Vector-valued Inequalities Research Articles

Matrix-weighted Besov–Triebel–Lizorkin spaces with logarithmic smoothness

In this article, the authors study the matrix-weighted Besov–Triebel–Lizorkin spaces with logarithmic smoothness. Via first obtaining the Lp(Rn)-boundedness and the Fefferman–Stein type vector-valued inequality of matrix-weighted Peetre-type maximal functions with the exquisite ranges of indices in terms of the Ap dimension of matrix weights under consideration, the authors establish an equivalent characterization of these spaces in terms of the matrix-weighted Peetre-type maximal functions, which further implies that these spaces are well defined. As an application, the authors obtain the boundedness of some pointwise multipliers on these spaces and, even back to classical Besov–Triebel–Lizorkin spaces, some of them are also new.

Vector-Valued Inequality of Fractional Integral Operator with Rough Kernel on Morrey-Adams Spaces

In 2019, Salim et al proved the vector-valued inequality for maximal operator with rough kernel on Lebesgue spaces and Morrey spaces. This results extend Fefferman-Stein inequality (1971). In 1970’s, Adams introduced another variant of Morrey spaces, which called as Morrey-Adams spaces. In this article, we prove vector-valued inequality for maximal operator and fractional integral operator with rough kernel on Morrey–Adams spaces.

Boundedness of the Vector-Valued Intrinsic Square Functions on Variable Exponents Herz Spaces

In this article, the authors study the boundedness of the vector-valued inequality for the intrinsic square function and the boundedness of the scalar-valued intrinsic square function on variable exponents Herz spaces K˙ρ(·)α,q(·)(Rn). In addition, the boundedness of commutators generated by the scalar-valued intrinsic square function and BMO class is also studied on K˙ρ(·)α,q(·)(Rn).

Extrapolation to product Herz spaces and some applications

Abstract This paper extends the extrapolation theory to product Herz spaces. To prove the main result, we first investigate the dual space of the product Herz space, and then show the boundedness of the strong maximal operator on product Herz spaces. By using this extrapolation theory, we establish the John–Nirenberg inequality, the characterization of little bmo, the Fefferman–Stein vector-valued inequality, the boundedness of the bi-parameter singular integral operator, the strong fractional maximal operator, and the bi-parameter fractional integral operator on product Herz spaces. We also give a new characterization of little bmo via the boundedness of the commutators of some bi-parameter operators on product Herz spaces. Even in the one-parameter setting, some of our results are new.

Boundedness of vector-valued sublinear operators on weighted Herz-Morrey spaces with variable exponents

Abstract If vector-valued sublinear operators satisfy the size condition and the vector-valued inequality on weighted Lebesgue spaces with variable exponent, then we obtain their boundedness on weighted Herz-Morrey spaces with variable exponents.

Martingale Musielak–Orlicz–Lorentz Hardy Spaces with Applications to Dyadic Fourier Analysis

Let $$(\Omega ,{\mathcal {F}},{\mathbb {P}})$$ be a probability space, $$\varphi :\ \Omega \times [0,\infty )\rightarrow [0,\infty )$$ a Musielak–Orlicz function, and $$q\in (0,\infty ]$$ . In this article, the authors introduce five martingale Musielak–Orlicz–Lorentz Hardy spaces and prove that these new spaces have some important features such as atomic characterizations, the boundedness of $$\sigma $$ -sublinear operators, and martingale inequalities. This new scale of martingale Hardy spaces requires the introduction of the Musielak–Orlicz–Lorentz space $$L^{\varphi ,q}(\Omega )$$ . In particular, the authors show that this Lorentz type space has some fundamental properties including the completeness, the convergence, real interpolations, and the Fefferman–Stein vector-valued inequality for the Doob maximal operator. As applications, the authors prove that the maximal Fejer operator is bounded from the martingale Musielak–Orlicz–Lorentz Hardy space $$H_{\varphi ,q}[0,1)$$ to $$L^{\varphi ,q}[0,1)$$ , which further implies some convergence results of the Fejer means. Moreover, all the above results are new even for Musielak–Orlicz functions with particular structure such as weight, weight Orlicz, and double-phase growth. The main approach used in this article can be viewed as a combination of the stopping time argument in probability theory and the real-variable technique of function spaces in harmonic analysis.

Local Muckenhoupt class for variable exponents

This work extends the theory of Rychkov, who developed the theory of A_{p}^{mathrm{loc}} weights. It also extends the work by Cruz-Uribe SFO, Fiorenza, and Neugebauer. The class A_{p(cdot )}^{mathrm{loc}} is defined. The weighted inequality for the local Hardy–Littlewood maximal operator on Lebesgue spaces with variable exponents is proven. Cruz-Uribe SFO, Fiorenza, and Neugebauer considered the Muckenhoupt class for Lebesgue spaces with variable exponents. However, due to the setting of variable exponents, a new method for extending weights is needed. The proposed extension method differs from that by Rychkov. A passage to the vector-valued inequality is realized by means of the extrapolation technique. This technique is an adaptation of the work by Cruz-Uribe and Wang. Additionally, a theory of extrapolation adapted to our class of weights is also obtained.

Strongly essential set of vector Ky Fan's points problem and its applications

In this paper, several existence results of strongly essential set of the solution set for Ky Fan's section problems and vector Ky Fan's point problems are obtained. Firstly, two kinds of strongly essential sets of Ky Fan's section problems are defined, and some further results on existence of the strongly essential component of solutions set of Ky Fan's section problems are proved, which generalize the conclusion in ^{[<span class="xref"><a href="#b22" ref-type="bibr">22</a></span>]}, and further generalize the conclusions in ^{[<span class="xref"><a href="#b21" ref-type="bibr">21</a></span>,<span class="xref"><a href="#b28" ref-type="bibr">28</a></span>]}. Secondly, based on the above results, two classes of stronger perturbations of vector-valued inequality functions are proposed respectively, and several existence results of the strongly essential component of set of vector Ky Fan's points are obtained. By comparing several metrics, we give some strong and weak relationships among the various metrics involved in the text. The main results of this paper actually generalize the relevant conclusions in the current literature. Finally, as an application, we obtain an existence result of the strongly essential component of weakly Pareto-Nash equilibrium for multiobjective games.

End-point estimates, extrapolation for multilinear Muckenhoupt classes, and applications

In this paper we present the results announced in the recent work by the first, second, and fourth authors of the current paper concerning Rubio de Francia extrapolation for the so-called multilinear Muckenhoupt classes. Here we consider the situations where some of the exponents of the Lebesgue spaces appearing in the hypotheses and/or in the conclusion can be possibly infinity. The scheme we follow is similar, but, in doing so, we need to develop a one-variable end-point off-diagonal extrapolation result. This complements the corresponding “finite” case obtained by Duoandikoetxea, which was one of the main tools in the aforementioned paper. The second goal of this paper is to present some applications. For example, we obtain the full range of mixed-norm estimates for tensor products of bilinear Calderón-Zygmund operators with a proof based on extrapolation and on some estimates with weights in some mixed-norm classes. The same occurs with the multilinear Calderón-Zygmund operators, the bilinear Hilbert transform, and the corresponding commutators with BMO functions. Extrapolation along with the already established weighted norm inequalities easily give scalar and vector-valued inequalities with multilinear weights and these include the end-point cases.

Extrapolation for multilinear Muckenhoupt classes and applications

In this paper we solve a long standing problem about the multivariable Rubio de Francia extrapolation theorem for the multilinear Muckenhoupt classes Ap→, which were extensively studied by Lerner et al. and which are the natural ones for the class of multilinear Calderón-Zygmund operators as well as for some bilinear rough singular integral operators. Furthermore, we go beyond the classes Ap→ and extrapolate within the classes Ap→,r→ which appear naturally associated to the weighted norm inequalities for multilinear sparse forms which control fundamental operators such as the bilinear Hilbert transform. We give several applications which can be easily obtained using extrapolation. First, weighted norm inequalities (scalar and vector-valued) for the bilinear rough singular integral operators are established. Second, for the bilinear Hilbert transform one can extrapolate from the recent result of Culiuc et al. who considered the Banach range, extend the estimates to the quasi-Banach range, and furthermore, prove trivially vector-valued inequalities. We also extend recent results of Carando et al. on Marcinkiewicz-Zygmund estimates for multilinear Calderón-Zygmund operators. Finally, our last application gives new weighted estimates (scalar and vector-valued) for the commutators of multilinear Calderón-Zygmund operators, bilinear rough singular integral operators, and for the bilinear Hilbert transform with BMO functions using ideas from Bényi et al.

Littlewood-Paley Operators on Spaces with Variable Exponent on Homogeneous Groups

In this paper, on homogeneous groups, we study the Littlewood-Paley operators in variable exponent spaces. First, we prove that the weighted Littlewood-Paley operators are controlled by the weighted Hardy-Littlewood maximal function, and obtain the vector-valued inequalities of the Littlewood-Paley operators, including the Lusin function, Littlewood-Paley g function and $$g_\lambda ^*$$ function. Second, we prove the boundedness of multilinear Littlewood-Paley $$g_{\psi, \lambda}^*$$ function.

Hardy Spaces on Homogeneous Groups and Littlewood-Paley Functions

Abstract We establish a characterization of the Hardy spaces on the homogeneous groups in terms of the Littlewood–Paley functions. The proof is based on vector-valued inequalities shown by applying the Peetre maximal function.

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.

A new approach to norm inequalities on weighted and variable Hardy spaces

We give new proofs of Hardy space estimates for fractional and singular integral operators on weighted and variable exponent Hardy spaces. Our proofs consist of several interlocking ideas: finite atomic decompositions in terms of $L^\infty$ atoms, vector-valued inequalities for maximal and other operators, and Rubio de Francia extrapolation. Many of these estimates are not new, but we give new and substantially simpler proofs, which in turn significantly simplifies the proofs of the Hardy spaces inequalities.

A Weak Type Vector-Valued Inequality for the Modified Hardy–Littlewood Maximal Operator for General Radon Measure on ℝ n

Abstract The aim of this paper is to prove the weak type vector-valued inequality for the modified Hardy– Littlewood maximal operator for general Radon measure on ℝ n . Earlier, the strong type vector-valued inequality for the same operator and the weak type vector-valued inequality for the dyadic maximal operator were obtained. This paper will supplement these existing results by proving a weak type counterpart.

An operator ideal generated by Orlicz spaces

Absolutely varphi -summing operators between Banach spaces generated by Orlicz spaces are investigated. A variant of Pietsch’s domination theorem is proved for these operators and applied to prove vector-valued inequalities. These results are used to prove asymptotic estimates of pi _varphi -summing norms of finite-dimensional operators and also diagonal operators between Banach sequence lattices for a wide class of Orlicz spaces based on exponential convex functions varphi . The key here is the description of a space of coefficients of the Rademacher series in this class of Orlicz spaces, proved via interpolation methods. As by-product, some absolutely varphi -summing operators on the Hilbert space ell _2 are characterized in terms of its approximation numbers.

Equivalent characterizations of Hardy spaces with variable exponent via wavelets

Via the boundedness of intrinsic g-functions from the Hardy spaces with variable exponent, ℋp(·)(ℝn), into Lebesgue spaces with variable exponent, Lp(·)(ℝn), and establishing some estimates on a discrete Littlewood-Paley g-function and a Peetre-type maximal function, we obtain several equivalent characterizations of ℋp(·)(ℝn) in terms of wavelets, which extend the wavelet characterizations of the classical Hardy spaces. The main ingredients are that, we overcome the difficulties of the quasi-norms of ℋp(·)(ℝn) by elaborately using an observation and the Fefferman-Stein vector-valued maximal inequality on Lp(·)(ℝn), and also overcome the difficulty of the failure of q = 2 in the atomic decomposition of ℋp(·)(ℝn) by a known idea.

Some Weighted Norm Inequalities on Manifolds

Let M be a complete non-compact Riemannian manifold satisfying the volume doubling property and the Gaussian upper bounds. Denote by Δ the Laplace-Beltrami operator and by ∇ the Riemannian gradient. In this paper, the author proves the weighted reverse inequality \(\left\| {{\Delta ^{\frac{1}{2}}}f} \right\|_{L^p(w)}\leq C\left\| {|\nabla f|} \right\|_{L^p(w)}\), for some range of p determined by M and w. Moreover, a weak type estimate is proved when p = 1. Some weighted vector-valued inequalities are also established.

A Vector-valued Weak Type Inequality for Ergodic Square Function

A Vector-valued Weak Type Inequality for Ergodic Square Function

Littlewood–Paley and Finite Atomic Characterizations of Anisotropic Variable Hardy–Lorentz Spaces and Their Applications

Let $$p(\cdot ):\ {{\mathbb {R}}}^n\rightarrow (0,\infty ]$$ be a variable exponent function satisfying the globally log-Holder continuous condition, $$q\in (0,\infty ]$$ and A be a general expansive matrix on $${\mathbb {R}}^n$$ . Let $$H_A^{p(\cdot ),q}({{\mathbb {R}}}^n)$$ be the anisotropic variable Hardy–Lorentz space associated with A defined via the radial grand maximal function. In this article, the authors characterize $$H_A^{p(\cdot ),q}({{\mathbb {R}}}^n)$$ by means of the Littlewood–Paley g-function or the Littlewood–Paley $$g_\lambda ^*$$ -function via first establishing an anisotropic Fefferman–Stein vector-valued inequality on the variable Lorentz space $$L^{p(\cdot ),q}({\mathbb {R}}^n)$$ . Moreover, the finite atomic characterization of $$H_A^{p(\cdot ),q}({{\mathbb {R}}}^n)$$ is also obtained. As applications, the authors then establish a criterion on the boundedness of sublinear operators from $$H^{p(\cdot ),q}_A({\mathbb {R}}^n)$$ into a quasi-Banach space. Applying this criterion, the authors show that the maximal operators of the Bochner–Riesz and the Weierstrass means are bounded from $$H^{p(\cdot ),q}_A({\mathbb {R}}^n)$$ to $$L^{p(\cdot ),q}({\mathbb {R}}^n)$$ and, as consequences, some almost everywhere and norm convergences of these Bochner–Riesz and Weierstrass means are also obtained. These results on the Bochner–Riesz and the Weierstrass means are new even in the isotropic case.