Norm Inequalities For Operators Research Articles

Operator norm inequalities in semi-Hilbertian spaces

In this work we extend Cordes inequality, McIntosh inequality and CPR-inequality for the operator seminorm defined by a positive semidefinite bounded linear operator A.

Sharp function boundedness for vector-valued multilinear singular integral operators with non-smooth kernels

Sharp function inequalities for several vector-valued, multilinear singular integral operators with non-smooth kernels are obtained. As an application, some weighted Lp (p > 1) norm inequalities for vector-valued multilinear operators are derived.

Weighted norm inequalities for pseudo-pseudodifferential operators defined by amplitudes

We prove weighted norm inequalities for pseudodifferential operators with amplitudes which are only measurable in the spatial variables. The result is sharp, even for smooth amplitudes. Nevertheless, in the case when the amplitude contains the oscillatory factor ξ ↦ e i | ξ | 1 − ρ , the result can be substantially improved. We extend the L p -boundedness of pseudo-pseudodifferential operators to certain weights. End-point results are obtained when the amplitude is either smooth or satisfies a homogeneity condition in the frequency variable. Our weighted norm inequalities also yield the boundedness of commutators of these pseudodifferential operators with functions of bounded mean oscillation.

Norm inequalities for the absolute value of Hilbert space operators

We use norm inequalities for operators, such as Cauchy–Schwarz inequality and Minkowski inequality, to establish sharp norm inequalities for the absolute value of operators. Among other inequalities, it is shown that if A, B and X are operators on a Hilbert space, such that X is self-adjoint, and r ≥ 1, then This inequality generalizes some earlier related results. Other related inequalities are also discussed.

Generalized Hilbert integral operators on the Herz spaces

This paper gives some necessary and sufficient conditions for the generalized Hilbert integral operators to be bounded on the Herz spaces. The corresponding new operator norm inequalities are obtained.

Lipschitz and BMO norm inequalities for operators

In this paper, we obtain Lipschitz norm inequalities and BMO norm inequalities for operators applied to differential forms satisfying the nonhomogeneous A -harmonic equation.

Numerical Radius and Operator Norm Inequalities

A general inequality involving powers of the numerical radius for sums and products of Hilbert space operators is given. This inequality generalizes several recent inequalities for the numerical radius, and includes that if and are operators on a complex Hilbert space , then for . It is also shown that if is normal , then . Related numerical radius and usual operator norm inequalities for sums and products of operators are also presented.

The norm inequalities for the weighted Cesaro mean operators

This paper gives some necessary and sufficient conditions for the weighted Cesaro mean operators to be bounded on Herz spaces. The corresponding new operator norm inequalities are obtained.

Two-weight norm inequalities for multilinear potential type integral operators

We give a condition which is sufficient for the two-weight (p, q) inequalities for multilinear potential type integral operators, where 1 < p ≤ q < ∞. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Weighted norm inequalities for fractional operators

We prove weighted norm inequalities for fractional powers of elliptic operators together with their commutators with BMO functions, encompassing what is known for the classical Riesz potentials and elliptic operators with Gaussian domination by the classical heat operator. The method relies upon a good-? method that does not use any size or smoothness estimates for the kernels.

Weighted norm inequalities and indices

We extend and simplify several classical results on weighted norm inequalities for classical operators acting on rearrangement invariant spaces using the theory of indices. As an application we obtain necessary and sufficient conditions for generalized Hardy type operators to be bounded on ?p(w), ?p,8(w), Gp(w) and Gp,8(w).

Weighted norm inequalities for convolution operators and links with the Weiss Conjecture

We study convolution operators on weighted Lebesgue spaces and obtain weight characterisations for boundedness of these operators with certain kernels. Our main result is Theorem 3 which enables us to obtain results for certain kernel functions supported on bounded intervals; in particular we get a direct proof of the known characterisations for Steklov operators in Section 3 by using the weighted Hardy inequality. Our methods also enable us to obtain new results for other kernel functions in Section 4. In Section 5 we demonstrate that these convolution operators are related to operators arising from the Weiss Conjecture (for scalar-valued observation functionals) in linear systems theory, so that results on convolution operators provide elementary examples of nearly bounded semigroups not satisfying the Weiss Conjecture. Also we apply results on the Weiss Conjecture for contraction semigroups to obtain boundedness results for certain convolution operators.

On some operator norm inequalities

Let B(H) be the C ∗ -algebra of all bounded linear operators on a complex Hilbert space H, S be an invertible and selfadjoint operator in B(H) and let ( I,∥.∥ I ) denote a norm ideal of B(H). In this note, we shall show the following inequality: ∀X∈I:∥SXS −1−S −1XS∥ I⩽(∥S∥∥S −1∥−1)∥SXS −1+S −1XS∥ I.

Weighted L p Boundedness of Marcinkiewicz Integral

A weighted norm inequality for the Marcinkiewicz integral operator $$ \mu_{\Omega} $$ is proved when Ω belongs to $$ H^1(S^{n-1}), n \geq 2 $$ . We also give the weighted L p -boundedness for a class of Marcinkiewicz integral operators with rough kernels $$ \mu^{*}_{\Omega, \lambda} $$ and $$ \mu_{\Omega, S} $$ related to the Littlewood-Paley $$ g^{*}_{\lambda} $$ -function and the area integral S, respectively.

Fractional integrals, potential operators and two-weight, weak type norm inequalities on spaces of homogeneous type

We prove two-weight, weak type norm inequalities for potential operators and fractional integrals defined on spaces of homogeneous type. We show that the operators in question are bounded from L p ( v ) to L q ,∞ ( u ), 1< p ⩽ q <∞, provided the pair of weights ( u , v ) verifies a Muckenhoupt condition with a “power-bump” on the weight u .

Fractional integrals, potential operators and two-weight, weak type norm inequalities on spaces of homogeneous type

We prove two-weight, weak type norm inequalities for potential operators and fractional integrals defined on spaces of homogeneous type. We show that the operators in question are bounded from L p ( v) to L q,∞ ( u), 1< p⩽ q<∞, provided the pair of weights ( u, v) verifies a Muckenhoupt condition with a “power-bump” on the weight u.

Extrapolation of Weighted norm inequalities for multivariable operators and applications

Two versions of Rubio de Francia’s extrapolation theorem for multivariable operators of functions are obtained. One version assumes an initial estimate with different weights in each space and implies boundedness on all products of Lebesgue spaces. Another version assumes an initial estimate with the same weight but yields boundedness on a product of Lebesgue spaces whose indices lie on a line. Applications are given in the context of multilinear Calderon-Zygmund operators. Vector-valued inequalities are automatically obtained for them without developing a multilinear Banach-valued theory. A multilinear extension of the Marcinkiewicz and Zygmund theorem on l2-valued extensions of bounded linear operators is also obtained.

Reduction of Opial-type inequalities to norm inequalities

Weighted Opial-type inequalities are shown to be equivalent to weighted norm inequalities for sublinear operators and for nearly positive operators. Examples involving the Hardy-Littlewood maximal function and the nonincreasing rearrangement are presented.

Sharp weighted estimates for vector-valued singular integral operators and commutators

We prove sharp weighted norm inequalities for vector-valued singular integral operators and commutators. We first consider the strong $(p,p)$ case with $p>1$ and then the weak-type $(1,1)$ estimate. Our results do not assume any condition on the weight function and involve iterations of the classical Hardy-Littlewood maximal function.

Directional operators and mixed norms

We present a survey of mixed norm inequalities for several directional operators, namely, directional Hardy-Littlewood maximal functions and Hilbert transforms (both appearing in the method of rotations of Calderon and Zygmund), X-ray transforms, and directional fractional operators related to Riesz type potentials with variable kernel. In dimensions higher than two several interesting questions remain unanswered. [Proceedings of the 6th International Conference on Harmonic Analysis and Partial Differential Equations, El Escorial (Madrid), 2002].