Sublinear Operators Research Articles

Boundedness of Some Commutators in Total Fofana Spaces

In this paper, we find necessary and sufficient conditions for the boundedness of the commutator of the Hardy-Littlewood maximal operator in total Fofana spaces. We also give in these spaces the boundedness of some sublinear operators and their commutators.

Publisher Correction: Strong and weak estimates for some sublinear operators in Herz spaces with power weights at indices beyond critical index

Publisher Correction: Strong and weak estimates for some sublinear operators in Herz spaces with power weights at indices beyond critical index

Homogeneous Grand Mixed Herz–Morrey Spaces and Their Applications

In this paper, we introduce the homogeneous grand mixed Herz–Morrey spaces MK˙q˜,λα,p),θ(Rn) and investigate their fundamental properties. We further explore the boundedness of sublinear operators and fractional-type operators on these spaces, establishing new results that contribute to the broader understanding of their applications.

Nikishin’s theorem and factorization through Marcinkiewicz spaces

Consider L 0 L^0 , the F F -space of all equivalence classes of measurable functions on a finite measure space equipped with the topology of convergence in measure. Inspired by Nikishin’s classical result on the factorization of sublinear continuous operators from a Banach space to L 0 L^0 , we prove a theorem that characterizes those maps from any quasi-metric space into L 0 L^0 that factor strongly through Marcinkiewicz weighted spaces. We show applications to sublinear operators on a certain class of quasi-Banach spaces with generalized Rademacher type generated by Orlicz sequence spaces.

Strong and weak estimates for some sublinear operators in Herz spaces with power weights at indices beyond critical index

Strong and weak estimates for some sublinear operators in Herz spaces with power weights at indices beyond critical index

Asymptotically periodic solutions of fractional order systems with applications to population models

Motivated by applications in population models, we consider S-asymptotically periodic solution of fractional differential equations with periodic environment forces or asymptotically periodic ones. The system is quasi-monotone, and the existence of positive S-asymptotically periodic solution is established by using upper and lower solutions. The sufficient conditions that ensure the uniqueness of positive S-asymptotically periodic solution are also established on the basis of theory of sublinear operator. The applications of the general conclusions to classical population models yield the global convergence of positive S-asymptotically periodic solution in logistic equation with or without weak Allee effect, and the model of two cooperative populations.

TWO-WEIGHT INEQUALITIES FOR THE SUBLINEAR OPERATOR IN WEIGHTED BANACH FUNCTION SPACES

Abstract. In this abstract we establish sufficient condition on weight functions for the boundedness of some sublinear operator

Extension properties for monotone sublinear operators

In this paper we obtain extension properties for monotone sublinear operators. The results obtained generalize those known for positive linear operators. Several examples illustrating the results obtained are given.

Products and commutators of martingales in H1 and BMO

Let f:=(fn)n∈Z+ and g:=(gn)n∈Z+ be two martingales related to the probability space (Ω,F,P) equipped with the filtration (Fn)n∈Z+. Assume that f is in the martingale Hardy space H1 and g is in its dual space, namely the martingale BMO. Then the semi-martingale f⋅g:=(fngn)n∈Z+ may be written as the sumf⋅g=G(f,g)+L(f,g). Here L(f,g):=(L(f,g)n)n∈Z+ with L(f,g)n:=∑k=0n(fk−fk−1)(gk−gk−1) for any n∈Z+, where f−1:=0=:g−1. The authors prove that L(f,g) is a process with bounded variation and limit in L1, while G(f,g) belongs to the martingale Hardy-Orlicz space Hlog associated with the Orlicz functionΦ(t):=tlog⁡(e+t),∀t∈[0,∞). The above bilinear decomposition L1+Hlog is sharp in the sense that, for particular martingales, the space L1+Hlog cannot be replaced by a smaller space having a larger dual. As an application, the authors characterize the largest subspace of H1, denoted by H1b with b∈BMO, such that the commutators [T,b] with classical sublinear operators T are bounded from H1b to L1. This endpoint boundedness of commutators allows the authors to give more applications. On the one hand, in the martingale setting, the authors obtain the endpoint estimates of commutators for both martingale transforms and martingale fractional integrals. On the other hand, in harmonic analysis, the authors establish the endpoint estimates of commutators both for the dyadic Hilbert transform beyond doubling measures and for the maximal operator of Cesàro means of Walsh–Fourier series.

Statistical Korovkin-type theorem for monotone and sublinear operators

"In this paper we generalize the result on statistical uniform convergence in the Korovkin theorem for positive and linear operators in C([a; b]), to the more general case of monotone and sublinear operators. Our result is illustrated by concrete examples."

Iterates of Monotone and Sublinear Operators on Spaces of Continuous Functions

Iterates of Monotone and Sublinear Operators on Spaces of Continuous Functions

The generalised Hausdorff measure of sets of Dirichlet non-improvable numbers

Let ψ : R + → R + \psi :\mathbb {R}_+\to \mathbb {R}_+ be a non-increasing function. A real number x x is said to be ψ \psi -Dirichlet improvable if the system | q x − p | > ψ ( t ) and | q | > t \begin{equation*} |qx-p|> \psi (t) \ \ {\text {and}} \ \ |q|>t \end{equation*} has a non-trivial integer solution for all large enough t t . Denote the collection of such points by D ( ψ ) D(\psi ) . In this paper, we prove a zero-infinity law valid for all dimension functions under natural non-restrictive conditions. Some of the consequences are zero-infinity laws, for all essentially sublinear dimension functions proved by Hussain-Kleinbock-Wadleigh-Wang [Mathematika 64 (2018), pp. 502–518], for some non-essentially sublinear dimension functions, and for all dimension functions but with a growth condition on the approximating function.

Korovkin-Type Theorems for Statistically Convergent Sequences of Monotone and Sublinear Operators

Korovkin-Type Theorems for Statistically Convergent Sequences of Monotone and Sublinear Operators

The Boundedness of Commutators of Sublinear Operators on Herz Triebel–Lizorkin Spaces with Variable Exponent

In this paper, the authors first discuss the characterization of Herz Triebel–Lizorkin spaces with variable exponent via two families of operators. By this characterization, the authors prove that the Lipschitz commutators of sublinear operators is bounded from Herz spaces with variable exponent to Herz Triebel–Lizorkin spaces with variable exponent. As applications, the corresponding boundedness estimates for the commutators of maximal operator, Riesz potential operator and Calderón–Zygmund operator are established.

Fixed point equations for superlinear operators with strong upper or strong lower solutions and applications

<abstract><p>It is well known that sublinear operators and superlinear operators are two classes of important nonlinear operators in nonlinear analysis and dynamical systems. Since sublinear operators have only weak nonlinearity, this advantage makes it easy to deal with them. However, superlinear operators have strong nonlinearity, and there are only a few results about them. In this paper, the convergence of Picard iteration for the superlinear operator $ A $ is obtained based on the conditions that the fixed point equation $ Ax = x $ has a strong upper solution and a lower solution (or alternatively, an upper solution and a strong lower solution). Besides, the uniqueness of the fixed point of strongly increasing operators as well as the global attractivity of strongly monotone dynamical systems are also discussed. In addition, the main results are applied to monotone dynamics of superlinear operators and nonlinear integral equations. The method used in our work develops the traditional method of upper and lower solutions. Since a strong upper (upper) solution and a lower (strong lower) solution are easily checked, the obtained results are effective and practicable in the study of nonlinear equations and dynamical systems. The main novelty is that this paper provides new fixed point results for increasing superlinear operators and the obtained results are applied to strongly monotone systems to investigate their global attractivity.</p></abstract>

Boundedness of sublinear operators on weighted grand Herz-Morrey spaces

<abstract><p>In this paper, we introduce weighted grand Herz-Morrey type spaces and prove the boundedness of sublinear operators and their multilinear commutators on these spaces. The results are still new even in the unweighted setting.</p></abstract>

Weighted Grand Herz-Type Spaces and Its Applications

In this paper, we introduce the weighted grand Herz spaces and weighted grand Herz-type Hardy spaces. The decompositional characterizations of these spaces are established. As its applications, the boundedness of some sublinear operators are established.

Weak solutions to triangular cross diffusion systems modeling chemotaxis with local sensing

New estimates and global existence results are provided for a class of systems of cross diffusion equations arising from the modeling of chemotaxis with local sensing, possibly featuring a growth term of logistic-type as well. For sublinear non-increasing motility functions, convergence to the spatially homogeneous steady state is shown, a dedicated Lyapunov functional being constructed for that purpose.

On Special Properties for Continuous Convex Operators and Related Linear Operators

This paper provides a uniform boundedness theorem for a class of convex operators, such as Banach–Steinhaus theorem for families of continuous linear operators. The case of continuous symmetric sublinear operators is outlined. Second, a general theorem characterizing the existence of the solution of the Markov moment problem is reviewed, and a related minimization problem is solved. Convexity is the common point of the two aims of the paper mentioned above.

Global dynamics and bifurcation analysis of an insect-borne plant disease model with two transmission routes

Huanglongbing (HLB) is a plant disease mainly spread by the insect-borne citrus psyllid. It is the most destructive citrus pathosystem worldwide. To understand the impact of sexual transmission on HLB dynamics, we propose a host–vector–HLB compartment model incorporating two transmission routes. The basic reproduction number [Formula: see text] is derived. Various interventions of the disease are assessed. We also investigate the effect of different incidence functions to simulate sexual transmission. For the case of sublinear incidence functions, the disease-free equilibrium is globally asymptotically stable (GAS) provided [Formula: see text]. For mass action incidence of sexual transmission, the endemic equilibrium is GAS provided [Formula: see text]. However, under nonlinear incidence, it is proved that the model may exhibit backward bifurcation. Theoretical and numerical studies reveal that (i) different forces of infection between heterosexual psyllids in the model may have a distinct impact on disease dynamics; (ii) sensitivity analysis shows that for [Formula: see text], the transmission rate between host and vector is more sensitive parameter than that between heterosexual psyllids; (iii) if the sexual transmission is ignored, the disease burden is likely to be underestimated in comparison with realistic scenarios; (iv) in the absence of chemical insecticides, the combined use of yellow sticky traps and injection of nutrient solutions can be more effective in suppressing the spread of HLB. These findings provide valuable insights for public policymakers to determine the long-term viability of implemented HLB management strategies and highlight the urgency of finding sustainable HLB solutions.