Strong Boundedness Research Articles

Strong boundedness of split Chevalley groups

Strong boundedness of split Chevalley groups

Explicit strong boundedness for higher rank symplectic groups

This paper gives an explicit argument to show strong boundedness for Sp2n(R) for R a ring of S-algebraic integers or a semi-local ring thereby giving a quantitative version of the abstract result in the paper [15]. The results presented further generalize older results regarding strong boundedness by Kedra, Libman and Martin [6] and Morris [9] from SLn to Sp2n. The results also completely solve the question of the asymptotic of strong boundedness for Sp2n(R) for R semi-local case with an argument that immediately generalizes to all other split Chevalley groups.

Forward-reflected-backward method with variance reduction

We propose a variance reduced algorithm for solving monotone variational inequalities. Without assuming strong monotonicity, cocoercivity, or boundedness of the domain, we prove almost sure convergence of the iterates generated by the algorithm to a solution. In the monotone case, the ergodic average converges with the optimal O(1/k) rate of convergence. When strong monotonicity is assumed, the algorithm converges linearly, without requiring the knowledge of strong monotonicity constant. We finalize with extensions and applications of our results to monotone inclusions, a class of non-monotone variational inequalities and Bregman projections.

Weak and strong boundedness for p-adic fractional Hausdorff operator and its commutator

Abstract In this paper, the boundedness of the Hausdorff operator on weak central Morrey space is obtained. Furthermore, we investigate the weak bounds of the p-adic fractional Hausdorff operator on weighted p-adic weak Lebesgue spaces. We also obtain the sufficient condition of commutators of the p-adic fractional Hausdorff operator by taking symbol function from Lipschitz spaces. Moreover, strong type estimates for fractional Hausdorff operator and its commutator on weighted p-adic Lorentz spaces are also acquired.

Regularity Of Spectral Stacks And Discreteness Of Weight-Hearts

Abstract We study regularity in the context of connective ring spectra and spectral stacks. Parallel to that, we construct a weight structure on the category of compact quasi-coherent sheaves on spectral quotient stacks of the form $X=[\operatorname{Spec} R/G]$ defined over a field, where R is a connective ${{\mathcal{E}}_\infty}$-k-algebra and G is a linearly reductive group acting on R. Under reasonable assumptions, we show that regularity of X is equivalent to regularity of R. We also show that if R is bounded, such a stack is discrete. This result can be interpreted in terms of weight structures and suggests a general phenomenon: for a symmetric monoidal stable $\infty$-category with a compatible bounded weight structure, the existence of an adjacent t-structure satisfying a strong boundedness condition should imply discreteness of the weight-heart. We also prove a gluing result for weight structures and adjacent t-structures, in the setting of a semi-orthogonal decomposition of stable $\infty$-categories.

Estimates of commutators on Herz-type spaces with variable exponent and applications

Let [b, T] be the commutator generated by b and T, where $$b\in \mathrm {BMO}({\mathbb {R}}^{n})$$ and T is a Calderon–Zygmund singular integral operator. In this paper, the authors establish some strong type and weak type boundedness estimates including the $$L\log L$$ type inequality for [b, T] on the Herz-type spaces with variable exponent. Meanwhile, the similar results for the commutators $$[b,I_l]$$ of fractional integral operator are also obtained. As applications, we consider the regularity in the Herz-type spaces with variable exponent of strong solutions to nondivergence elliptic equations with $$\mathrm {VMO}$$ coefficients.

Some characterizations of the generalized BMO spaces via the boundedness of maximal function commutators

In this paper, we investigated the commutators of Hardy–Littlewood maximal function with symbol function b in generalized BMO spaces. Some characterizations of generalized BMO spaces are obtained via the strong and weak boundedness of the commutators on Orlicz–Morrey space.

Weak and Strong Boundedness for p-ADIC Fractional Hausdorff Operator and Its Commutator

In this paper, boundedness of Hausdorff operator on weak central Morrey space is obtained. Furthermore, we investigate the weak bounds of p- adic fractional Hausdorff Operator on weighted p-adic weak Lebesgue Space. We also obtain the sufficient condition of commutators of p-adic fractional Hausdorff Operator by taking symbol function from Lipschitz space. Moreover, strong type estimates for fractional Hausdorff Operator and its commutator on weighted p-adic Lorentz space are also acquired.

Checking weak optimality and strong boundedness in interval linear programming

Interval programming provides a mathematical tool for dealing with uncertainty in optimization problems. In this paper, we study two properties of interval linear programs: weak optimality and strong boundedness. The former property refers to the existence of a scenario possessing an optimal solution, or the problem of deciding non-emptiness of the optimal set. We investigate the problem from a complexity-theoretic point of view and prove that checking weak optimality is NP-hard for all types of programs, even if the variables are restricted to a single orthant. The property of strong boundedness implies that each feasible scenario of the program has a bounded objective function. It is co-NP-hard to decide for inequality-constrained interval linear programs. For this class of programs, we derive a sufficient and necessary condition for testing strong boundedness using the orthant decomposition method. We also discuss the open problem of checking strong boundedness of programs described by equations with nonnegative variables.

Pursuing polynomial bounds on torsion

We show that for all ϵ > 0, there is a constant C(ϵ) > 0 such that for all elliptic curves E defined over a number field F with j(E) ∈ Q we have $$\# E\left( F \right)\left[ {tors} \right] \leq C\left( \in \right){\left[ {F:\mathbb{Q}} \right]^{5 + 2 + \in }}$$ . We pursue further bounds on the size of the torsion subgroup of an elliptic curve over a number field E/F that are polynomial in [F: Q] under restrictions on j(E). We give an unconditional result for j(E) lying in a fixed quadratic field that is not imaginary of class number one as well as two further results, one conditional on GRH and one conditional on the strong boundedness of isogenies of prime degree for non-CM elliptic curves.

Weak and Strong Type A_1\u2013A_infty Estimates for Sparsely Dominated Operators

We consider operators T satisfying a sparse domination property |⟨Tf,g⟩|≤c∑Q∈S⟨f⟩p0,Q⟨g⟩q0′,Q|Q|\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} |\\langle Tf,g\\rangle |\\le c\\sum _{Q\\in \\mathscr {S}}\\langle f\\rangle _{p_0,Q}\\langle g\\rangle _{q_0',Q}|Q| \\end{aligned}$$\\end{document}with averaging exponents 1le p_0<q_0le infty . We prove weighted strong type boundedness for p_0<p<q_0 and use new techniques to prove weighted weak type (p_0,p_0) boundedness with quantitative mixed A_1–A_infty estimates, generalizing results of Lerner, Ombrosi, and Pérez and Hytönen and Pérez. Even in the case p_0=1 we improve upon their results as we do not make use of a Hörmander condition of the operator T. Moreover, we also establish a dual weak type (q_0',q_0') estimate. In a last part, we give a result on the optimality of the weighted strong type bounds including those previously obtained by Bernicot, Frey, and Petermichl.

Analysis on the Transformation of Marketing Strategy and the Countermeasures Under the Network Economic Times

In this paper, we theoretically analyze the transformation of marketing strategy and the countermeasures under the network economic times. Network marketing is based on the network technology, including the whole process of marketing activities as a new form of marketing. In this paper, we analyze the issue from the listed perspectives. (1) Ultra space-and-time. The traditional marketing has very strong boundedness to the region, and this boundedness displays, the specific transaction can only be closed in the specific region, if the enterprise wants to expand the market share, only then establishes the retailing organization. (2) Interactivity. Network as a media, with the one-on-one interaction, this feature allows companies to basic communicate with consumers, and strengthen consumer participation, to determine the product form, function and the price. (3) The symmetry of information. Consumers can find all kinds of that related products on the Internet information, there are plenty of time to judge the various products and prices. Companies will also be released in a timely manner on the Internet the company's latest product information. Under this basis, we propose the new idea on the network marketing development direction that will help to build up the more efficient marketing system.

Strong boundedness, strong convergence and generalized variation

A trigonometric series strongly bounded at two points and with coefficients forming a log-quasidecreasing sequence is necessarily the Fourier series of a function belonging to all $L^{p}$ spaces, $1\leq p < \infty$. We obtain new results on strong convergence of Fourier series for functions of generalized bounded variation.

Riesz potential and its commutators on Orlicz spaces

In the present paper, we shall give necessary and sufficient conditions for the strong and weak boundedness of the Riesz potential operator I_{alpha} on Orlicz spaces. Cianchi (J. Lond. Math. Soc. 60(1):247-286, 2011) found necessary and sufficient conditions on general Young functions Φ and Ψ ensuring that this operator is of weak or strong type from L^{Phi} into L^{Psi}. Our characterizations for the boundedness of the above-mentioned operator are different from the ones in (Cianchi in J. Lond. Math. Soc. 60(1):247-286, 2011). As an application of these results, we consider the boundedness of the commutators of Riesz potential operator [b,I_{alpha }] on Orlicz spaces when b belongs to the BMO and Lipschitz spaces, respectively.

Multilinear Square Functions with Kernels of Dini’s Type

LetTbe a multilinear square function with a kernel satisfying Dini(1) condition and letT⁎be the corresponding multilinear maximal square function. In this paper, first, we showed thatTis bounded fromL1×⋯×L1toL1/m,∞.Secondly, we obtained that if eachpi&gt;1, thenTandT⁎are bounded fromLp1(ω1)×⋯×Lpm(ωm)toLp(νω→)and if there ispi=1, thenTandT⁎are bounded fromLp1(ω1)×⋯×Lpm(ωm)toLp,∞(νω→), whereνω→=∏i=1mωip/pi.Furthermore, we established the weighted strong and weak type boundedness forTandT⁎on weighted Morrey type spaces, respectively.

Characterization of two-weighted inequalities for multilinear fractional maximal operator

In this paper, we restudied the two-weight problem of multilinear fractional maximal operator Mα. First, we gave a characterization of two-weight inequalities for Mα related to a multilinear analogue of Sawyer’s two-weight condition S(p→,q), which essentially improved and extended some known results before. This was done mainly by using the technique of the well known atomic decomposition of tent space. Then we obtained the strong boundedness of Mα associated with multiple weight A(p→,q) class, which removed the power bump condition assumed in the known results before. Finally, a new two-weight B(p→,q) condition was introduced and the two-weight inequality of Mα with B(p→,q) condition was established.

Ideal Limit Theorems and Their Equivalence in $(\ell)$-Group Setting

We prove some equivalence results between limit theorems for sequences of $(\ell)$-group-valued measures, with respect to order ideal convergence. A fundamental role is played by the tool of uniform ideal exhaustiveness of a measure sequence already introduced for the real case or more generally for the Banach space case in our recent papers, to get some results on uniform strong boundedness and uniform countable additivity. We consider both the case in which strong boundedness, countable additivity and the related concepts are formulated with respect to a common order sequence and the context in which these notions are given in a classical like setting, that is not necessarily with respect to a same $(O)$-sequence. We show that, in general, uniform ideal exhaustiveness cannot be omitted. Finally we pose some open problems.

Boundedness of oscillatory integral operators and their commu- tators on weighted Morrey spaces

Boundedness of oscillatory integral operators play a key role in Harmonic Analysis. In this paper, we obtain the strong and weak estimates for a class of oscillatory integral operators, due to Ricci and Stein, on weighted Morrey spaces. Based on the above result, we study the strong boundedness of commutators generated by these oscillatory integral operators and BMO functions. Meanwhile, the corresponding result for the fractional oscillatory integral operators are established.

Limit Theorems in (&lt;em&gt;l&lt;/em&gt;)-Groups with Respect to (&lt;em&gt;D&lt;/em&gt;)-Convergence

Some Schur, Vitali-Hahn-Saks and Nikodým convergence theorems for \((l)\)-group-valued measures are given in the context of \((D)\)-convergence. We consider both the \(\sigma\)-additive and the finitely additive case. Here the notions of strong boundedness, countable additivity and absolute continuity are formulated not necessarily with respect to a same regulator, while the pointwise convergence of the measures is intended relatively to a common \((D)\)-sequence. Among the tools, we use the Fremlin lemma, which allows us to replace a countable family of \((D)\)-sequence with one regulator, and the Maeda-Ogasawara-Vulikh representation theorem for Archimedean lattice groups.

Weighted estimates for the k-plane transform of radial functions on euclidean spaces

We discuss the L p − L q mapping property of k-plane transforms acting on radial functions in certain weighted L p spaces with power weight. We show that for all admissible power weights it is not always possible to get strong (p, q) boundedness of the k-plane transform. However, we prove the best possible estimates with respect to the Lorentz norms.