One-sided Operators Research Articles

On the Calderón problem for nonlocal Schrödinger equations with homogeneous, directionally antilocal principal symbols

In this article we consider direct and inverse problems for α-stable, elliptic nonlocal operators whose kernels are possibly only supported on cones and which satisfy the structural condition of directional antilocality as introduced by Y. Ishikawa in the 80s. We consider the Dirichlet problem for these operators on the respective “domain of dependence of the operator” and in several, adapted function spaces. This formulation allows one to avoid natural “gauges” which would else have to be considered in the study of the associated inverse problems. Exploiting the directional antilocality of these operators we complement the investigation of the direct problem with infinite data and single measurement uniqueness results for the associated inverse problems. Here, due to the only directional antilocality, new geometric conditions arise on the measurement domains. We discuss both the setting of symmetric and a particular class of non-symmetric nonlocal elliptic operators, and contrast the corresponding results for the direct and inverse problems. In particular for only “one-sided operators” new phenomena emerge both in the direct and inverse problems: For instance, it is possible to study the problem in data spaces involving local and nonlocal data, the unique continuation property may not hold in general and further restrictions on the measurement set for the inverse problem arise.

Asymmetric Hysteresis Modeling Approach Featuring "Inertial System + Shape Function" for Magnetostrictive Actuators.

Hysteresis of the actuators based on magnetostrictive materials influences the control performance of the application systems. It is of importance and significance to establish an effective hysteresis model for the magnetostrictive actuators for precision engineering. In this paper, based on the analysis of the Duhem model, a first-order inertial system with hysteresis characteristic under harmonic input is used to describe the hysteresis caused by the inertia of the magnetic domains of magnetostrictive materials. Shape function is employed to describe the pinning of domain walls, the interactions of different magnetic domains of magnetostrictive materials, and the saturation properties of the hysteresis. Specifically, under an architecture of “inertial system + shape function” (ISSF-Duhem model), firstly a new hysteresis model is proposed for magnetostrictive actuators. The formulation of the inertial system is constructed based on its general expression, which is capable of describing the hysteresis characteristics of magnetostrictive actuators. Then, the developed models with a Grompertz function-based shape function, a modified hyperbolic tangent function-based shape function employing an exponential function as an offset function, a one-sided dead-zone operator-based shape function are compared with each other, and further compared with the classic modified Prandtl–Ishlinskii model with a one-sided dead-zone operator. Sequentially, feasibility and capability of the proposed hysteresis model are verified and evaluated by describing and predicting the hysteresis characteristics of a commercial magnetostrictive actuator.

Fourier Method for Solving Two-Sided Convolution Equations on Finite Noncommutative Groups

The Fourier method on commutative groups is used in many fields of mathematics, physics, and engineering. Nowadays, this method finds increasingly wide application to non-commutative groups. Along with the one-sided convolution operators and the corresponding convolution equations, two-sided convolution operators on noncommutative groups are studied. Two-sided convolution operators have a number of applications in complex analysis and are used in quantum mechanics. In this paper, two-sided convolutions on arbitrary finite noncommutative groups are considered. A criterion for the inversibility of the two-sided convolution operator is obtained. An algorithm for solving the two-sided convolution equation on an arbitrary finite noncommutative group, using the Fourier transform, is developed. Estimates of the computational complexity of the algorithm developed are given. It is shown that the complexity of solving the two-sided convolution equation depends both on the type of the group representation and on the computational complexity of the Fourier transform. The algorithm is considered in detail on the example of the finite dihedral group $${{\mathbb{D}}_{m}}$$ and the Heisenberg group $$\mathbb{H}({{\mathbb{F}}_{p}})$$ over a simple Galois field, and the results of numerical experiments are presented.

A Note on the Regularity of the Two-Dimensional One-Sided Hardy-Littlewood Maximal Function

We investigate the regularity properties of the two-dimensional one-sided Hardy-Littlewood maximal operator. We point out that the above operator is bounded and continuous on the Sobolev spaces Ws,p(R2) for 0≤s≤1 and 1<p<∞. More importantly, we establish the sharp boundedness and continuity for the discrete two-dimensional one-sided Hardy-Littlewood maximal operator from l1(Z2) to BV(Z2). Here BV(Z2) denotes the set of all functions of bounded variation on Z2.

One-Sided Operators in Grand Variable Exponent Lebesgue Spaces

The boundedness of one-sided integral operators in grand variable exponent Lebesgue spaces unifying grand Lebesgue spaces and variable exponent Lebesgue spaces are established. The conditions on variable exponent is weaker than the log-Hölder continuity condition.

Sharp weighted bounds for one-sided operators

Abstract In this paper, we establish sharp weighted bounds (Buckley-type theorems) for one-sided maximal and fractional integral operators in terms of one-sided A p {A_{p}} characteristics.

Operator Inequalities of Morrey Spaces Associated with Karamata Regular Variation

Karamata regular variation is a basic tool in stochastic process and the boundary blow-up problems for partial differential equations (PDEs). Morrey space is closely related to study of the regularity of solutions to elliptic PDEs. The aim of this paper is trying to bring together these two areas and this paper is intended as an attempt at motivating some further research on these areas. A version of Morrey space associated with Karamata regular variation is introduced. As application, some estimates of operators, especially one-sided operators, on these spaces are considered.

Sharp bounds for fractional one-sided operators

In this paper, we characterize the sharp boundedness of the one-sided fractional maximal function for one-weight and two-weight inequalities. Also a new two-weight testing condition for the one-sided fractional maximal function is introduced extending the work of Martin-Reyes and de la Torre. Improving some extrapolation result for the one-sided case, we get weak sharp bounded estimates for one-sided fractional maximal function and weak and strong sharp bounded estimates for one-sided fractional integral.

Norm inequalities for higher-order commutators of one-sided oscillatory singular integrals

In the present paper, we study the weighted norm inequalities for higher-order commutators formed by a class of one-sided oscillatory singular integrals and $BMO$ functions. We obtain that the boundedness of these commutators can be deduced by that of one-sided Calderon-Zygmund singular integral operators.

Two-Weight Norm Inequality for the One-Sided Hardy-Littlewood Maximal Operators in Variable Lebesgue Spaces

The authors establish the two-weight norm inequalities for the one-sided Hardy-Littlewood maximal operators in variable Lebesgue spaces. As application, they obtain the two-weight norm inequalities of variable Riemann-Liouville operator and variable Weyl operator in variable Lebesgue spaces on bounded intervals.

Scalar Discrete Convolutions in Spaces of Sequences Summed with Exponential Weights—Part 1: One-Sided Invertibility

The theory of the one-sided invertibility of discrete composite convolution operators in the space of sequences summed with exponential weight is constructed. In this part we consider those situations when Φ-theory of the operators is the same as the theory of their one-sided invertibility. The kernels and co-kernels of the operators are described. Constructions of the one-sided inverse operators are given.

The one-sided dyadic Hardy—Littlewood maximal operator

The main aim of this paper is to introduce an appropriate dyadic one-sided maximal operator , smaller than the one-sided Hardy–Littlewood maximal operator M+ but such that it controls M+ in a similar way to how the usual dyadic maximal operator controls the Hardy-Littlewood maximal operator. We characterize the weighted inequalities for this dyadic one-sided maximal operator.

Study of the boundedness of one-sided operators and the well-posedness of KdV type dispersion equations

Many important issues in Mathematics and Physics can often attribute to the boundedness of operators on some function spaces. The study for the boundedness of singular integral is one of the important components in Harmonic analysis. A variety of methods and techniques in Harmonic analysis have been widely used in the study of partial differential equations. By the space-time estimates for oscillatory type integral operators on Lebesgue spaces or Morrey type spaces and semigroup theory, one can obtain the well-posedness for the Cauchy problem of nonlinear dispersion equations in the low order Sobolev spaces. We give the definitions of one kind of one-sided oscillatory singular integral operators and study some classical boundedness of these operators. Inspired by the classical results for the characterizations via commutators, we introduce the idea of characterizations via commutators on Morrey spaces and obtain characterizations via commutators with symbol belonging to two kinds of function spaces on these spaces. The results obtained above offer some new tools to the study of partial differential equations. Finally, combining energy method and Number theory, we establish the well-posedness of the Cauchy problem for a class of KdV type dispersion equations.

Weighted Lipschitz estimates for commutators of one-sided operators on one-sided Triebel-Lizorkin spaces

Weighted Lipschitz estimates for commutators of one-sided operators on one-sided Triebel-Lizorkin spaces

Estimates of Some Operators on One-Sided Weighted Morrey Spaces

A version of one-sided weighted Morrey space is introduced. The boundedness of some classical one-sided operators in harmonic analysis and PDE on these spaces are discussed, including the Riemann-Liouville fractional integral.

Norm inequalities for some one-sided operators

We show that the one-sided maximal operators associated with Borel measures are of strong type (p, p) , 1 < p < ∞ , with constant p∗ , and the related one-sided geometric maximal operators are of strong type (p, p) , 0 < p < ∞ , with constant e1/p . We also investigate norm inequalities for integral operators with three measures on the cone of nonnegative nonincreasing functions. Our results show that if we restrict the measures in the inequalities to some particular classes, then a simple characterization for these inequalities to hold can be obtained. Mathematics subject classification (2010): Primary 26D15; Secondary 26D10, 42B25.

Weighted estimates in a limited range with applications to the Bochner-Riesz operators

From weighted inequalities for weights in subsets of Ap classes, we deduce two kinds of weighted estimates, from L (u) to Lq,∞(v) and from the weighted Lorentz space Λ(w) to Λp,∞(w). The applications include the Bochner-Riesz operators and some others. We also consider the results in the case of one-sided operators.

Weighted estimates for commutators of one-sided oscillatory integral operators

In this paper, we study the weighted norm inequalities for commutators formed by a class of one-sided oscillatory integral operators and functions in one-sided BMO spaces.

单边Triebel-Lizorkin 空间及其应用

The authors introduced a class of spaces, called one-sided Triebel-Lizorkin spaces. Using extrapolation of one-sided weights, they studied the boundedness of commutators generated by one-sided singular integral operators and Lipschitz functions from weighted Lebesgue spaces to weighted one-sided Triebel-Lizorkin spaces. Moreover, the corresponding results of commutators of one-sided fractional integrals are established.

MIXED WEAK TYPE INEQUALITIES FOR ONE-SIDED OPERATORS

We discuss mixed weak type inequalities in weighted spaces for one-sided operators. In particular, we prove that if Tcf(x) = (x − c)−1 ∫ x c f (y) dy, x > c, is the Hardy averaging operator, u ∈ A1 (one-sided Muckenhoupt A1 class) and v ∈ A1 (the another one-sided Muckenhoupt A1 class) then there exists a constant C such that supc∈R ∫ {x:|Tcf(x)|>v(x)} uv ≤ C ∫ R |f |u.