Sublinear Operators Research Articles

Two weight estimates for a class of (p, q) type sublinear operators and their commutators

Abstract In the present paper, the authors investigate the two weight, weak-(p, q) type norm inequalities for a class of sublinear operators 𝓣γ and their commutators [b, 𝓣γ] on weighted Morrey and Amalgam spaces. What should be stressed is that we introduce the new BMO type space and our results generalize known results before.

Two boundedness criteria for a class of operators on Musielak–Orlicz Hardy spaces and applications

AbstractLet φ : ℝn × [0, ∞) → [0, ∞) satisfy that φ(x, · ), for any given x ∈ ℝn, is an Orlicz function and φ( · , t) is a Muckenhoupt A∞ weight uniformly in t ∈ (0, ∞). The (weak) Musielak–Orlicz Hardy space Hφ(ℝn) (WHφ(ℝn)) generalizes both the weighted (weak) Hardy space and the (weak) Orlicz Hardy space and hence has wide generality. In this paper, two boundedness criteria for both linear operators and positive sublinear operators from Hφ(ℝn) to Hφ(ℝn) or from Hφ(ℝn) to WHφ(ℝn) are obtained. As applications, we establish the boundedness of Bochner–Riesz means from Hφ(ℝn) to Hφ(ℝn), or from Hφ(ℝn) to WHφ(ℝn) in the critical case. These results are new even when φ(x, t): = Φ(t) for all (x, t) ∈ ℝn × [0, ∞), where Φ is an Orlicz function.

The minimum mean square estimator of integrable variables under sublinear operators

In this paper, we study the minimum mean square estimator for non-bounded random variables under sublinear operators. The existence and uniqueness of the minimum mean square estimator are obtained. Several properties of the minimum mean square estimator for non-bounded random variables are proved under some mild assumptions.

A variant of Yano\u2019s extrapolation theorem on Hardy spaces

In this note, a variant of Yano’s classical extrapolation theorem for sublinear operators acting on analytic Hardy spaces over the torus is obtained.

Real-variable characterizations of Orlicz-slice Hardy spaces

In this paper, the authors first introduce a class of Orlicz-slice spaces which generalize the slice spaces recently studied by Auscher et al. Based on these Orlicz-slice spaces, the authors then introduce a new kind of Hardy-type spaces, the Orlicz-slice Hardy spaces, via the radial maximal functions. This new scale of Orlicz-slice Hardy spaces contains the variant of the Orlicz–Hardy space of Bonami and Feuto as well as the Hardy-amalgam space of de Paul Ablé and Feuto as special cases. Their characterizations via the atom, the molecule, various maximal functions, the Poisson integral and the Littlewood–Paley functions are also obtained. As an application of these characterizations, the authors establish their finite atomic characterizations, which further induce a description of their dual spaces and a criterion on the boundedness of sublinear operators from these Orlicz-slice Hardy spaces into a quasi-Banach space. Then, applying this criterion, the authors obtain the boundedness of [Formula: see text]-type Calderón–Zygmund operators on these Orlicz-slice Hardy spaces. All these results are new even for slice Hardy spaces and, moreover, for Hardy-amalgam spaces, the Littlewood–Paley function characterizations, the dual spaces and the boundedness of [Formula: see text]-type Calderón–Zygmund operators on these Hardy-type spaces are also new.

Commutators of sublinear operators in grand morrey spaces

Abstract In this paper we establish the boundedness of commutators of sublinear operators in weighted grand Morrey spaces. The sublinear operators under consideration contain integral operators such as Hardy-Littlewood and fractional maximal operators, Calderón-Zygmund operators, potential operators etc. The operators and spaces are defined on quasi-metric measure spaces with doubling measure.

Interpolation of sublinear operators which map into Riesz spaces and applications

We establish an interpolation result for sublinear operators which map into Riesz spaces. This result applies to all interpolation functors including the real interpolation and the complex interpolation. One component of our proof which may be of independent interest is the perhaps already known fact that the generalized versions of the Hahn-Banach theorem due to L. V. Kantorovich and M. M. Day also hold for complex vector spaces.

Maximal operators and singular integrals on the weighted Lorentz and Morrey spaces

In this paper, we first give some new characterizations of Muckenhoupt type weights through establishing the boundedness of maximal operators on the weighted Lorentz and Morrey spaces. Secondly, we establish the boundedness of sublinear operators including many interesting in harmonic analysis and its commutators on the weighted Morrey spaces. Finally, as an application, the boundedness of strongly singular integral operators and commutators with symbols in BMO space are also given.

Boundedness for Sublinear Operators with Rough Kernels on Weighted Grand Morrey Spaces

In this paper, we study the boundedness of some sublinear operators with rough kernels, satisfied by most of the operators in classical harmonic analysis, on the generalized weighted grand Morrey spaces. More specifically, we show that the sublinear operators with rough kernels are bounded on these spaces under the conditions that the operators and the kernel functions satisfy some size conditions, and the operators are bounded on Lebesgue spaces. This is done by exploiting the well-known boundedness of sublinear operators with rough kernels on Lebesgue spaces, a more explicit decomposition of the generalized weighted grand Morrey spaces and the good properties of the weight functions and the kernel functions. Through combining some properties of <i>A_p</i> weight with the relevant lemmas of operators with rough kernel, we obtain the boundedness for sublinear operators with rough kernels on weighted grand morrey spaces. Furthermore, using the equivalent norm and the properties of <i>BMO</i> functions, an application of the boundedness of the sublinear operators with rough kernels to the corresponding commutators formed by certain operators and <i>BMO</i> functions are also considered. And the boundedness of commutator is obtained by the lemma of function <i>BMO</i>.

On the Behaviors of Rough Fractional Type Sublinear Operators on Vanishing Generalized Weighted Morrey Spaces

The aim of this paper is to get the boundedness of rough sublinear operators generated by fractional integral operators on vanishing generalized weighted Morrey spaces under generic size conditions which are satisfied by most of the operators in harmonic analysis. Also, rough fractional integral operator and a related rough fractional maximal operator which satisfy the conditions of our main result can be considered as some examples.

Spaces generated by the cone of sublinear operators

This paper deals with a study on classes of non linear operators. Let $SL(X,Y)$ be the set of all sublinear operators between two Riesz spaces $X$ and $Y$. It is a convex cone of the space $H(X,Y)$ of all positively homogeneous operators. In this paper we study some spaces generated by this cone, therefore we study several properties, which are well known in the theory of Riesz spaces, like order continuity, order boundedness etc. Finally, we try to generalise the concept of adjoint operator. First, by using the analytic form of Hahn-Banach theorem, we adapt the notion of adjoint operator to the category of positively homogeneous operators. Then we apply it to the class of operators generated by the sublinear operators.

Caputo fractional approximation by sublinear operators

Here we consider the approximation of functions by sublinear positive operators with applications to a big variety of Max-Product operators under Caputo fractional differentiability.&#x0D; Our study is based on our general fractional results about positive sublinear operators. We produce Jackson type inequalities under simple initial conditions. So our approach is quantitative by producing inequalities with their right hand sides involving the modulus of continuity of fractional derivative of the function underapproximation.

Sublinear operators on weighted Hardy spaces with variable exponents

Abstract We establish the mapping properties for some sublinear operators on weighted Hardy spaces with variable exponents by using extrapolation. In particular, we study the Calderón–Zygmund operators, the maximal Bochner–Riesz means, the intrinsic square functions and the Marcinkiewicz integrals on weighted Hardy spaces with variable exponents.

High-throughput laser generation of Si-nanoparticle based surface coatings for antibacterial applications

High-productivity regime of nanosecond IR-laser ablative generation of silicon colloidal solutions in water for anti-bacterial applications was found in terms of GW/cm2-level laser intensity and scanning velocity by measuring multi-shot ablative mass loss and extinction coefficients of the colloids as sub-linear and third-power intensity functions, respectively. This advantageous regime implies sub-linear mass loss versus laser intensity at the simultaneous third-power yield of nanoparticles, resulting from the subcritical-density, opaque ablative plasma regulating the sample ablation rate and the related plasma-mediated dissociation (dispergation) of the ablation products. In contrast, at higher intensities, there is a drastic increase in mass loss with the corresponding increased yield of (sub) micrometer-sized particles owing to intense plasma-driven expulsion of micro-scale melt droplets and the corresponding saturation of the extinction coefficient of the colloidal solutions because of their dynamic local “self-limiting” effect during the high-rate ablation. The optimal low-intensity regime for Si nanoparticle production demonstrates the monotonous correlated increase of mass loss and extinction coefficient in terms of increasing laser scanning velocity, indicating the diminished cumulative effects. Surface coatings prepared from the generated Si nanoparticles exhibit minor surface oxidation, as acquired as their elemental composition via energy-dispersive X-ray spectroscopy, making their contact angle for water droplets (≈51°) close to that of bare Si wafer (≈58°) with its nanometer-thick native oxide layer. Owing to good wetting, the nanoparticle-based surface coatings show strong antibacterial response regarding Gram-negative Pseudomonas auereginosa bacteria even despite their minor oxidative passivation.

A Complete Real-Variable Theory of Hardy Spaces on Spaces of Homogeneous Type

Let $(X,d,\mu)$ be a space of homogeneous type, with the upper dimension $\omega$, in the sense of R. R. Coifman and G. Weiss. Assume that $\eta$ is the smoothness index of the wavelets on $X$ constructed by P. Auscher and T. Hyt\"onen. In this article, when $p\in(\omega/(\omega+\eta),1]$, for the atomic Hardy spaces $H_{\mathrm{cw}}^p(X)$ introduced by Coifman and Weiss, the authors establish their various real-variable characterizations, respectively, in terms of the grand maximal function, the radial maximal function, the non-tangential maximal functions, the various Littlewood-Paley functions and wavelet functions. This completely answers the question of R. R. Coifman and G. Weiss by showing that no any additional (geometrical) condition is necessary to guarantee the radial maximal function characterization of $H_{\mathrm{cw}}^1(X)$ and even of $H_{\mathrm{cw}}^p(X)$ with $p$ as above. As applications, the authors obtain the finite atomic characterizations of $H^p_{\mathrm{cw}}(X)$, which further induce some criteria for the boundedness of sublinear operators on $H^p_{\mathrm{cw}}(X)$. Compared with the known results, the novelty of this article is that $\mu$ is not assumed to satisfy the reverse doubling condition and $d$ is only a quasi-metric, moreover, the range $p\in(\omega/(\omega+\eta),1]$ is natural and optimal.

Quantitative approximation by shift invariant multivariate sublinear-Choquet operators

Abstract A very general multivariate positive sublinear Choquet integral type operator is given through a convolution-like iteration of another multivariate general positive sublinear operator with a multivariate scaling type function. For it, sufficient conditions are given for shift invariance, preservation of global smoothness, convergence to the unit with rates. Furthermore, two examples of very general multivariate specialized operators are presented fulfilling all the above properties; the higher order of multivariate approximation of these operators is also studied.

Pointwise multiple averages for sublinear functions

For any measure-preserving system $(X,{\mathcal{B}},\unicode[STIX]{x1D707},T_{1},\ldots ,T_{d})$ with no commutativity assumptions on the transformations $T_{i},$$1\leq i\leq d,$ we study the pointwise convergence of multiple ergodic averages with iterates of different growth coming from a large class of sublinear functions. This class properly contains important subclasses of Hardy field functions of order zero and of Fejér functions, i.e., tempered functions of order zero. We show that the convergence of the single average, via an invariant property, implies the convergence of the multiple one. We also provide examples of sublinear functions which are, in general, bad for convergence on arbitrary systems, but good for uniquely ergodic systems. The case where the fastest function is linear is addressed as well, and we provide, in all the cases, an explicit formula of the limit function.

Minimal convex majorants of functions and Demyanov–Rubinov exhaustive super(sub)differentials

ABSTRACTThe primary goal of the paper is to establish characteristic properties of (extended) real-valued functions defined on normed vector spaces that admit the representation as the lower envelope (the pointwise infimum) of their minimal (with respect of the pointwise ordering) convex majorants. The results presented in the paper generalize and extend the well-known Demyanov-Rubinov characterization of upper semicontinuous positively homogeneous functions as the lower envelope of exhaustive families of continuous sublinear functions to larger classes of (not necessarily positively homogeneous) functions defined on arbitrary normed spaces. As applications of the above results, we introduce, for nonsmooth functions, a new notion of the Demyanov-Rubinov exhaustive subdifferential at a given point, and show that it generalizes a number of known notions of subdifferentiability, in particular, the Fenchel-Moreau subdifferential of convex functions, the Dini-Hadamard (directional) subdifferential of directionally differentiable functions, and the Φ-subdifferential in the sense of the abstract convexity theory. Some applications of Demyanov-Rubinov exhaustive subdifferentials to extremal problems are considered.

Iterated Integral Operators on the Cone of Monotone Functions

Criteria for the boundedness of sublinear integral two-kernel operators of iterated type on cones of monotone functions in Lebesgue spaces on the real semiaxis are given.

A Virtual-Queue-Based Algorithm for Constrained Online Convex Optimization With Applications to Data Center Resource Allocation

In this paper, online convex optimization (OCO) problems with time-varying objective and constraint functions are studied from the perspective of an agent who takes actions in real time. Information about the current objective and constraint functions is revealed only after the corresponding action is already chosen. Inspired by a fast converging algorithm for time-invariant optimization in the very recent work [1] , we develop a novel online algorithm based on virtual queues for constrained OCO. Optimal points of the dynamic optimization problems with full knowledge of the current objective and constraint functions are used as a dynamic benchmark sequence. Upper bounds on the regrets with respect to the dynamic benchmark and the constraint violations are derived for the presented algorithm in terms of the temporal variations of the underlying dynamic optimization problems. It is observed that the proposed algorithm possesses sublinear regret and sublinear constraint violations, as long as the temporal variations of the optimization problems are sublinear, i.e., the objective and constraint functions do not vary too drastically across time. The performance bounds of the proposed algorithm are superior to those of the state-of-the-art OCO method in most scenarios. Besides, different from the saddle point methods widely used in constrained OCO, the stepsize of the proposed algorithm does not rely on the total time horizon, which may be unknown in practice. Finally, the algorithm is applied to a dynamic resource allocation problem in data center networks. Numerical experiments are conducted to corroborate the merit of the developed algorithm and its advantage over the state-of-the-art.