Equality cases in matrix norm inequalities of golden-thompson type

The equality cases in several matrix norm (trace) inequalities are characterized. Norm inequalities of Golden-Thompson type and of its complement type are treated as well as a logarithmic trace inequality and norm inequalities for matrix exponentials.     

Strict anisotropic norm bounded real lemma in terms of matrix inequalities

This paper is aimed at extending the H _∞ Bounded Real Lemma to stochastic systems under random disturbances with imprecisely known probability distributions. The statistical uncertainty is measured in the terms of information theory using the mean anisotropy functional. The disturbance attenuation capabilities of the system are quantified by the anisotropic norm which is a stochastic counterpart of the H _∞ norm. A state-space sufficient criterion for the anisotropic norm of a linear discrete time invariant system to be bounded by a given threshold value is derived. The resulting Strict Anisotropic Norm Bounded Real Lemma involves an inequality on the determinant of a positive definite matrix and a linear matrix inequality. These convex constraints can be approximated by two linear matrix inequalities.     

Two-weight norm inequalities for commutators of potential type integral operators

We give a condition which is sufficient for the two-weight (p,q) inequalities for commutators of potential type integral operators.     

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.

     

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.     

Weighted norm inequalities for the maximal function on spaces of homogeneous type

A characterization is obtained for weight function v for which the Hardy-Littlewood operator relative to a metric d is bounded from L^P(X, wdμ) to L^P(X, vdμ) for some nontrivial w, where (X, d, μ) is a space of homogeneous type. Supported by Natural Science Foundation of China.     

Weighted norm inequalities for operators of potential type and fractional maximal functions

In this paper, we study two-weight norm inequalities for operators of potential type in homogeneous spaces. We improve some of the results given in [6] and [8] by significantly weakening their hypotheses and by enlarging the class of operators to which they apply. We also show that corresponding results of Carleson type for upper half-spaces can be derived as corollaries of those for homogeneous spaces. As an application, we obtain some necessary and sufficient conditions for a large class of weighted norm inequalities for maximal functions under various assumptions on the measures or spaces involved.Research of the first author was supported in part by NSERC grant A5149.Research of the second author was supported in part by NSF grant DMS93-02991.     

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 of Bochner-Riesz means

Let w be a Muckenhoupt weight and be the weighted Hardy spaces. We use the atomic decomposition of and their molecular characters to show that the Bochner-Riesz means are bounded on for 0<p?1 and δ>max{n/p−(n+1)/2,[n/p]rw⁻¹(rw−1)−(n+1)/2}, where rw is the critical index of w for the reverse Hölder condition. We also prove the boundedness of the maximal Bochner-Riesz means for 0<p?1 and δ>n/p−(n+1)/2.     

Two-weight norm inequalities for fractional maximal operators on spaces of generalized homogeneous type

In this paper a characterization is given for a pairs of weights (w,v) for which the fractional maximal operator is bounded from when is a space of generalized homogeneous type introduced by A. Carbery et al. [4].     

Normal operators and inequalities in norm ideals

In this work we characterize normal invertible operators via inequalities with unitarily invariant norm of elementary operators.     

Nonlinear equations and weighted norm inequalities

We study connections between the problem of the existence of positive solutions for certain nonlinear equations and weighted norm inequalities. In particular, we obtain explicit criteria for the solvability of the Dirichlet problem

on a regular domain in in the ``superlinear case' . The coefficients are arbitrary positive measurable functions (or measures) on . We also consider more general nonlinear differential and integral equations, and study the spaces of coefficients and solutions naturally associated with these problems, as well as the corresponding capacities.

Our characterizations of the existence of positive solutions take into account the interplay between , , and the corresponding Green's kernel. They are not only sufficient, but also necessary, and are established without any a priori regularity assumptions on and ; we also obtain sharp two-sided estimates of solutions up to the boundary. Some of our results are new even if and is a ball or half-space.

The corresponding weighted norm inequalities are proved for integral operators with kernels satisfying a refined version of the so-called -inequality by an elementary ``integration by parts' argument. This also gives a new unified proof for some classical inequalities including the Carleson measure theorem for Poisson integrals and trace inequalities for Riesz potentials and Green potentials.

     

Weighted norm inequalities for integral operators

We consider a large class of positive integral operators acting on functions which are defined on a space of homogeneous type with a group structure. We show that any such operator has a discrete (dyadic) version which is always essentially equivalent in norm to the original operator. As an application, we study conditions of ``testing type,' like those initially introduced by E. Sawyer in relation to the Hardy-Littlewood maximal function, which determine when a positive integral operator satisfies two-weight weak-type or strong-type estimates. We show that in such a space it is possible to characterize these estimates by testing them only over ``cubes'. We also study some pointwise conditions which are sufficient for strong-type estimates and have applications to solvability of certain nonlinear equations.

     

Weighted norm inequalities for oscillatory integrals with finite type phases on the line

We obtain two-weighted $L^2$ norm inequalities for oscillatory integral operators of convolution type on the line whose phases are of finite type. The conditions imposed on the weights involve geometrically-defined maximal functions, and the inequalities are best-possible in the sense that they imply the full $L^p(\mathbb{R})\to L^q(\mathbb{R})$ mapping properties of the oscillatory integrals. Our results build on work of Carbery, Soria, Vargas and the first author.     

A note on operator norm inequalities

If P is a positive operator on a Hilbert space H whose range is dense, then a theorem of Foias, Ong, and Rosenthal says that: [(P)]^–1T[(P)]<-12 max {T, P^–1TP} for any bounded operator T on H, where is a continuous, concave, nonnegative, nondecreasing function on [0, P]. This inequality is extended to the class of normal operators with dense range to obtain the inequality [(N)]^–1T[(N)]<-12c² max {tT, N^–1TN} where is a complex valued function in a class of functions called vase-like, and c is a constant which is associated with by the definition of vase-like. As a corollary, it is shown that the reflexive lattice of operator ranges generated by the range NH of a normal operator N consists of the ranges of all operators of the form (N), where is vase-like. Similar results are obtained for scalar-type spectral operators on a Hilbert space.This author gratefully acknowledges the support of Central Michigan University in the form of a Research Professorship.