Riesz means for the sublaplacian on the Heisenberg group

The uniform boundedness of the Riesz means for the sublaplacian on the Heisenberg groupH ⁿ is considered. It is proved thatS _R ^α are uniformly bounded onL ^p(Hⁿ) for 1≤p≤2 provided α>α(p)=(2n+1)[(1/p)−(1/2)].     

A combinatorial proof of the Borsuk-Ulam antipodal point theorem

We give a proof of Tucker’s Combinatorial Lemma that proves the fundamental nonexistence theorem: There exists no continuous map fromB ⁿ toS ^{n − 1} that maps antipodal points of∂B ⁿ to antipodal points ofS ^{n − 1}.     

Stable ℱ-harmonic maps between Finsler manifolds

In this paper, we derive the first and second variation formulas for JC-harmonic maps between Finsler manifolds, and when F″≤ 0 and n ≥ 3, we prove that there is no nondegenerate stable F-harmonic map between a Riemannian unit sphere Sn and any compact Finsler manifold.     

On the Construction of <Emphasis Type="Italic">p</Emphasis>-Harmonic Morphisms and Conformal Actions

We produce p-harmonic morphisms by conformal foliations and Clifford systems. First, we give a useful criterion for a foliation on an m-dimensional Riemannian manifold locally generated by conformal fields to produce p-harmonic morphisms. By using this criterion we manufacture conformal foliations, with codimension not equal to p, which are locally the fibres of p-harmonic morphisms. Then we give a new approach for the construction of p-harmonic morphisms from R^m／{0} to R^n. By the well-known representation of Clifford algebras, we find an abundance of the new 2/3 （m ＋ 1）-harmonic morphism Ф： R^m／{0} → R^n where m = 2κδ（n - 1）.     

BOUNDEDNESS OF VECTOR-VALUED CALDERON-ZYGMUND OPERATORS ON HERZ SPACES WITH NON-DOUBLING MEASURES

In the paper we obtain vector-valued inequalities for Calderon-Zygmund operator,simply CZO On Herz space and weak Herz space.In particular,we obtain vector-valued inequalities for CZO on Lq(Rd,│x│αdμ)space,with 1＜q＜∞,-n＜α＜n(q-1),and on L1,∞(Rd,│x│αdμ)space,with -n＜α＜0.     

The minimality of the map \frac{x}{\|{x}\|} for weighted energy

In this paper, we investigate the minimality of the map from the Euclidean unit ball Bⁿ to its boundary 핊ⁿ⁻¹ for weighted energy functionals of the type E_p,f = ∫_Bⁿ f(r)‖∇ u‖^p dx, where f is a non-negative function. We prove that in each of the two following cases:

On the chromatic polynomial of a graph

Let P(G,λ) be the chromatic polynomial of a graph G with n vertices, independence number α and clique number ω. We show that for every λ≥n, ()^α≤≤ () ^{n −ω}. We characterize the graphs that yield the lower bound or the upper bound.?These results give new bounds on the mean colour number μ(G) of G: n− (n−ω)() ^{n −ω}≤μ(G)≤n−α() ^α. Received: December 12, 2000 / Accepted: October 18, 2001?Published online February 14, 2002     

Carleson measures for the area Nevanlinna spaces and applications

Let 1 ≤ p < ∞ and let μ be a finite positive Borel measure on the unit disk D. The area Nevanlinna-Lebesgue space N ^p (μ) consists of all measurable functions h on D such that log⁺ |h| ∈ L ^p (μ), and the area Nevanlinna space N _α ^p is the subspace consisting of all holomorphic functions, in N ^p ((1−|z|²)^α dv(z)), where α > −1 and ν is area measure on D. We characterize Carleson measures for N _α ^p , defined to be those measures μ for which N _α ^p ⊂ N ^p (μ). As an application, we show that the spaces N _α ^p are closed under both differentiation and integration. This is in contrast to the classical Nevanlinna space, defined by integration on circles centered at the origin, which is closed under neither. Applications to composition operators and to integral operators are also given. The second author was supported in part by KRF-2004-015-C00019.     

Convergence of an implicit, constraint preserving finite element discretization of p-harmonic heat flow into spheres

We propose an implicit discretization of the p-harmonic map heat flow into the sphere S ² that enjoys a discrete energy inequality and converges under only a mild mesh constraint to a weak solution. A fully practical iterative scheme that approximates the solution of the nonlinear system of equations in each time step is proposed and analyzed. Computational studies to motivate possible finite-time blow-up behavior of solutions for p ≠ 2 are included. Supported by Deutsche Forschungsgemeinschaft through the DFG Research Center Matheon “Mathematics for key technologies” in Berlin.     

On the convex approximation property and the asymptotic behavior of nonlinear contractions in Banach spaces

We prove that ifC is a bounded closed convex subset of a uniformly convex Banach space,T:C→C is a nonlinear contraction, andS _n =(I+T+…+T ⁿ⁻¹ )/n, then lim _n ‖S _n (x)−TS _n (x)‖=0 uniformly inx inC. T also satisfies an inequality analogous to Zarantonello’s Hilbert space inequality. which permits the study of the structure of the weak ω-limit set of an orbit. These results are valid forB-convex spaces if some additional condition is imposed on the mapping. Partially supported by NSF Grant MCS-7802305A01.     

Conjecture of Li and Praeger concerning the isomorphisms of Cayley graphs of A5

LetG be a finite group, andS a subset ofG \ |1| withS =S ⁻¹. We useX = Cay(G,S) to denote the Cayley graph ofG with respect toS. We callS a Cl-subset ofG, if for any isomorphism Cay(G,S) ≈ Cay(G,T) there is an α∈ Aut(G) such thatS ^α =T. Assume that m is a positive integer.G is called anm-Cl-group if every subsetS ofG withS =S ⁻¹ and | S | ≤m is Cl. In this paper we prove that the alternating groupA ₅ is a 4-Cl-group, which was a conjecture posed by Li and Praeger.     

Pressure and equilibrium states for countable state markov shifts

We give a general definition of the topological pressureP _top (f, S) for continuous real valued functionsf: X→ℝ on transitive countable state Markov shifts (X, S). A variational principle holds for functions satisfying a mild distortion property. We introduce a new notion of Z-recurrent functions. Given any such functionf, we show a general method how to obtain tight sequences of invariant probability measures supported on periodic points such that a weak accumulation pointμ is an equilibrium state forf if and only if εf ⁻ dμ<∞. We discuss some conditions that ensure this integrability. As an application we obtain the Gauss measure as a weak limit of measures supported on periodic points.     

The Dissipative Quasi-geostrophic Equation in Weak Morrey Spaces

The author studies the Cauchy problem of the dissipative quasi-geostrophic equation in weak Morrey spaces. The global well-posedness is established for any small initial data in the weak space Mp^＊,γ（R^n）, with 1〈p〈∞and A = n-（2α-1）p, and for a small external force in a time-weighted weak Morrey space.     

Partial regularity for weak solutions of a nonlinear elliptic equation

For scalar non-linear elliptic equations, stationary solutions are defined to be critical points of a functional with respect to the variations of the domain. We consideru a weak positive solution of −Δu=u ^α in -Δu=u ^α in Ω ⊂ ℝ ⁿ , which is stationary. We prove that the Hausdorff dimension of the singular set ofu is less thann−2α+1/α−1, if α≥n+2/n−2.     

The continuity of the inversion and the structure of maximal subgroups in countably compact topological semigroups

We search for conditions on a countably compact (pseudocompact) topological semigroup under which: (i) each maximal subgroup H(e) in S is a (closed) topological subgroup in S; (ii) the Clifford part H(S) (i.e. the union of all maximal subgroups) of the semigroup S is a closed subset in S; (iii) the inversion inv: H(S) → H(S) is continuous; and (iv) the projection π: H(S) → E(S), π: x ↦ xx ⁻¹, onto the subset of idempotents E(S) of S, is continuous.      

Some estimates related to fractal measures and Laplacians on manifolds

Let Δ be the Laplace-Beltrami operator on ann dimensional completeC ^∞ manifoldM In this paper we establish an estimate ofe ^tΔ (dμ) valid for allt>0 wheredμ is a locally uniformly α dimensional measure onM 0≤α≤n The result is used to study the mapping properties of (I-tΔ)^-β considered as an operator fromL ^p (M dμ) toL ^p (M dx) wheredx is the Riemannian measure onM β>(n−α)/2p′ 1/p+1/p′=1 1≤p≤∞     

On the pontryagin-steenrod-wu theorem

We present a short and direct proof (based on the Pontryagin-Thom construction) of the following Pontryagin-Steenrod-Wu theorem: (a) LetM be a connected orientable closed smooth (n + 1)-manifold,n≥3. Define the degree map deg: π ⁿ (M) →H ⁿ (M; ℤ) by the formula degf =f*[S ⁿ ], where [S ⁿ ] εH ⁿ (M; ℤ) is the fundamental class. The degree map is bijective, if there existsβ εH ₂(M, ℤ/2ℤ) such thatβ ·w ₂(M) ε 0. If suchβ does not exist, then deg is a 2-1 map; and (b) LetM be an orientable closed smooth (n+2)-manifold,n≥3. An elementα lies in the image of the degree map if and only ifρ ₂ α ·w ₂(M)=0, whereρ ₂: ℤ → ℤ/2ℤ is reduction modulo 2.     

UNIFORM CONVERGENCE OF CESARO MEANS OF NEGATIVE ORDER OF DOUBLE TRIGONOMETRIC FOURIER SERIES

In this paper we prove that iff ∈ C([-π,π]2) and the function f is bounded partial p-variation for some p ∈ [1, ∞), then the double trigonometric Fourier series of a function f is uniformly (C;-α,-β) summable (α β＜ 1/p,α,β＞ 0) in the sense of Pringsheim. If α β≥ 1/p, then there exists a continuous function f0 of bounded partial double trigonometric Fourier series of fo diverge over cubes.     

L p − Lq estimates for convolution operators with n-Dimensional singular measures

Let S ⊂ ℜⁿ⁺¹ be the graph of the function ϕ :[−1, 1] ⁿ → ℜ defined by ϕ (x ₁ , …, x_n) = ∑ _j=1 ⁿ |x_j|^αj, with1<α ₁ ≤ … ≤ α_n, let σ the Euclidean area measure on S. In this article we study the L^p − L^q boundedness of convolution operators with the singular Borel measure on Rⁿ⁺¹ given by μ (E)=σ (E ∩ S)     

Onl p-complemented copies in Orlicz spaces

Let 1<α≦β<∞ andF be an arbitrary closed subset of the interval [α,β]. An Orlicz sequence spacel ^φ (resp. an Orlicz function spaceL ^φ(μ)) with associated indices α and β is found in such a way that the set of valuesp for which thel ^p-space is isomorphic to a complemented subspace ofl ^φ (resp.L ^φ(μ)) is precisely the given setF (resp.F ∪ {2}). Also, a recent result of Hernández and Peirats [1] is extended showing that, even for the case in which the indices satisfy α_φ ^∞<2<β_φ ^∞, there exist minimal Orlicz function spacesL ^φ(μ) with no complemented copy ofl ^p for anyp ≠ 2. Supported in part by CAICYT grant 0338-84.