Approximate convexity and submonotonicity

It is shown that a locally Lipschitz function is approximately convex if, and only if, its Clarke subdifferential is a submonotone operator. Consequently, in finite dimensions, the class of locally Lipschitz approximately convex functions coincides with the class of lower-C¹ functions. Directional approximate convexity is introduced and shown to be a natural extension of the class of lower-C¹ functions in infinite dimensions. The following characterization is established: a multivalued operator is maximal cyclically submonotone if, and only if, it coincides with the Clarke subdifferential of a locally Lipschitz directionally approximately convex function, which is unique up to a constant. Furthermore, it is shown that in Asplund spaces, every regular function is generically approximately convex.     

Bessel Potentials in Ahlfors Regular Metric Spaces

In this paper we introduce Bessel potentials and the Sobolev potential spaces resulting from them in the context of Ahlfors regular metric spaces. The Bessel kernel is defined using a Coifman type approximation of the identity, and we show integration against it improves the regularity of Lipschitz, Besov and Sobolev-type functions. For potential spaces, we prove density of Lipschitz functions, and several embedding results, including Sobolev-type embedding theorems. Finally, using singular integrals techniques such as the T1 theorem, we find that for small orders of regularity Bessel potentials are inversible, its inverse in terms of the fractional derivative, and show a way to characterize potential spaces, concluding that a function belongs to the Sobolev potential space if and only if itself and its fractional derivative are in L^p. Moreover, this characterization allows us to prove these spaces in fact coincide with the classical potential Sobolev spaces in the Euclidean case.     

Characterizing Convexity of a Function by Its Fréchet and Limiting Second-Order Subdifferentials

The Fréchet and limiting second-order subdifferentials of a proper lower semicontinuous convex function \(\varphi: \mathbb R^n\rightarrow\bar{\mathbb R}\) have a property called the positive semi-definiteness (PSD)—in analogy with the notion of positive semi-definiteness of symmetric real matrices. In general, the PSD is insufficient for ensuring the convexity of an arbitrary lower semicontinuous function φ. However, if φ is a C ^1,1 function then the PSD property of one of the second-order subdifferentials is a complete characterization of the convexity of φ. The same assertion is valid for C ¹ functions of one variable. The limiting second-order subdifferential can recognize the convexity/nonconvexity of piecewise linear functions and of separable piecewise C ² functions, while its Fréchet counterpart cannot.     

Abstract convexity of extended real-valued ICR functions

The theory of non-negative increasing and co-radiant (ICR) functions defined on ordered topological vector spaces has been well developed. In this article, we present the theory of extended real-valued ICR functions defined on an ordered topological vector space X. We first give a characterization for non-positive ICR functions and examine abstract convexity of this class of functions. We also investigate polar function and subdifferential of these functions. Finally, we characterize abstract convexity, support set and subdifferential of extended real-valued ICR functions.     

Generalized gradients of monotone type

We examine the properties of the subdifferential in the sense of Clarke of certain locally Lipschitz, quasi-convex functions. We prove that, even if they may not possess a pseudomonotone-type subdifferential, if we consider the operator A+∂f, where A is an operator of type (S)₊, then the sum is pseudomonotone. A new type of subdifferential for Lipschitz functions is also presented. We prove some calculus rules and we establish that in the context of reflexive Banach spaces is an operator of type (M).     

Resolvent characterisation of generators of cosine functions and C 0-groups

The paper provides new characterisations of generators of cosine functions and C ₀-groups on UMD spaces and their applications to some classical problems in cosine function theory. In particular, we show that on UMD spaces, generators of cosine functions and C ₀-groups can be characterised by means of a complex inversion formula. This allows us to provide a strikingly elementary proof of Fattorini’s result on square root reduction for cosine function generators on UMD spaces. Moreover, we give a cosine function analogue of McIntosh’s characterisation of the boundedness of the H ^∞ functional calculus for sectorial operators in terms of square function estimates. Another result says that the class of cosine function generators on a Hilbert space is exactly the class of operators which possess a dilation to a multiplication operator on a vector-valued L ₂ space. Finally, we prove a cosine function analogue of the Gomilko-Feng-Shi characterisation of C ₀-semigroup generators and apply it to answer in the affirmative a question by Fattorini on the growth bounds of perturbed cosine functions on Hilbert spaces.     

Algorithmic Construction of the Subdifferential from Directional Derivatives

The subdifferential of a function is a generalization for nonsmooth functions of the concept of gradient. It is frequently used in variational analysis, particularly in the context of nonsmooth optimization. The present work proposes algorithms to reconstruct a polyhedral subdifferential of a function from the computation of finitely many directional derivatives. We provide upper bounds on the required number of directional derivatives when the space is ?¹ and ?², as well as in ? ⁿ where subdifferential is known to possess at most three vertices.     

Invariant subspaces of finite codimension in Banach spaces of analytic functions

We give a characterization of invariant subspaces of finite codimension in Banach spaces of vector-valued analytic functions in several variables, where invariant refers to invariance under multiplication by any polynomial. We obtain very weak conditions under which our characterization applies, that unifies and improves a number of previous results. In the vector-valued case, the results are new even for one complex variable. As a concrete application in several variables, we consider the Bergman space on a strictly pseudo-convex domain, and we improve previous results (assuming C^∞-boundary) to the case of C²-boundary.     

Exponential laws for ultrametric partially differentiable functions and applications

We establish exponential laws for certain spaces of differentiable functions over a valued field $\mathbb{K}$ . For example, we show that $$C^{(\alpha ,\beta )} \left( {U \times V,E} \right) \cong C^\alpha \left( {U,C^\beta \left( {V,E} \right)} \right)$$ if α ∈ (?₀ ∪ {∞}) ⁿ , β ∈ (?₀ ∪ {∞}) ^m , $U \subseteq \mathbb{K}^n$ and $V \subseteq \mathbb{K}^m$ are open (or suitable more general) subsets, and E is a topological vector space. As a first application, we study the density of locally polynomial functions in spaces of partially differentiable functions over an ultrametric field (thus solving an open problem by Enno Nagel), and also global approximations by polynomial functions. As a second application, we obtain a new proof for the characterization of C ^r -functions on (? _p ) ⁿ in terms of the decay of their Mahler expansions. In both applications, the exponential laws enable simple inductive proofs via a reduction to the one-dimensional, vector-valued case.     

Minimal convex majorants of functions and Demyanov–Rubinov exhaustive super(sub)differentials

ABSTRACT

The primary goal of the paper is to establish characteristic properties of (extended) real-valued functions defined on normed vector spaces that admit the representation as the lower envelope (the pointwise infimum) of their minimal (with respect of the pointwise ordering) convex majorants. The results presented in the paper generalize and extend the well-known Demyanov-Rubinov characterization of upper semicontinuous positively homogeneous functions as the lower envelope of exhaustive families of continuous sublinear functions to larger classes of (not necessarily positively homogeneous) functions defined on arbitrary normed spaces. As applications of the above results, we introduce, for nonsmooth functions, a new notion of the Demyanov-Rubinov exhaustive subdifferential at a given point, and show that it generalizes a number of known notions of subdifferentiability, in particular, the Fenchel-Moreau subdifferential of convex functions, the Dini-Hadamard (directional) subdifferential of directionally differentiable functions, and the Φ-subdifferential in the sense of the abstract convexity theory. Some applications of Demyanov-Rubinov exhaustive subdifferentials to extremal problems are considered.     

Universal spaces for the subdifferentials of sublinear operators with values in the spaces of continuous functions

Given a continuous sublinear operator P: V → C(X) from a Hausdorff separable locally convex space V to the Banach space C(X) of continuous functions on a compact set X we prove that the subdifferential ∂P at zero is operator-affinely homeomorphic to the compact subdifferential ∂ ^c Q, i.e., the subdifferential consisting only of compact linear operators, of some compact sublinear operator Q: ł² → C(X) from a separable Hilbert space ł², where the spaces of operators are endowed with the pointwise convergence topology. From the topological viewpoint, this means that the space L ^c (ł², C(X)) of compact linear operators with the pointwise convergence topology is universal with respect to the embedding of the subdifferentials of sublinear operators of the class under consideration.     

Lipschitz-type functions on metric spaces

In order to find metric spaces X for which the algebra Lip^∗(X) of bounded Lipschitz functions on X determines the Lipschitz structure of X, we introduce the class of small-determined spaces. We show that this class includes precompact and quasi-convex metric spaces. We obtain several metric characterizations of this property, as well as some other characterizations given in terms of the uniform approximation and the extension of uniformly continuous functions. In particular we show that X is small-determined if and only if every uniformly continuous real function on X can be uniformly approximated by Lipschitz functions.     

Subdifferential characterization of s-lower regular functions

As for Moreau envelopes of primal lower nice as well as prox-regular functions, Moreau s-envelopes of s-lower regular functions have been proved recently to have several remarkable differential properties and to have many important applications. Here, we provide a subdifferential characterization of extended real-valued s-lower regular functions on Banach spaces in terms of a hypomonotonicity-like property of the subdifferential.     

An Extension of the Notion of Orthogonality to Banach Spaces

We extend the usual notion of orthogonality to Banach spaces. We show that the extension is quite rich in structure by establishing some of its main properties and consequences. Geometric characterizations and comparison results with other extensions are established. Also, we establish a characterization of compact operators on Banach spaces that admit orthonormal Schauder bases. Finally, we characterize orthogonality in the spaces l₂^p(C).     

James' Theorem Fails for Starlike Bodies

Starlike bodies are interesting in nonlinear functional analysis because they are strongly related to bump functions and to n-homogeneous polynomials on Banach spaces, and their geometrical properties are thus worth studying. In this paper we deal with the question whether James' theorem on the characterization of reflexivity holds for (smooth) starlike bodies, and we establish that a feeble form of this result is trivially true for starlike bodies in nonreflexive Banach spaces, but a reasonable strong version of James' theorem for starlike bodies is never true, even in the smooth case. We also study the related question as to how large the set of gradients of a bump function can be, and among other results we obtain the following new characterization of smoothness in Banach spaces: a Banach space X has a C¹ Lipschitz bump function if and only if there exists another C¹ smooth Lipschitz bump function whose set of gradients contains the unit ball of the dual space X*. This result might also be relevant to the problem of finding an Asplund space with no smooth bump functions.     

A characterization of Lévy probability distribution functions on Euclidean spaces

In 1937, Paul Lévy proved two theorems that characterize one-dimensional distribution functions of class L. In 1972, Urbanik generalized Lévy's first theorem. In this note, we generalize Lévy's second theorem and obtain a new characterization of Lévy probability distribution functions on Euclidean spaces. This result is used to obtain a new characterization of operator stable distribution functions on Euclidean spaces and to show that symmetric Lévy distribution functions on Euclidean spaces need not be symmetric unimodal.     

Hardy–Carleson Measures and Their Dual Poisson–Szegö Transforms

Recently Choe et al. have introduced the notion of dual Berezin transforms and used it to obtain new characterizations of the Carleson measures for the weighted Bergman spaces over the unit ball in C ⁿ . Continuing our investigation on the Hardy spaces, we obtain new characterizations of the Carleson measures for the Hardy spaces by means of the dual Poisson–Szegö transforms introduced by Koosis. Compared with the results for the weighted Bergman spaces, our results for the Hardy spaces not only show an similarity, but also reveal a new characterization.     

Explicit characterizations of orthogonality in lS(C)

In a previous paper, the author used a notion of orthogonality introduced in another article to establish characterizations for orthogonality in the spaces l_S^p(C), 1?p<∞, thus obtaining generalizations of the usual characterization of orthogonality in the Hilbert spaces l_S²(C), via inner products. In this paper we make explicit these characterizations for some of the spaces l_S^p(C). We finish by presenting some remarks and open problems.     

Hypoellipticity in spaces of ultradistributions—Study of a model case

In this work we study C ^??-hypoellipticity in spaces of ultradistributions for analytic linear partial differential operators. Our main tool is a new a-priori inequality, which is stated in terms of the behaviour of holomorphic functions on appropriate wedges. In particular, for sum of squares operators satisfying H?rmander??s condition, we thus obtain a new method for studying analytic hypoellipticity for such a class. We also show how this method can be explicitly applied by studying a model operator, which is constructed as a perturbation of the so-called Baouendi-Goulaouic operator.     

Smooth robustness of parameterized perturbations of exponential dichotomies

We establish the robustness of nonuniform exponential dichotomies in Banach spaces, under sufficiently small C¹-parameterized perturbations. Moreover, we show that the stable and unstable subspaces of the exponential dichotomies obtained from the perturbation are also of class C¹ on the parameter, thus yielding an optimal smoothness.