Hölder–Lipschitz Norms and Their Duals on Spaces with Semigroups,with Applications to Earth Mover’s Distance

We introduce a family of bounded, multiscale distances on any space equipped with an operator semigroup. In many examples, these distances are equivalent to a snowflake of the natural distance on the space. Under weak regularity assumptions on the kernels defining the semigroup, we derive simple characterizations of the Hölder–Lipschitz norm and its dual with respect to these distances. As the dual norm of the difference of two probability measures is the Earth Mover’s Distance (EMD) between these measures, our characterizations give simple formulas for a metric equivalent to EMD. We extend these results to the mixed Hölder–Lipschitz norm and its dual on the product of spaces, each of which is equipped with its own semigroup. Additionally, we derive an approximation theorem for mixed Lipschitz functions in this setting.     

Vector Variational Inequalities Involving Set-valued Mappings via Scalarization with Applications to Error Bounds for Gap Functions

In this paper, by using the scalarization approach of Konnov, several kinds of strong and weak scalar variational inequalities (SVI and WVI) are introduced for studying strong and weak vector variational inequalities (SVVI and WVVI) with set-valued mappings, and their gap functions are suggested. The equivalence among SVVI, WVVI, SVI, WVI is then established under suitable conditions and the relations among their gap functions are analyzed. These results are finally applied to the error bounds for gap functions. Some existence theorems of global error bounds for gap functions are obtained under strong monotonicity and several characterizations of global (respectively local) error bounds for the gap functions are derived.     

Hölder Flow and Differentiability for SDEs with Nonregular Drift

We prove the existence of a stochastic flow of Hölder homeomorphisms for solutions of SDEs with singular time dependent drift having only certain integrability properties. We also show that the solution map x → X ^x is differentiable in a weak sense.     

Estimates of Variation with Respect to a Set and Applications to Optimization Problems

A variational norm that plays a role in functional optimization and learning from data is investigated. For sets of functions obtained by varying some parameters in fixed-structure computational units (e.g., Gaussians with variable centers and widths), upper bounds on the variational norms associated with such units are derived. The results are applied to functional optimization problems arising in nonlinear approximation by variable-basis functions and in learning from data. They are also applied to the construction of minimizing sequences by an extension of the Ritz method.     

Preface to “Optimization,Convex and Variational Analysis”

Set-Valued and Variational Analysis -     

Hölder categories

Hölder categories are invented to provide an axiomatic foundation for the study of categories of archimedean lattice-ordered algebraic structures. The basis of such a study is Hölder’s Theorem (1908), stating that the archimedean totally ordered groups are precisely the subgroups of the additive real numbers ? with the usual addition and ordering, which remains the single most consequential result in the studies of lattice-ordered algebraic systems since Birkhoff and Fuchs to the present. This study originated with interest in W*, the category of all archimedean lattice-ordered groups with a designated strong order unit, and the ?-homomorphisms which preserve those units, and, more precisely, with interest in the epireflections on W*. In the course of this study, certain abstract notions jumped to the forefront. Two of these, in particular, seem to have been mostly overlooked; some notion of simplicity appears to be essential to any kind of categorical study of W*, as are the quasi-initial objects in a category. Once these two notions have been brought into the conversation, a Hölder category may then be defined as one which is complete, well powered, and in which (a) the initial object I is simple, and (b) there is a simple quasi-initial coseparator R. In this framework it is shown that the epireflective hull of R is the least monoreflective class. And, when I = R — that is, the initial element is simple and a coseparator — a theorem of Bezhanishvili, Morandi, and Olberding, for bounded archimedean f-algebras with identity, can be be generalized, as follows: for any Hölder category subject to the stipulation that the initial object is a simple coseparator, every uniformly nontrivial reflection — meaning that the reflection of each non-terminal object is non-terminal — is a monoreflection. Also shown here is the fact that the atoms in the class of epireflective classes are the epireflective hulls of the simple quasi-initial objects. From this observation one easily deduces a converse to the result of Bezhanishvili, Morandi, and Olberding: if in a Hölder category every epireflection is a monoreflection, then the initial object is a coseparator.     

Inequalities for the generalized elliptic integrals with respect to Hölder means

In this paper, the authors prove some inequalities for the generalized elliptic integrals with respect to Hölder means.     

Besicovitch cylindrical transformation with a Hölder function

For any γ ∈ (0, 1) and ε > 0, we construct a cylindrical cascade with a γ-Hölder function over some rotation of the circle. This transformation has the Besicovitch property; i.e., it is topologically transitive and has discrete orbits. The Hausdorff dimension of the set of points of the circle that have discrete orbits is greater than 1 ? γ ? ε.     

Hölder Continuous Solutions to Complex Hessian Equations

We prove the Hölder continuity of the solution to complex Hessian equation with the right hand side in L ^p , \(p>\frac {n}{m}\) , 1 < m < n, in a m-strongly pseudoconvex domain in ? ⁿ under some additional conditions on the density near the boundary and on the boundary data.     

Hölder Continuity to Hamilton-Jacobi Equations with Superquadratic Growth in the Gradient and Unbounded Right-hand Side

We show that solutions of time-dependent degenerate parabolic equations with super-quadratic growth in the gradient variable and possibly unbounded right-hand side are locally 𝒞^{0, α}. Unlike the existing (and more involved) proofs for equations with bounded right-hand side, our arguments rely on constructions of sub- and supersolutions combined with improvement of oscillation techniques.     

Inverse Hlder Inequality Related to Metrics and It’s Applications to Nonlinear Subelliptic Systems

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Hölder type quasicontinuity

It is proved that a functionuL ^m,p (R ⁿ ) (which coincides with the Sobolev spaceW ^1,p (R ⁿ ) ifm=1) coincides with a Hölder continuous functionw outside a set of smallm,q-capacity, whereq<p. Moreover, ifm=1, then the functionw can be chosen to be close tou in theW ^1,p -norm.     

Hölder Continuity of Harmonic Functions with Respect to Operators of Variable Order

ABSTRACT

We consider a class of integrodifferential operators and their corresponding harmonic functions. Under mild assumptions on the family of jump measures we prove a priori estimates and establish Hölder continuity of bounded functions that are harmonic in a domain.     

Hölder Continuity of Solutions to Parametric Vector Equilibrium Problems with Nonlinear Scalarization

In this article, using the nonlinear scalarization approach by virtue of the nonlinear scalarization function, commonly known as the Gerstewitz function in the theory of vector optimization, Hölder continuity of solution mappings for both set-valued and single-valued cases to parametric vector equilibrium problems is studied. The nonlinear scalarization function is a powerful tool that plays a key role in the proofs, and its main properties (such as sublinearity, continuity, convexity) are fully employed. Especially, its locally and globally Lipschitz properties are provided and the Lipschitz property is first exploited to investigate the Hölder continuity of solutions.     

Oscillatory Singular Integral Operators with Hölder Class Kernels

Journal of Fourier Analysis and Applications - We establish the boundedness on $$L^p({\mathbb {R}}^n)$$ of oscillatory singular integral operators whose kernels are the products of an oscillatory...