathcal {M}})$$ . These spaces share many properties with their classical counterparts. In particular, new multiplication operator spaces and complex interpolation spaces of such spaces are given under a wide range of conditions. Moreover, letting $${\mathcal {A}}$$ be a maximal subdiagonal algebra of $${\mathcal {M}}$$ , we introduce the noncommutative Calderon–Lozanovskiĭ–Hardy spaces $$H^\varphi ({\mathcal {A}})$$ and transfer the recent results of the noncommutative Hardy spaces to the noncommutative Calderon–Lozanovskiĭ–Hardy spaces.

Orlicz spaces associated to a quasi-Banach function space: applications to vector measures and interpolation

The Orlicz spaces $$X^{\varPhi }$$ associated to a quasi-Banach function space X are defined by replacing the role of the space $$L^1$$ by X in the classical construction of Orlicz spaces. Given a vector measure m, we can apply this construction to the spaces $$L^1_w(m),$$ $$L^1(m)$$ and $$L^1(\Vert m\Vert )$$ of integrable functions (in the weak, strong and Choquet sense, respectively) in order to obtain the known Orlicz spaces $$L^{\varPhi }_w(m)$$ and $$L^{\varPhi }(m)$$ and the new ones $$L^{\varPhi }(\Vert m\Vert ).$$ Therefore, we are providing a framework where dealing with different kind of Orlicz spaces in a unified way. Some applications to complex interpolation are also given.

Stability of Euler–Lagrange-Type Cubic Functional Equations in Quasi-Banach Spaces

In this paper, we study the generalized Hyers–Ulam stability of Euler–Lagrange-type cubic functional equation of the form $$\begin{aligned} 2mf(x + my) + 2f(mx - y) = (m^3 + m)[f(x+ y) + f(x - y)] + 2(m^4 - 1)f(y) \end{aligned}$$for all \(x,y \in X\), where m is a fixed scalar such that \(m \ne 0,1\), and f is a map from a quasi-normed space X to a quasi-Banach space Y over the same field with X by applying the alternative fixed point theorem.

Compactness in quasi-Banach function spaces with applications to $$L^1$$ of the semivariation of a vector measure

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.

Linear operators, Fourier integral operators and k-plane transforms on rearrangement-invariant quasi-Banach function spaces

We establish the mapping properties of linear operators on rearrangement-invariant quasi-Banach function spaces. Our result applies to those linear operators that map $$L^{p}$$ to $$L^{q}$$ with $$p\not =q$$ . Therefore, it can be used to study the mapping properties of the fractional integral operators, the Fourier integral operators and the k-plane transforms on rearrangement-invariant quasi-Banach function spaces.

Stability of the Fréchet Equation in Quasi-Banach Spaces

We investigate the Hyers–Ulam stability of the well-known Fréchet functional equation that comes from a characterization of inner product spaces. We also show its hyperstability on a restricted domain. We work in the framework of quasi-Banach spaces. In the proof, a fixed point theorem due to Dung and Hang, which is an extension of a fixed point theorem in Banach spaces, plays a main role.

Littlewood-Paley Characterizations of Hardy-Type Spaces Associated with Ball Quasi-Banach Function Spaces

Let X be a ball quasi-Banach function space on \({{\mathbb {R}}}^n\). In this article, assuming that the powered Hardy–Littlewood maximal operator satisfies some Fefferman–Stein vector-valued maximal inequality on X and is bounded on the associated space, the authors establish various Littlewood–Paley function characterizations of the Hardy space \(H_X({{\mathbb {R}}}^n)\) associated with X, under some weak assumptions on the Littlewood–Paley functions. To this end, the authors also establish a useful estimate on the change of angles in tent spaces associated with X. All these results have wide applications. Particularly, when \(X:=M_r^p({{\mathbb {R}}}^n)\) (the Morrey space), \(X:=L^{\vec {p}}({{\mathbb {R}}}^n)\) (the mixed-norm Lebesgue space), \(X:=L^{p(\cdot )}({{\mathbb {R}}}^n)\) (the variable Lebesgue space), \(X:=L_\omega ^p({{\mathbb {R}}}^n)\) (the weighted Lebesgue space) and \(X:=(E_\Phi ^r)_t({{\mathbb {R}}}^n)\) (the Orlicz-slice space), the Littlewood–Paley function characterizations of \(H_X({{\mathbb {R}}}^n)\) obtained in this article improve the existing results via weakening the assumptions on the Littlewood–Paley functions and widening the range of \(\lambda \) in the Littlewood–Paley \(g_\lambda ^*\)-function characterization of \(H_X({\mathbb {R}}^n)\).

On relatively compact sets in quasi-Banach function spaces

This paper is devoted to the study of the relatively compact sets in Quasi-Banach function spaces, providing an important improvement of the known results. As an application, we take the final step in establishing a relative compactness criteria for function spaces with any weight without any assumption.

On the Limit Regularity in Sobolev and Besov Scales Related to Approximation Theory

We study the interrelation between the limit $$L_p(\Omega )$$-Sobolev regularity $$\overline{s}_p$$ of (classes of) functions on bounded Lipschitz domains $$\Omega \subseteq \mathbb {R}^d$$, $$d\ge 2$$, and the limit regularity $$\overline{\alpha }_p$$ within the corresponding adaptivity scale of Besov spaces $$B^\alpha _{\tau ,\tau }(\Omega )$$, where $$1/\tau =\alpha /d+1/p$$ and $$\alpha >0$$ ($$p>1$$ fixed). The former determines the convergence rate of uniform numerical methods, whereas the latter corresponds to the convergence rate of best N-term approximation. We show how additional information on the Besov or Triebel–Lizorkin regularity may be used to deduce upper bounds for $$\overline{\alpha }_p$$ in terms of $$\overline{s}_p$$ simply by means of classical embeddings and the extension of complex interpolation to suitable classes of quasi-Banach spaces due to Kalton et al. (in: De Carli and Milman (ed) Interpolation theory and applications, American Mathematical Society, Providence, 2007). The results are applied to the Poisson equation, to the p-Poisson problem, and to the inhomogeneous stationary Stokes problem. In particular, we show that already established results on the Besov regularity for the Poisson equation are sharp.

Applications of Hardy Spaces Associated with Ball Quasi-Banach Function Spaces

Let $X$ be a ball quasi-Banach function space satisfying some minor assumptions. In this article, the authors establish the characterizations of $H_X(\mathbb{R}^n)$, the Hardy space associated with $X$, via the Littlewood--Paley $g$-functions and $g_\lambda^\ast$-functions. Moreover, the authors obtain the boundedness of Calder\'on--Zygmund operators on $H_X(\mathbb{R}^n)$. For the local Hardy-type space $h_X(\mathbb{R}^n)$ associated with $X$, the authors also obtain the boundedness of $S^0_{1,0}(\mathbb{R}^n)$ pseudo-differential operators on $h_X(\mathbb{R}^n)$ via first establishing the atomic characterization of $h_X(\mathbb{R}^n)$. Furthermore, the characterizations of $h_X(\mathbb{R}^n)$ by means of local molecules and local Littlewood--Paley functions are also given. 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 Morrey--Hardy space, the variable Hardy space, the Orlicz-slice Hardy space and their local versions. Some special cases of these applications are even new and, particularly, in the case of the variable Hardy space, the $g_\lambda^\ast$-function characterization obtained in this article improves the known results via widening the range of $\lambda$.

Hyers-Ulam stability of a finite variable mixed type quadratic-additive functional equation in quasi-Banach spaces

In this paper, we introduce a mixed type finite variable functional equation deriving from quadratic and additive functions and obtain the general solution of the functional equation and investigate the Hyers-Ulam stability for the functional equation in quasi-Banach spaces.

Rosenthal’s inequalities: ${\Delta }$-norms and quasi-Banach symmetric sequence spaces

Let $X$ be a symmetric quasi-Banach function space with the Fatou property and let $E$ be an arbitrary symmetric quasi-Banach sequence space. Suppose that $(f_k)_{k\geq 0}\subset X$ is a sequence of independent random variables. We present a necessary and

Real-variable characterizations of new anisotropic mixed-norm Hardy spaces

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 Calder\'{o}n-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 Calder\'on-Zygmund operators on the mixed-norm Lebesgue space $L^{\vec{p}}(\mathbb{R}^n)$ with $\vec{p}\in(1,\infty)^n$ is also presented.

LOGARITHMIC INTERPOLATION METHODS AND MEASURE OF NON-COMPACTNESS

Abstract We derive interpolation formulae for the measure of non-compactness of operators interpolated by logarithmic methods with $\theta = 0,1$ between quasi-Banach spaces. Applications are given to operators between Lorentz–Zygmund spaces.

A unified method for maximal truncated Calderón–Zygmund operators in general function spaces by sparse domination

AbstractIn this note we give simple proofs of several results involving maximal truncated Calderón–Zygmund operators in the general setting of rearrangement-invariant quasi-Banach function spaces by sparse domination. Our techniques allow us to track the dependence of the constants in weighted norm inequalities; additionally, our results hold in ℝnas well as in many spaces of homogeneous type.

The Definition of a Self-Similar function in Quasi-Banach Spaces

The definition of a self-similar function is extended to quasi-Banach Lebesgue spaces. A sufficient condition for a function with given self-similarity parameters to lie in some such space is given.

Weak Hardy-Type Spaces Associated with Ball Quasi-Banach Function Spaces II: Littlewood–Paley Characterizations and Real Interpolation

Let X be a ball quasi-Banach function space on $${\mathbb R}^n$$ . In this article, assuming that the powered Hardy–Littlewood maximal operator satisfies some Fefferman–Stein vector-valued maximal inequality on X as well as it is bounded on both the weak ball quasi-Banach function space WX and the associated space, the authors establish various Littlewood–Paley function characterizations of $$WH_X({{\mathbb {R}}}^n)$$ under some weak assumptions on the Littlewood–Paley functions. The authors also prove that the real interpolation intermediate space $$(H_{X}({{\mathbb {R}}}^n),L^\infty ({{\mathbb {R}}}^n))_{\theta ,\infty }$$ , between the Hardy space associated with X, $$H_{X}({{\mathbb {R}}}^n)$$ , and the Lebesgue space $$L^\infty ({\mathbb R}^n)$$ , is $$WH_{X^{{1}/{(1-\theta )}}}({{\mathbb {R}}}^n)$$ , where $$\theta \in (0, 1)$$ . All these results are of wide applications. Particularly, when $$X:=M_q^p({{\mathbb {R}}}^n)$$ (the Morrey space), $$X:=L^{\vec {p}}({{\mathbb {R}}}^n)$$ (the mixed-norm Lebesgue space) and $$X:=(E_\Phi ^q)_t({{\mathbb {R}}}^n)$$ (the Orlicz-slice space), all these results are even new; when $$X:=L_\omega ^\Phi ({\mathbb R}^n)$$ (the weighted Orlicz space), the result on the real interpolation is new and, when $$X:=L^{p(\cdot )}({{\mathbb {R}}}^n)$$ (the variable Lebesgue space) and $$X:=L_\omega ^\Phi ({{\mathbb {R}}}^n)$$ , the Littlewood–Paley function characterizations of $$WH_X({{\mathbb {R}}}^n)$$ obtained in this article improves the existing results via weakening the assumptions on the Littlewood–Paley functions.

The optimal symmetric quasi-Banach range of the discrete Hilbert transform

We identify symmetric quasi-Banach range of the discrete Calder\'{o}n operator and Hilbert transform acting on a symmetric quasi-Banach sequence space. As an application we present an example of optimal range in the case when the domain of those operators is the weak-$\ell_{1}$ space of sequences.

On the bounded approximation property on subspaces of ℓ p when 0 < p < 1 and related issues

Abstract This paper studies the bounded approximation property (BAP) in quasi-Banach spaces. In the first part of the paper, we show that the kernel of any surjective operator ℓ p → X {\ell_{p}\to X} has the BAP when X has it and 0 < p ≤ 1 {0<p\leq 1} , which is an analogue of the corresponding result of Lusky for Banach spaces. We then obtain and study nonlocally convex versions of the Kadec–Pełczyński–Wojtaszczyk complementably universal spaces for Banach spaces with the BAP.

On new hyperstability results for the generalized $p$-radical functional equation in quasi-Banach spaces with the illustrative example

On new hyperstability results for the generalized $p$-radical functional equation in quasi-Banach spaces with the illustrative example