Oscillation of Functions in the Hölder Class

Potential Analysis - We study the size of the set of points where the α-divided difference of a function in the Hölder class Λα is bounded below by a fixed positive constant....     

The Hörmander index of symmetric periodic orbits

A symmetric periodic orbit is a special kind of periodic orbit that can also be regarded as a Lagrangian intersection point. Therefore it has two Maslov indices whose difference is the Hörmander index. In this paper we provide a formula for the Hörmander index of a symmetric periodic orbit and its iterates in terms of Chebyshev polynomials.     

Smoothing of Commutators for a Hörmander Class of Bilinear Pseudodifferential Operators

Commutators of bilinear pseudodifferential operators with symbols in the Hörmander class $BS_{1, 0}^{1}$ and multiplication by Lipschitz functions are shown to be bilinear Calderón-Zygmund operators. A connection with a notion of compactness in the bilinear setting for the iteration of the commutators is also made.     

On the Hörmander Classes of Bilinear Pseudodifferential Operators

Bilinear pseudodifferential operators with symbols in the bilinear analog of all the Hörmander classes are considered and the possibility of a symbolic calculus for the transposes of the operators in such classes is investigated. Precise results about which classes are closed under transposition and can be characterized in terms of asymptotic expansions are presented. This work extends the results for more limited classes studied before in the literature and, hence, allows the use of the symbolic calculus (when it exists) as an alternative way to recover the boundedness on products of Lebesgue spaces for the classes that yield operators with bilinear Calderón–Zygmund kernels. Some boundedness properties for other classes with estimates in the form of Leibniz’ rule are presented as well.     

Hölder Continuity of the Saddle Point Set for Real-Valued Functions

This paper is concerned with Hölder continuity of the solution to a saddle point problem. Some new su?cient conditions for the uniqueness and Hölder continuity of the solution for a perturbed saddle point problem are established. Applications of the result on Hölder continuity of the solution for perturbed constrained optimization problems are presented under mild conditions. Examples are given to illustrate the obtained results.     

Approximation of Functions from the Hölder Class on a Half-Axis

Let (0 < < 1) be the Hölder class on the semi-axis [0,). We characterize the class by rates of approximation of its functions by entire functions of order 1/2 belonging to a special class (similarly to the classical JacksonBernstein theorem). Bibliography: 4 titles.     

On the Phragmén–Lindelöf Indicator for Random Entire Functions

We establish that, for the majority of entire functions of finite order, their generalized Phragmén–Lindelöf indicators are identically equal to constants.     

Turán Inequalities and Complete Monotonicity for a Class of Entire Functions

Analysis Mathematica - The aim of this work is to derive several analytical properties for the class of entire functions defined by $${{\cal I}_\phi}(t) = \sum\limits_{n = 0}^\infty {{1 \over...     

On the Compactness of the Hypercomplex Commutator in Hölder Continuous Functions Spaces

This paper is devoted to the compactness of the hypercomplex commutator S _γ M _a ? M _a S _γ, where S _γ is the Cauchy singular integral operator (in the Douglis sense), a is a Hölder continuous hypercomplex function and M _a is the multiplication operator given by M _a f = a f. We extend a known compactness sufficient condition for the commutator of the Cauchy singular integral operator to the frame of the hypercomplex analysis, where γ is merely required to be an arbitrary regular closed Jordan curve.     

Generalized Hölder Spaces of Holomorphic Functions in Domains in the Complex Plane

We study some nonstandard spaces of functions holomorphic in domains on the complex plain with certain smoothness conditions up to the boundary. The first type is the space of Hölder-type holomorphic functions with prescribed modulus of continuity \(\omega =\omega (h)\), and the second is the variable exponent holomorphic Hölder space with the modulus of continuity \(|h|^{\lambda (z)}\). We give a characterization of functions in these spaces in terms of the behavior of their derivatives near the boundary.     

The Cauchy Problem for the Fractional Diffusion Equation in a Weighted Hölder Space

Under study is the Cauchy problem for the fractional diffusion equation with a Caputo derivative. The existence and uniqueness theorems for a smooth solution are proven in a weighted H¨older space.     

Regularity results for the minimum time function with Hörmander vector fields

In a bounded domain of ${\mathbb{R}}^n$ with boundary given by a smooth $(n?1)$-dimensional manifold, we consider the homogeneous Dirichlet problem for the eikonal equation associated with a family of smooth vector fields $\{X_1,\dots ,X_N\}$ subject to Hörmander's bracket generating condition. We investigate the regularity of the viscosity solution T of such problem. Due to the presence of characteristic boundary points, singular trajectories may occur. First, we characterize these trajectories as the closed set of all points at which the solution loses point-wise Lipschitz continuity. Then, we prove that the local Lipschitz continuity of T, the local semiconcavity of T, and the absence of singular trajectories are equivalent properties. Finally, we show that the last condition is satisfied whenever the characteristic set of $\{X_1,\dots ,X_N\}$ is a symplectic manifold. We apply our results to several examples.     

Multiplicity of solutions for a class of Schrödinger–Maxwell systems

In this paper, we study existence and multiplicity of nontrivial solutions for a class of Schrödinger–Maxwell systems via variational methods. Some new existence results of nontrivial solutions are obtained.