Estimates for harmonic conjugate functions and Riesz-system

In this paper we generalize a theorem of Hardy and Littlewood concerning the growth of conjugate harmonic functions. Moreover, we obtain an estimate for Riesz systems in the upper half-space R ₊ ⁿ⁺¹ of R ⁿ⁺¹.     

On the Boundary Behaviour of Poisson Integrals of Finite Measures in (n + 1)-dimensional Half Space

It is well known that the harmonic functions u on the Euclidean upper (n + 1)-dimensional half-space E⁺ _n+1 = {(x, y) = (x ₁,…,x_n,y) ? E _n+1: y > 0} satisfying sup _y>0 ||u(·,y)||₁ < ∞ are precisely the Poisson-integrals u(x,y) = ∫ _En P(x - t,y)dμ(t) with respect to a measure μ of finite variation on E_n , and that (Fatou's theorem in E⁺ _n+1) in almost every point x ? E_n the non-tangential boundary limit of u exists and coincides with du/dλ. While this is a special case of a general assertion in potential theory, it is shown that the proof of Fatou's theorem for harmonic functions on a ball may readily be transferred to the given setup and that the influence of a singular component of μ on the boundary behaviour of u may also be established without recourse to the existence of the derivative dμ/dλ. Finally the C ₀-property of u is characterized by suitable conditions on μ.     

Yet another variation on the theme of decomposition of transvections

The unipotent decomposition method consists in representing elementary matrices as products of factors belonging to proper parabolic subgroups whose images under endomorphisms (e.g., conjugations) remain in proper parabolic subgroup. For the complete linear group, this method was suggested in 1987 by Stepanov, who applied it to simplify the proof of Souslin’s normality theorem. Soon after this, Vavilov and Plotkin transferred the method to other classical groups and the Chevalley groups. Since then, many results in the same spirit have been obtained. The paper suggests yet another variation on this theme. Namely, let R be a commutative ring with identity, and let g ∈ GL(n, R), where n ≥ 4. Then, the elementary group E(n, R) is generated by transvections e + uv, where u ∈ R ⁿ , v ∈ ⁿ R, and vu = 0, such that v, gu, and vg ^?1 have at least one zero component each. This result is related to a simplified proof of theorems of Waterhouse, Golubchik, Mikhalev, Zel’manov, and Petechuk about the automorphisms of the complete linear group being standard, which uses unipotent elements.     

L 2 Harmonic 1-Forms on Minimal Submanifolds in Spheres

We study a complete noncompact minimal submanifold M ⁿ in a sphere S ^n+p . We prove there is no nontrivial L ² harmonic 1-form and at most one nonparabolic end on M if the total curvature is bounded from above by a constant depending only on n. The rigidity theorem is a generalized version of Ni’s, Yun’s and the second author’s results on submanifolds in Euclidean spaces and Seo’s result on minimal submanifolds in hyperbolic spaces.     

Representations for eigenfunctions of expected value operators in the Wishart distribution

Recent articles by Kushner and Meisner (1980) and Kushner, Lebow and Meisner (1981) have posed the problem of characterising the ‘EP’ functions f(S) for which Ef(S) for which E(f(S)) = λ_nf(Σ) for some λ_n ? $\text{R}$, whenever the m × m matrix S has the Wishart distribution W(m, n, Σ). In this article we obtain integral representations for all nonnegative EP functions. It is also shown that any bounded EP function is harmonic, and that EP polynomials may be used to approximate the functions in certain L^p spaces.     

Sets of Harmonicity for Finely Harmonic Functions

Given an open set U in R ⁿ (n3) and a dense open subset V of U, it is shown that there is a finely harmonic function u on U such that V is the largest open subset of U on which u is harmonic. This result, which establishes the sharpness of a theorem of Fuglede, is obtained following a consideration of fine cluster sets of arbitrary functions.     

A Note on Commutators with Power Central Values on Lie Ideals

This note proves that, if R is a prime ring of characteristic 2 with d a derivation of R and L a noncentral Lie ideal of R such that [d（u）,u]^n is central, for all u ∈ L, then R must satisfy s4, the standard identity in 4 variables. The case where R is a semiprime ring is also examined by the authors. The results of the note improve Carini and Filippis＇s results.     

Eigenschaften der Lösungsmenge eines Systems von Volterra-Integralgleichungen

By a well known theorem of H. Kneser [4] the set U of all solutions of the initial value problem $$u' = f(x,u)forx\varepsilon [0,a],u(0) = u_o $$ has the following property: If f is continuous and bounded then U(x₀)={u(x₀): u∈U} is a continuum (i.e. a compact and connected subset) for every x₀∈[0,a]. In the present paper we claim to extend this theorem to a system of Volterra integral equations in several variables of the form x∈B∞R^m, ν=1,...,n that had been investigated in [8]. In fact we shall prove that U is a continuum of the Banach space Cⁿ(B) of all ‘vector functions’ u(x)=(u₁(x),...,u_n(x)), continuous on B. It is an immediate consequence from this that U(x₀) is a continuum of Rⁿ. These results will be established by the help of a suitable modification of a method used by M. Müller [5] to prove Kneser's theorem. Especially, we obtain new theorems for some initial value problems for hyperbolic equations.     

Square area integral estimates for subharmonic functions in NTA domains

In this paper, we prove a good-λ inequality between the nontangential maximal function and the square area integral of a subharmonic functionu in a bounded NTA domainD inR ⁿ . We achieve this by showing that a weighted Riesz measure ofu is a Carleson measure, with the Carleson norm bounded by a constant independent ofu. As consequences of the good-λ inequality, we obtain McConnell-Uchiyama's inequality and an analogue of Murai-Uchiyama's inequality for subharmonic functions inD.     

Density rate of a set, application to rectifiability results for measurable jets

In this work, we present some new results concerning the sets of h-density points and their application in the context of the Whitney extension theorem. In particular, we establish a simple relation between the amount of density of a set E at x and the property that u φ _E has derivative of order h at x in the L ^p sense whenever u is continuous at x. Moreover we prove Whitney-type rectifiability results for measurable jets restricted to sets of high density points in its domain. It’s worth mentioning the case when the domain is a locally finite perimeter set.     

Oscillatory radial solutions for subcritical biharmonic equations

It is well known that the biharmonic equation Δ²u=u|u|^p−1 with p∈(1,∞) has positive solutions on Rn if and only if the growth of the nonlinearity is critical or supercritical. We close a gap in the existing literature by proving the existence and uniqueness, up to scaling and symmetry, of oscillatory radial solutions on Rn in the subcritical case. Analyzing the nodal properties of these solutions, we also obtain precise information about sign-changing large radial solutions and radial solutions of the Dirichlet problem on a ball.     

On univalence and P-matrices

Suppose F is a differentiable mapping from a rectangle R?Eⁿ into Eⁿ. Gale and Nikaido proved that if the Jacobian of F is a P-matrix in R, then F is univalent in R. Their paper has served as the basis of numerous results on univalence. Recently H. Scarf conjectured a significant extension: that the Jacobian of F need not be a P-matrix everywhere in the rectangle R, but merely on its boundary. This paper proves Scarf's conjecture, and to do so utilizes a conceptually different method of proof than that of Gale and Nikaido. The proof is presented in such a way as to demonstrate a suggestion of Scarf that orientation arguments may provide an alternative proof of the Gale-Nikaido theorem.     

On the dependence of the reflection operator on boundary conditions for biharmonic functions

The biharmonic equation arises in areas of continuum mechanics including linear elasticity theory and the Stokes flows, as well as in a radar imaging problem. We discuss the reflection formulas for the biharmonic functions u(x,y)∈R² subject to different boundary conditions on a real-analytic curve in the plane. The obtained formulas, generalizing the celebrated Schwarz symmetry principle for harmonic functions, have different structures. In particular, in the special case of the boundary, Γ₀:={y=0}, reflections are point-to-point when the given on Γ₀ conditions are u=n_∂u=0, u=Δu=0 or n_∂u=n_∂Δu=0, and point to a continuous set when u=n_∂Δu=0 or n_∂u=Δu=0 on Γ₀.     

A new proof of the theorem: Harmonic manifolds with minimal horospheres are flat

In this note we reprove the known theorem: Harmonic manifolds with minimal horospheres are flat. It turns out that our proof is simpler and more direct than the original one. We also reprove the theorem: Ricci flat harmonic manifolds are flat, which is generally affirmed by appealing to Cheeger–Gromov splitting theorem. We also confirm that if a harmonic manifold M has same volume density function as ? ⁿ , then M is flat.     

Subharmonicity of the modulus of quasiregular harmonic mappings

In this note we determine all numbers q∈R such that q|u| is a subharmonic function, provided that u is a K-quasiregular harmonic mappings in an open subset Ω of the Euclidean space Rn.     

Idempotent elements determined matrix algebras

Let M_n(R) be the algebra of all n×n matrices over a unital commutative ring R with 2 invertible, V be an R-module. It is shown in this article that, if a symmetric bilinear map {·,·} from M_n(R)×M_n(R) to V satisfies the condition that {u,u}={e,u} whenever u²=u, then there exists a linear map f from M_n(R) to V such that . Applying the main result we prove that an invertible linear transformation θ on M_n(R) preserves idempotent matrices if and only if it is a Jordan automorphism, and a linear transformation δ on M_n(R) is a Jordan derivation if and only if it is Jordan derivable at all idempotent points.     

Multiple walsh series and Zygmund sets

The classical Zygmund theorem claims that, for any sequence of positive numbers {? _n } monotonically tending to zero and any δ > 0, there exists a set of uniqueness for the class of trigonometric serieswhose coefficients aremajorized by the sequence {? _n } whosemeasure is greater than 2π ?δ. In this paper, we prove the analog of Zygmund’s theorem for multiple series in the Walsh system on whose coefficients rather weak constraints are imposed.     

Splitting theorem, Poincaré-Hopf theorem and jumping nonlinear problems

In this paper, we establish a Gromoll-Meyer splitting theorem and a shifting theorem for J∈C^2-0(E,R) and by using the finite-dimensional approximation, mollifiers and Morse theory we generalize the Poincaré-Hopf theorem to J∈C¹(E,R) case. By combining the Poincaré-Hopf theorem and the splitting theorem, we study the existence of multiple solutions for jumping nonlinear elliptic equations.