The commutator of the Kato square root for second order elliptic operators on ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

Let L=-div(A▽) be a second order divergence form elliptic operator, and A be an accretive, n×n matrix with bounded measurable complex coefficients in Rⁿ. We obtain the L^p bounds for the commutator generated by the Kato square root √L and a Lipschitz function, which recovers a previous result of Calderón, by a different method. In this work, we develop a new theory for the commutators associated to elliptic operators with Lipschitz function. The theory of the commutator with Lipschitz function is distinguished from the analogous elliptic operator theory.     

Lipschitz compact operators

We introduce the notion of Lipschitz compact (weakly compact, finite-rank, approximable) operators from a pointed metric space X into a Banach space E. We prove that every strongly Lipschitz p-nuclear operator is Lipschitz compact and every strongly Lipschitz p-integral operator is Lipschitz weakly compact. A theory of Lipschitz compact (weakly compact, finite-rank) operators which closely parallels the theory for linear operators is developed. In terms of the Lipschitz transpose map of a Lipschitz operator, we state Lipschitz versions of Schauder type theorems on the (weak) compactness of the adjoint of a (weakly) compact linear operator.     

Isotonic Regression under Lipschitz Constraint

The pool adjacent violators (PAV) algorithm is an efficient technique for the class of isotonic regression problems with complete ordering. The algorithm yields a stepwise isotonic estimate which approximates the function and assigns maximum likelihood to the data. However, if one has reasons to believe that the data were generated by a continuous function, a smoother estimate may provide a better approximation to that function. In this paper, we consider the formulation which assumes that the data were generated by a continuous monotonic function obeying the Lipschitz condition. We propose a new algorithm, the Lipschitz pool adjacent violators (LPAV) algorithm, which approximates that function; we prove the convergence of the algorithm and examine its complexity. The authors were supported by the Intramural Research Program of NIH, National Library of Medicine.     

Smoothness of Subharmonic Functions and Potentials of the Bergman Metric in the Unit Ball

We consider Lipschitz smoothness of an arbitray invariant potential U on the unit ball B in . We establish some Lipschitz estimates for both U and its gradient vector field U with respect to the Bergman metric. These estimates are taken with respect an invariant distance on B and shown to hold outside on open sets with arbitrarily small Hausdorff conttent. We also prove that for an M-subharmonic function u which satisfies Littelwood's integrability condition, there are such open sets , such that u is Lipschitz smooth on B\.     

The SC property of the squared norm of the SOC Fischer-Burmeister function

We show that the gradient mapping of the squared norm of Fischer-Burmeister function is globally Lipschitz continuous and semismooth, which provides a theoretical basis for solving nonlinear second-order cone complementarity problems via the conjugate gradient method and the semismooth Newton’s method.     

Distinct differentiable functions may share the same Clarke subdifferential at all points

We construct, using Zahorski's Theorem, two everywhere differentiable real-valued Lipschitz functions differing by more than a constant but sharing the same Clarke subdifferential and the same approximate subdifferential.

     

“Two Nontrivial Critical Points for Nonsmooth Functionals via Local Linking and Applications”

In this paper, we extend to nonsmooth locally Lipschitz functionals the multiplicity result of Brezis–Nirenberg (Communication Pure Applied Mathematics and 44 (1991)) based on a local linking condition. Our approach is based on the nonsmooth critical point theory for locally Lipschitz functions which uses the Clarke subdifferential. We present two applications. This first concerns periodic systems driven by the ordinary vector p-Laplacian. The second concerns elliptic equations at resonance driven by the partial p-Laplacian with Dirichlet boundary condition. In both cases the potential function is nonsmooth, locally Lipschitz.     

Semiconcavity of the value function for the Bolza control problem

In this paper we investigate the regularity of the value function of the Bolza control problem. We propose sufficient conditions for the value function to be semiconcave or locally Lipschitz when the controls are unbounded.     

Lipschitz函数空间的John-Nirenberg不等式及其应用

本文证明了R~n上Lipschitz函数空间的John-Nirenberg不等式,由此得到了Lipschitz函数空间的一些新的范数等价刻划。此外还对Lipschitz函数空间的定义进行了弱化。     

On the structure of the solution set of a third kind boundary value problem

We show that given any closed subset C of a real Banach space E, there is a continuous function f(t, x) which is Lipschitz continuous in its second variable such that the solution set of the corresponding third kind boundary value problem is homeomorphic to C (Theorem 1.1). In the special problem we give the infimum of Lipschitz constants L_f of such functions f(t, x) (Theorem 1.3).     

New types of Lipschitz summing maps between metric spaces

Building upon the results of M. C. Matos and extending previous work of J. D. Farmer, W. B. Johnson and J. A. Chávez‐Domínguez we define a Lipschitz mixed summable sequence as the pointwise product of a strongly summable sequence and a weakly Lipschitz summable one. Then we introduce classes of Lipschitz maps satisfying inequalities between Lipschitz mixed summable sequence and strongly summable sequences analogously to the linear case. These classes generalize the classes of Lipschitz summable maps considered earlier in the literature. We use standard techniques to establish several basic properties, showing that these classes of maps are ideals and some relationships between them. We establish various composition and inclusion theorems between different classes of Lipschitz summing maps and several characterizations. Furthermore, we prove that the classes of Lipschitz p‐summing maps coincide and the nonlinear “Pietsch Domination Theorem” for the case . We also identify cases where all Lipschitz maps are in the aforementioned classes of Lipschitz maps and discuss a sufficient condition for a Lipschitz composition formula as in the linear case.     

Lipschitz functions with maximal Clarke subdifferentials are generic

We show that on a separable Banach space most Lipschitz functions have maximal Clarke subdifferential mappings. In particular, the generic nonexpansive function has the dual unit ball as its Clarke subdifferential at every point. Diverse corollaries are given.

     

Quotients of little Lipschitz algebras

We prove a Tietze type theorem which provides extensions of little Lipschitz functions defined on closed subsets. As a consequence, we get that the quotient of any little Lipschitz algebra by any norm-closed ideal is another little Lipschitz algebra.

     

Smooth approximation for intrinsic Lipschitz functions in the Heisenberg group

We characterize intrinsic Lipschitz functions as maps which can be approximated by a sequence of smooth maps, with pointwise convergent intrinsic gradient. We also provide an estimate of the Lipschitz constant of an intrinsic Lipschitz function in terms of the $L^{\infty }$ -norm of its intrinsic gradient.     

Lipschitz continuity of the gradient of a one-parametric class of SOC merit functions

In this article, we show that a one-parametric class of SOC merit functions has a Lipschitz continuous gradient; and moreover, the Lipschitz constant is related to the parameter in this class of SOC merit functions. This fact will lay a building block when the merit function approach as well as the Newton-type method are employed for solving the second-order cone complementarity problem with this class of merit functions.     

Extending Lipschitz and Hölder maps between metric spaces

We introduce a stochastic generalization of Lipschitz retracts, and apply it to the extension problems of Lipschitz, Hölder, large-scale Lipschitz and large-scale Hölder maps into barycentric metric spaces. Our discussion gives an appropriate interpretation of a work of Lee and Naor.     

Porosity,Differentiability and Pansu’s Theorem

We use porosity to study differentiability of Lipschitz maps on Carnot groups. Our first result states that directional derivatives of a Lipschitz function act linearly outside a \(\sigma \)-porous set. The second result states that irregular points of a Lipschitz function form a \(\sigma \)-porous set. We use these observations to give a new proof of Pansu’s theorem for Lipschitz maps from a general Carnot group to a Euclidean space.     

Error Bounds of Regularized Gap Functions for Nonsmooth Variational Inequality Problems

We study the Clarke–Rockafellar directional derivatives of the regularized gap functions (and of some modified ones) for the variational inequality problem (VIP) defined by a locally Lipschitz but not necessarily differentiable function on a closed convex set in an Euclidean space. As applications we show that, under the strong monotonicity assumption, the regularized gap functions have fractional exponent error bounds and consequently that the sequences provided by an algorithm of Armijo type converge to the solution of the (VIP). The research of this author was supported by an Earmarked Grant from the Research Council of Hong Kong.     

Hardy spaces of the conjugate Beltrami equation

We study Hardy spaces of solutions to the conjugate Beltrami equation with Lipschitz coefficient on Dini-smooth simply connected planar domains, in the range of exponents 1<p<∞. We analyse their boundary behaviour and certain density properties of their traces. We derive on the way an analog of the Fatou theorem for the Dirichlet and Neumann problems associated with the equation div(σ∇u)=0 with Lp boundary data.     

Asymptotic error distribution for the Euler scheme with locally Lipschitz coefficients

In traditional works on numerical schemes for solving stochastic differential equations (SDEs), the globally Lipschitz assumption is often assumed to ensure different types of convergence. In practice, this is often too strong a condition. Brownian motion driven SDEs used in applications sometimes have coefficients which are only Lipschitz on compact sets, but the paths of the SDE solutions can be arbitrarily large. In this paper, we prove convergence in probability and a weak convergence result under a less restrictive assumption, that is, locally Lipschitz and with no finite time explosion. We prove if a numerical scheme converges in probability uniformly on any compact time set (UCP) with a certain rate under a global Lipschitz condition, then the UCP with the same rate holds when a globally Lipschitz condition is replaced with a locally Lipschitz plus no finite explosion condition. For the Euler scheme, weak convergence of the error process is also established. The main contribution of this paper is the proof of $\sqrt{n}$ weak convergence of the normalized error process and the limit process is also provided. We further study the boundedness of the second moments of the weak limit process and its running supremum under both global Lipschitz and locally Lipschitz conditions.