A Characterization of Some Classes of Harmonic Functions

In this paper we investigate harmonic Hardy-Orlicz and Bergman-Orlicz b _φ,α (B) spaces, using an identity of Hardy-Stein type. We also extend the notion of the Lusin property by introducing (φ, α)-Lusin property with respect to a Stoltz domain. The main result in the paper is as follows: Let be a nonnegative increasing convex function twice differentiable on (0, ∞), and u a harmonic function on the unit ball B in . Then the following statements are equivalent:

On the Multiplicity of Nodal Solutions to a Singularly Perturbed Neumann Problem

We consider the problem

Nonregular Boundary Problems for Elliptic Systems

We discuss explicit boundary value problems for solutions of the Fueter equation in which are normally solvable. The results extend to nonlinear first order elliptic systems. Received: October, 2007, Accepted: February, 2008.     

Singular Integral Operators with Coefficients of a Special Structure Related to Operator Equalities

In our previous works we have constructed operator equalities which transform scalar singular integral operators with shift to matrix characteristic singular integral operators without shift and found some of their applications to problems with shift. In this article the operator equalities are used for the study of matrix characteristic singular integral operators. Conditions for the invertibility of the singular integral operators with orientation preserving shift and coefficients with a special structure generated by piecewise constant functions, t, t ⁻¹, were found. Conditions for the invertibility of the matrix characteristic singular integral operators with four-valued piecewise constant coefficients of a special structure were likewise obtained. Submitted: June 15, 2007. Revised: October 25, 2007. Accepted: November 5, 2007.     

Gradient Estimates for Solutions of Ultraparabolic Equations

We establish a priori estimates for solutions to ultraparabolic equations which play a crucial role in the solvability of the initial value problem. A class of these equations came from population dynamics, namely from a fish larvae model.      

Representation of Contractive Solutions of a Class of Algebraic Riccati Equations as Characteristic Functions of Maximal Dissipative Operators

Let

On the Integral Representation of Spherical k-Regular Functions in Clifford Analysis

We study the null solutions of iterated applications of the spherical (Atiyah-Singer) Dirac operator on locally defined polynomial forms on the unit sphere of ; functions valued in the universal Clifford algebra , here called spherical k-regular functions. We construct the kernel functions, get the integral representation formula and Cauchy integral formula of spherical k-regular functions, and as applications, the weak solutions of higher order inhomogeneous spherical (Atiyah-Singer) Dirac equations . We obtain, in particular, the weak solution of an inhomogeneous spherical Poisson equation Δ _s g = f. This work was partially supported by NNSF of China (No.10471107) and RFDP of Higher Education (No.20060486001).     

Optimization by the Homogenization Method for Nonlinear Elliptic Dirichlet Systems

We study the existence of solutions of control problems relative to a nonlinear elliptic system with Dirichlet boundary conditions. In this problem, the control variables are the coefficients of the equations and the open set where they are posed. It is known that this class of problems has no solution in general, but using homogenization results about elliptic systems we show the existence of solutions when the controls are searched in a bigger set. These results are related to the selection of optimal materials and shapes.     

On Toeplitz-type Operators Related to Wavelets

Let G be the “ax + b”-group with the left invariant Haar measure dν and ψ be a fixed real-valued admissible wavelet on . The structure of the space of Calderón (wavelet) transforms inside is described. Using this result some representations, properties and the Wick calculus of the Calderón-Toeplitz operators T _α acting on whose symbols a = a(ζ) depend on for are investigated. This paper was supported by Grant VEGA 2/0097/08.     

On the Positive Parts of Second Order Symmetric Pseudodifferential Operators

Using microlocalization, the positive and the negative parts for a class of second order formally self-adjoint pseudodifferential operators are constructed.      

Resolvent Estimates for Operators Belonging to Exponential Classes

For a, α > 0 let E(a, α) be the set of all compact operators A on a separable Hilbert space such that s _n (A) = O(exp(-an^α)), where s _n (A) denotes the n-th singular number of A. We provide upper bounds for the norm of the resolvent (zI − A)⁻¹ of A in terms of a quantity describing the departure from normality of A and the distance of z to the spectrum of A. As a consequence we obtain upper bounds for the Hausdorff distance of the spectra of two operators in E(a, α).      

Hausdorff Dimension of Ruptures for Solutions of a Semilinear Elliptic Equation with Singular Nonlinearity

We consider the following semilinear elliptic equation with singular nonlinearity:

The Strong Variant of a Barbashin Theorem on Stability of Solutions for Non-autonomous Differential Equations in Banach Spaces

Among others we shall prove that an exponentially bounded evolution family U = {U(t, s)} _t≥s≥0 of bounded linear operators acting on a Banach space X is uniformly exponentially stable if and only if there exists q [1, ∞) such that

Meromorphic Factorization Revisited and Application to Some Groups of Matrix Functions

Some properties and applications of meromorphic factorization of matrix functions are studied. It is shown that a meromorphic factorization of a matrix function G allows one to characterize the kernel of the Toeplitz operator with symbol G without actually having to previously obtain a Wiener–Hopf factorization. A method to turn a meromorphic factorization into a Wiener–Hopf one which avoids having to factorize a rational matrix that appears, in general, when each meromorphic factor is treated separately, is also presented. The results are applied to some classes of matrix functions for which the existence of a canonical factorization is studied and the factors of a Wiener–Hopf factorization are explicitly determined. Submitted: April 15, 2007. Revised: October 26, 2007. Accepted: December 12, 2007.     

Representation of nonnegative convex polynomials

We provide a specific representation of convex polynomials nonnegative on a convex (not necessarily compact) basic closed semi-algebraic set . Namely, they belong to a specific subset of the quadratic module generated by the concave polynomials that define . Received: 15 December 2007     

Syzygies of Multi-Variable Higher Spin Dirac Operators on $$\mathbb{R}^{3}$$

This paper is a short report on the generalization of some results of our previous paper [12] to the case of spin j/2 Dirac operators in real dimension three for arbitrary odd integer j. We use an explicit formula for the local expression of such operators to study their algebraic properties, construct the compatibility conditions of the overdetermined system associated to the operator in several spatial variables, and we prove that its associated algebraic complex, dual do the BGG sequence coming from representation theory, has substantially the same pattern as the Cauchy-Fueter complex. The author is a member of the Eduard Čech Center and his research is supported by the relative grants.     

Some Nonlinear Elliptic Problems in Unbounded Domains

In this paper the results of some investigations concerning nonlinear elliptic problems in unbounded domains are summarized and the main difficulties and ideas related to these researches are described. The model problem

Some algebraic invariants related to mixed product ideals

We compute some algebraic invariants (e.g. depth, Castelnuovo-Mumford regularity) for a special class of monomial ideals, namely the ideals of mixed products. As a consequence, we characterize the Cohen-Macaulay ideals of mixed products. Received: 25 October 2007     

Xia Spectrum for Some Class of Operators

In this paper, we introduce Xia spectra of n-tuples of operators satisfying |T ²| ≥ U|T ²|U* for the polar decomposition of T = U|T| and we extend Putnam’s inequality to these tuples [7]. This research is partially supported by Grant-in-Aid Research No.17540176.     

The Hermitean Hilbert–Dirac Connection

Hermitean Clifford analysis is a recent branch of Clifford analysis, refining the Euclidean case; it focusses on the simultaneous null solutions, called Hermitean monogenic functions, of two complex Dirac operators which are invariant under the action of the unitary group. The specificity of the framework, introduced by means of a complex structure creating a Hermitean space, forces the underlying vector space to be even dimensional. Thus, any Hilbert convolution kernel in should originate from the non-tangential boundary limits of a corresponding Cauchy kernel in . In this paper we show that the difficulties posed by this inevitable dimensional jump can be overcome by following a matrix approach. The resulting matrix Hermitean Hilbert transform also gives rise, through composition with the matrix Dirac operator, to a Hermitean Hilbert–Dirac convolution operator “factorizing” the Laplacian and being closely related to Riesz potentials. Received: October, 2007. Accepted: February, 2008.