Let v be a smooth vector field on the plane, that is a map from the plane to the unit circle. We study sufficient conditions for the boundedness of the Hilbert transform Hv,ǫ f(x) := p.v. ∫ ǫ −ǫ f(x− yv(x)) dy y where ǫ is a suitably chosen parameter, determined by the smoothness properties of the vector field. It is a conjecture, due to E.M. Stein, that if v is Lipschitz, there is a positive ǫ...

Let v be a smooth vector field on the plane, that is a map from the plane to the unit circle. We study sufficient conditions for the boundedness of the Hilbert transform Hv,ǫ f(x) := p.v. ∫ ǫ −ǫ f(x− yv(x)) dy y where ǫ is a suitably chosen parameter, determined by the smoothness properties of the vector field. It is a conjecture, due to E.M. Stein, that if v is Lipschitz, there is a positive ǫ...

We characterize compact composition operators on real Banach spaces of complex-valued bounded Lipschitz functions on metric spaces, not necessarily compact, with Lipschitz involutions and determine their spectra.

In this article, we study the existence of piecewise-continuous mild solutions for the initial value problems for a class of semilinear evolution equations. These equations have non-instantaneous impulses in Banach spaces and the corresponding solution semigroup is noncompact. We assume that the nonlinear term satisfies certain local growth condition and a noncompactness measure condition. Also...

Exclusion algorithms have been used recently to find all solutions of a system of nonlinear equations or to find the global minimum of a function over a compact domain. These algorithms are based on a minimization condition that can be applied to each cell in the domain. In this paper, we consider Lipschitz functions of order and give a new minimization condition for the exclusion algorithm. Fu...

Let/be a locally Lipschitz function on a Banach space X, and S a subset of X. We define regular (i.e. noncritical) points for / relative to S, and give a sufficient condition for a point z e S to be regular. This condition is then expressed in the particular case when/is C1, and is used to obtain a new proof of Hoffman's inequality in linear programming.

We consider the control system ẋ = Ax+α(t)bu, where the pair (A, b) is controllable, x ∈ R2, u is a scalar control, and the unknown signal α satisfies a persistent excitation condition. We study the stabilization of this system, and we prove that it is globally asymptotically stable with arbitrarily large exponential rate uniformly with respect to all signals satisfying a common persistent exci...

The Skorokhod oblique reflection problem is studied in the case of n-dimensional convex polyhedral domains. The natural sufficient condition on the reflection directions is found, which together with the Lipschitz condition on the coefficients gives the existence and uniqueness of the solution. The continuity of the corresponding solution mapping is established. This property enables one to con...

Two-stage stochastic programs with random right-hand side are considered. Optimal values and solution sets are regarded as mappings of the expected recourse functions and their perturbations, respectively. Conditions are identiied implying that these mappings are directionally diierentiable and semidiieren-tiable on appropriate functional spaces. Explicit formulas for the derivatives are derive...

Piecewise linear convex functions arise as integrands in stochastic programs. They are Lipschitz continuous on their domain, but do not belong to tensor product Sobolev spaces. Motivated by applying Quasi-Monte Carlo methods we show that all terms of their ANOVA decomposition, except the one of highest order, are smooth if the underlying densities are smooth and a certain geometric condition is...