Norm Inequalities Research Articles

Numerical radius, Berezin number, and Berezin norm inequalities for sums of operators

Numerical radius, Berezin number, and Berezin norm inequalities for sums of operators

A norm inequality for three real matrices

A norm inequality for three real matrices

Weighted Characterization of Parabolic Fractional Maximal Operator

In this paper, we introduce the parabolic off-diagonal Muckenhoupt A + weights. We reveal the properties of these weights, and our main results characterize them through weak-type and strong-type weighted norm inequalities for the fractional forward-in-time maximal operator. Moreover, our results cover some well-known results.

Some norm inequalities for some positive block matrices

Some norm inequalities for some positive block matrices

An Optimal Control Problem for the Navier–Stokes Equations with Point Sources

We analyze, in two dimensions, an optimal control problem for the Navier–Stokes equations where the control variable corresponds to the amplitude of forces modeled as point sources; control constraints are also considered. This particular setting leads to solutions to the state equation exhibiting reduced regularity properties. We operate under the framework of Muckenhoupt weights, Muckenhoupt-weighted Sobolev spaces, and the corresponding weighted norm inequalities and derive the existence of optimal solutions and first- and, necessary and sufficient, second-order optimality conditions.

Tighter sum uncertainty relations via metric-adjusted skew information

In this paper, we first provide three general norm inequalities, which are used to give new uncertainty relations of any finite observables and quantum channels via metric-adjusted skew information. The results are applicable to its special cases as Wigner-Yanase-Dyson skew information. In quantifying the uncertainty of channels, we discuss two types of lower bounds and compare the tightness between them, meanwhile, a tight lower bound is given. The uncertainty relations obtained by us are stronger than the existing ones. To illustrate our results, we give several specific examples.

Norm Inequalities via Convex and Log-Convex Functions

Norm Inequalities via Convex and Log-Convex Functions

Some Excitation Independent Bounds for the Total Active Reflection Coefficient of Antenna Arrays

The total active reflection coefficient (TARC) is a key parameter in the characterization of multiport antenna arrays used in multiple-input multiple-output (MIMO) platforms. The TARC dependency on scattering parameters and port excitations is expressed using matrix <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p$ </tex-math></inline-formula> - and Frobenius norms. It is demonstrated how coupling and excitation effects can be separated from each other by mathematical manipulations of the TARC equation and represented in the form of inequalities. The derived inequalities are used to determine excitation-independent upper bounds or approximations for the TARC. It is shown that the proposed bounds can be used to define proper margins for efficient prediction of the TARC variations without a priori information about port excitations. In addition to the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p$ </tex-math></inline-formula> - and Frobenius norm inequalities, the extrema of the TARC as a function of the port phases are analytically calculated using the Hessian matrix. Finally, the derived bounds are verified and compared numerically via simulating different antenna arrays in both random and uniform configurations with different port excitations corresponding to different beamsteering directions. It is shown that the TARC values are fluctuating very close to the tightest proposed bounds which are defined based on the Frobenius and 2-norm of the array <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$S$ </tex-math></inline-formula> -matrix.

Refined Hermite–Hadamard Inequalities and Some Norm Inequalities

It is well known that the Hermite–Hadamard inequality (called the HH inequality) refines the definition of convexity of function f(x) defined on [a,b] by using the integral of f(x) from a to b. There are many generalizations or refinements of HH inequality. Furthermore HH inequality has many applications to several fields of mathematics, including numerical analysis, functional analysis, and operator inequality. Recently, we gave several types of refined HH inequalities and obtained inequalities which were satisfied by weighted logarithmic means. In this article, we give an N-variable Hermite–Hadamard inequality and apply to some norm inequalities under certain conditions. As applications, we obtain several inequalities which are satisfied by means defined by symmetry. Finally, we obtain detailed integral values.

Weighted norm inequalities for multilinearfractional integral operators on metric measure spaces

Let $({\mathcal~X},~d,~\mu)$ be a metric measurespace satisfying the geometrically doubling condition. We establish some weak type weighted norm inequalitiesfor the multilinear fractional integral operatorand the multilinear fractional maximal function.As an application, we also establish a weak type weighted norm inequalityon Morrey spaces for the multilinear fractional integral.

New Brezis–Van Schaftingen–Yung–Sobolev type inequalities connected with maximal inequalities and one parameter families of operators

Motivated by the recent characterization of Sobolev spaces due to Brezis–Van Schaftingen–Yung we prove new weak-type inequalities for one parameter families of operators connected with mixed norm inequalities. The novelty of our approach comes from the fact that the underlying measure space incorporates the parameter as a variable. We also show that our framework can be adapted to treat related characterizations of Sobolev spaces obtained earlier by Bourgain–Nguyen. The connection to classical and fractional order Sobolev spaces is shown through the use of generalized Riesz potential spaces and the Caffarelli–Silvestre extension principle. Higher order inequalities are also considered. We indicate many examples and applications to PDE's and different areas of Analysis, suggesting a vast potential for future research. In a different direction, and inspired by methods originally due to Gagliardo and Garsia, we obtain new maximal inequalities which combined with mixed norm inequalities are applied to obtain Brezis–Van Schaftingen–Yung type inequalities in the context of Calderón–Campanato spaces. In particular, Log versions of the Gagliardo–Brezis–Van Schaftingen–Yung spaces are introduced and compared with corresponding limiting versions of Calderón–Campanato spaces, resulting in a sharpening of recent inequalities due to Crippa–De Lellis and Brué–Nguyen.

Further norm inequalities for sums of operators

We give several further norm inequalities for sums of bounded linear operators on a complex Hilbert space. These inequalities refine and generalize earlier inequalities. Among other applications, it is shown that for A∈B(H) and r∈[0,1],w(A)≤12‖A‖+12‖|A|r|A⁎|r‖12‖|A|1−r|A⁎|1−r‖12, which refines a result of Kittaneh (2003) [5]. Here, w(⋅) and ‖⋅‖ denote the numerical radius and the usual operator norm, respectively.

Norm and numerical radius inequalities for Hilbert space operators

Norm and numerical radius inequalities for Hilbert space operators

Adaptive Appointed-Time Consensus Control of Networked Euler–Lagrange Systems With Connectivity Preservation

With consideration of motion control performance and efficient information communication, the synchronization problem on communication connectivity preservation and guaranteed consensus performance for networked mechanical systems has attracted considerable attention in recent years. Different from the existing works, this article investigates a brand-new appointed-time consensus control approach for uncertain networked Euler-Lagrange systems on a directed graph via exploring the prescribed performance control structure. First, a two-layer prescribed performance envelope is formulated via using an appointed-time convergent function for position-related and velocity-related consensus errors, respectively. Then, a simple state-feedback virtual controller with online adaptive performance adjustment is developed to preserve the communication connectivity. Moreover, to guarantee the velocity consensus of the networked systems and improve the position consensus accuracy, an appointed-time adaptive controller is designed by applying the norm inequality to the system uncertainties and external disturbances. Compared to the existing consensus control approaches, the prime advantage of the proposed one is that the constraints generated from the communication ranges are approximated by a time-varying contractive performance envelope, wherein, the appointed-time convergence and steady-state tracking accuracy are preassigned a priori. Meanwhile, no repeated logarithmic error transformations are required in the relevant controller design, which implies that the complexity of the devised control laws has decreased dramatically. Finally, two groups of illustrative examples are organized to validate the effectiveness of the proposed consensus control approach.

Norm inequalities for maximal operators

In this paper, we introduce a family of one-dimensional maximal operators mathscr{M}_{kappa ,m}, kappa geq 0 and min mathbb{N}setminus {0}, which includes the Hardy–Littlewood maximal operator as a special case (kappa =0, m=1). We establish the weak type (1,1) and the strong type (p,p) inequalities for mathscr{M}_{kappa ,m}, p>1. To do so, we prove a technical covering lemma for a finite collection of intervals.

Norm inequalities involving the weighted numerical radii of operators

It is shown, among other inequalities, that if A and B are Hilbert space operators which belong to the Schatten p-class, then, for p≥1 and 0<v<1, we have‖A+B‖p≤max⁡(2,2p−1)wp,vp([0AB⁎0])−|1−2v|p‖A−B‖pppandmin⁡(2,2p−1)wp,vp([0AB⁎0])−|1−2v|p|‖A‖p−‖B‖p|pp≤‖A‖p+‖B‖p,where ‖⋅‖p denotes the Schatten p-norm and wp,v is the generalized weighted numerical radius induced by the Schatten p-norm.

On weighted norm inequalities for oscillatory integral operators

We prove weighted norm inequalities with Muckenhoupt’s A_p-weights, for a wide class of oscillatory integral operators. As a consequence, one also obtains the boundedness of commutators of the aforementioned operators with functions in BMO.

Some norm inequalities involving convex functions of operators

Some norm inequalities involving convex functions of operators

Properties of Certain Volterra type ABC Fractional Integral Equations

In this paper we study existence, uniqueness and other properties of solutions of Volterra type ABC fractional integral equations. We have used Banach fixed point theorem with Bielecki type norm and Gronwall inequality in the frame of ABC fractional integral for proving our results.

Hypercontractive Inequalities for the Second Norm of Highly Concentrated Functions, and Mrs. Gerber's-Type Inequalities for the Second Rényi Entropy.

Let , , be the noise operator acting on functions on the boolean cube . Let f be a distribution on and let . We prove tight Mrs. Gerber-type results for the second Rényi entropy of which take into account the value of the Rényi entropy of f. For a general function f on we prove tight hypercontractive inequalities for the norm of which take into account the ratio between and norms of f.