On generalized Nash games and variational inequalities

We show that for a large class of problems a generalized Nash equilibrium can be calculated by solving a variational inequality. We analyze what solutions are found by this reduction procedure and hint at possible applications.     

Fourier multiplier theorem for atomic Hardy spaces on unbounded Vilenkin groups

We characterize atomic Hardy spaces on unbounded locally compact Vilenkin groups by means of a modified maximal function. The obtained Fourier multiplier theorem is more general than the corresponding results due to Kitada, Onneweer and Quek, Daly and Phillips that were proved under the boundedness assumption on the underlying group.     

An abstract form of a theorem of Helson and applications to sets of synthesis and sets of uniqueness

Let E be a compact perfect subset of the real line R such that the restriction of the Fourier transform from L¹(R) into C(E) is onto. Helson proved that then, for μ∈M(E), is possible only if μ=0. In this paper we present an abstract version of this theorem of Helson and provide some applications of it to the study of sets of spectral synthesis and sets of uniqueness.     

Hadamard products with power functions and multipliers of Hardy spaces

We consider Hadamard products of power functions P(z)=(1−z)^−b with functions analytic in the open unit disk in the complex plane, and an integral representation is obtained when 0<Reb<2. Let where μ is a complex-valued measure on the closed unit disk Such sequences are shown to be multipliers of H^p for 1?p?∞. Moreover, if the support of μ is contained in a finite set of Stolz angles with vertices on the unit circle, we prove that {μ_n} is a multiplier of H^p for every p>0. When the support of μ is [0,1] we get the multiplier sequence which provides more concrete applications. We show that if the sequences {μ_n} and {ν_n} are related by an asymptotic expansion

     

Characterizations of Arveson's Hardy Space

A Hardy-type space H ² _d in the unit ball B_d of C^d , which was recently introduced by Arveson [W. Arveson (1998). Subalgebras of C*-algebras III: multivariable operator theory. Acta Math., 181, 159-228.], is appropriate for the operator theory of d-contractions. In this article, it is proved that H ² _d actually coincides with a Hardy-Sobolev space. This yields almost immediately some of the related results obtained in [W. Arveson (1998). Subalgebras of C*-algebras III: multivariable operator theory. Acta Math., 181, 159-228.], including the facts that H ² _d is not associated with any measure on C ^d ; and that the corresponding algebra of multipliers M ? H ^∞(B_d ) and the inclusion is proper. Finally, a function-theoretic version of von Neumann's inequality for the d-contractions is presented.     

On boundedness of discrete multilinear singular integral operators

Let m(ξ,η) be a measurable locally bounded function defined in R². Let 1?p₁,q₁,p₂,q₂<∞ such that pi=1 implies qi=∞. Let also 0<p₃,q₃<∞ and 1/p=1/p₁+1/p₂−1/p₃. We prove the following transference result: the operator

     

Multipliers, paramultipliers, and weak-strong uniqueness for the Navier-Stokes equations

In this article, we describe spaces P such that: if u is a weak (in the sense of Leray [J. Leray, Sur le mouvement d'un fluide visqueux remplissant l'espace, Acta Math. 63 (1934) 193-248]) solution of the Navier-Stokes system for some initial data u₀, and if u belongs to P, then u is unique in the class of weak solutions. We say then that weak-strong uniqueness holds. It turns out that the proof of such results relies on the boundedness of a trilinear functional , where α, β belong to [0,1]. In order to find optimal conditions for the boundedness of F, we are led to describing spaces of multipliers and of paramultipliers (that is, functions which map, by classical pointwise product or by paraproduct, a given Sobolev spaces in another given Sobolev space). The study of these spaces enables us to give conditions for weak-strong uniqueness which generalise all previously known results, from the famous Serrin criterion [J. Serrin, The initial value problem for the Navier-Stokes equations, in: R.E. Langer (Ed.), Nonlinear Problems, Univ. of Wisconsin Press, 1963, pp. 69-98], to the recent conditions formulated by Lemarié-Rieusset [P.-G. Lemarié-Rieusset, Recent Developments in the Navier-Stokes Problem, Chapman and Hall, 2003].     

Universal and chaotic multipliers on spaces of operators

We use tensor product techniques to study universality, hypercyclicity and chaos of multipliers defined on operator ideals and of multiplication operators on the space of all continuous and linear operators, thus continuing the work of Kit Chan. We also obtain the first examples of outer multipliers on a Banach algebra which are chaotic in the sense of Devaney, and prove sufficient conditions for the existence of closed subspaces of universal vectors for operators between Fréchet spaces.     

One-parameter groups of regular quasimultipliers

Groups of unbounded operators are approached in the setting of the Esterle quasimultiplier theory. We introduce groups of regular quasimultipliers of growth ω, or ω-groups for short, where ω is a continuous weight on the real line. We study the relationship of ω-groups with families of operators and homomorphisms such as regularized, distribution and integrated groups, holomorphic semigroups, and functional calculi. Some convolution Banach algebras of functions with derivatives to fractional order are needed, which we construct using the Weyl fractional calculus.     

On conditions to have bounded multipliers in locally lipschitz programming

For a nonlinear programming problem with locally Lipschitz objective and inequality constraint functions and continuously differentiable equality constraint functions, a necessary and sufficient condition is presented for the set of multiplier vectors to be nonempty and bounded.This author's work was supported by the National Science Foundation under Grant No. MCS 78-06716.