Partial regularity for stationary harmonic maps at a free boundary

Let and be smooth Riemannian manifolds, of the dimension n≥2 with nonempty boundary, and compact without boundary. We consider stationary harmonic maps u ∈ H¹(, ) with a free boundary condition of the type u(∂) ⊂ Γ, given a submanifold Γ⊂. We prove partial boundary regularity, namely (sing(u))=0, a result that was until now only known in the interior of the domain (see [B]). The key of the proof is a new lemma that allows an extension of u by a reflection construction. Once the partial regularity theorem is known, it is possible to reduce the dimension of the singular set further under additional assumptions on the target manifold and the submanifold Γ.     

Mean-value theorems for multiplicative functions on Beurling’s generalized integers (I)

Halász’s general mean-value theorem for multiplicative functions on ℕ is classical in probabilistic number theory. We extend this theorem to functions f, defined on a set of generalized integers associated with a set of generalized primes in Beurling’s sense, which satisfies Halász’s conditions, in particular,Assume that the distribution function N(x) of satisfieswith γ>γ₀, where ρ₁<ρ₂<···<ρ_m are constants with ρ_m≥1 and A₁,···,A_m are real constants with A_m>0. Also, assume that the Chebyshev function ψ(x) of satisfieswith M>M₀. Then the asymptoticimplieswhere τ is a positive constant with τ≥1 and L(u) is a slowly oscillating function with |L(u)|=1.     

A Springer Theorem for Hermitian forms

Throughout this paper, we let (D,σ) be a central F -division algebra with involution σ such that F^σ={d ∈ F|σ(d)=d} is a Henselian valued field. By [11], the valuation on F^σ extends uniquely to a valuation on D. We denote this valuation by v. Moreover, we assume that the characteristic of the residue field, , is not 2. If the valuation on F is discrete, then any quadratic form q can be written as q= q₁⊥πq₂, where π is a uniformizer and q_i are unit forms. Springer's Theorem states that q is isotropic if and only if at least one of the residue forms and is isotropic. In this paper we generalize this result to ɛ -Hermitian forms. In Section 4, we use the connection between involutions on algebras and ɛ-Hermitian forms to prove an analog of the Springer Theorem for involutions. This paper was part of the author's doctoral dissertation at New Mexico State University. The author wishes to thank his advisor Pat Morandi for his tireless help.     

Square functions and <Emphasis Type="Italic">H</Emphasis><Superscript>∞</Superscript> calculus on subspaces of <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> and on Hardy spaces

Let X be a (closed) subspace of L^p with 1≤p<∞, and let A be any sectorial operator on X. We consider associated square functions on X, of the form and we show that if A admits a bounded H^∞ functional calculus on X, then these square functions are equivalent to the original norm of X. Then we deduce a similar result when X=H¹(ℝ^N) is the usual Hardy space, for an appropriate choice of || ||_F. For example if N=1, the right choice is the sum for h ∈ H¹(ℝ), where H denotes the Hilbert transform.     

Complexity of nilpotent orbits and the Kostant-Sekiguchi correspondence

Let G be a connected linear semisimple Lie group with Lie algebra , and let be the complexified isotropy representation at the identity coset of the corresponding symmetric space. Suppose that Ω is a nilpotent G-orbit in and is the nilpotent -orbit in associated to Ω by the Kostant-Sekiguchi correspondence. We show that the corank of the Hamiltonian K-space Ω is twice the complexity of the variety .     

Density results for the <Emphasis Type="Italic">W</Emphasis><Superscript>1/2</Superscript> energy of maps into a manifold

We show that maps from Bⁿ to a smooth compact boundaryless manifold which are smooth out of a singular set of dimension n−2 are dense for the strong topology in W^1/2(Bⁿ,). We also prove that for n≥2 smooth maps from Bⁿ to are dense in W^1/2(Bⁿ,) if and only if π₁()=0, i.e. the first homotopy group of is trivial.     

Exceptional points of an endomorphism of the projective plane

Let f an endomorphism of of degree >1. We show how to obtain a bound (depending only on n) on the number of codimension-two subspaces in which are completely invariant for f (where L is completely invariant for f means that f^–1(L)=L set-theoretically).     

C k -estimates for the -equation on convex domains of finite type

For a bounded convex domain with C^∞ smooth boundary of finite type m and q=1, . . . ,n−1, we construct a -solving integral operator T*_q such that for all k ∈ ℕ and the usual C^k and -norms the operator is continuous.     

On the Rellich inequality with magnetic potentials

In lectures given in 1953 at New York University, Franz Rellich proved that for all f∈C₀^∞(Rⁿ \{0}) and n≠2where the constant C(n):=n²(n−4)²/16 is sharp. For n=2 extra conditions were required for f, and for n=4, C(4)=0, producing a trivial inequality. Influenced by recent work of Laptev-Weidl on Hardy-type inequalities in R², the authors show that for n≥2, the inclusion of a magnetic field B=curl(A) of Aharonov-Bohm type yields non-trivial Rellich-type inequalities of the formwhere Δ_A=(∇−iA)² is the magnetic Laplacian. As in the Laptev-Weidl inequality, the constant C(n,α) depends upon the distance of the magnetic flux to the integers Z. When the flux is an integer and α=0, the inequalities reduce to Rellich’s inequality.The first author gratefully acknowledges the hospitality and support of the Mathematics Department at UAB where much of this work was done.     

Continuity of eigenvalues of subordinate processes in domains

Let X={X_t,t≥0} be a symmetric Markov process in a state space E and D an open set of E. Let S⁽ⁿ⁾={S⁽ⁿ⁾_t, t ≥ 0} be a subordinator with Laplace exponent ϕ_n and S={S_t,t≥0} a subordinator with Laplace exponent ϕ. Suppose that X is independent of S and S⁽ⁿ⁾. In this paper we consider the subordinate processes and and their subprocesses and X^ϕ,D killed upon leaving D. Suppose that the spectra of the semigroups of and X^ϕ,D are all discrete, with being the eigenvalues of the generator of and being the eigenvalues of the generator of X^ϕ,D. We show that, if lim_n→∞ϕ_n(λ)=ϕ(λ) for every λ>0, then The research of this author is supported in part by NSF Grant DMS-0303310. The research of this author is supported in part by a joint US-Croatia grant INT 0302167.     

Area density and regularity for soap film-like surfaces spanning graphs

For a given boundary Γ in Rⁿ consisting of arcs and vertices, with two or more arcs meeting at each vertex, we treat the problem of estimating the area density of a soap film-like surface Σ spanning Γ. Σ is assumed locally to minimize area, or more generally, to be strongly stationary for area with respect to Γ. We introduce a notion of total curvature (Γ) for such graphs, or nets, Γ. We show that 2π times the area density of Σ at any point is less than or equal to (Γ). For n=3, these density estimates imply, for example, that if (Γ)≤3.649π, then the only possible singularities of a piecewise smooth (M,0,δ)-minimizing set Σ are curves, along which three smooth sheets of Σ meet with equal angles of 120^°. Second author supported in part by NSF grant 00-71862.     

Non-abelian class field theory for arithmetic surfaces

Let be a regular arithmetic surface. Assume that for all irreducible curves there are given open normal subgroups of ₁(C), which fulfill a compatibility condition at all closed points x . We then show that these data uniquely determine a normal subgroup of ₁(). This is used to construct abelian class field theory for arithmetic surfaces using only K₀ and K₁ groups of local and global fields.Mathematics Subject Classification (2000): 14G40, 11R37, 14H25     

Isoperimetric inequalities and random walks on quotients of graphs and buildings

Let be a Euclidean or hyperbolic building and let GAut be a locally compact unimodular group, which acts strongly transitively on . We use graphs , quasi-isometric to , to study asymptotic properties of quotients , where is a discrete subgroup of G. If G has Kazhdans property (T) we show that such quotients satisfy strong isoperimetric inequalities. This yields new examples of graphs with positive Cheeger constant. Such graphs cannot be bi-Lipschitz embedded into Hilbert space. Moreover, simple random walks on such quotients are shown to be recurrent if and only if is a uniform lattice in G.Mathematics Subject Classification (1991): 11E95, 22E40, 22E50, 51E24, 60G50in final form: 10 October 2003     

Order and algebra isomorphisms of spaces of regular operators

If E and F are real Banach lattices and there is an algebra and order isomorphism Φ:(E) →(F) between their respective ordered Banach algebras of regular operators then there is a linear order isomorphism U:E→F such that Φ(T) =UTU⁻¹ for all T ∈ (E).     

Global -homotopy with <Emphasis Type="Italic">C</Emphasis><Superscript><Emphasis Type="Italic">k</Emphasis></Superscript> estimates for a family of compact,regular <Emphasis Type="Italic">q</Emphasis>-pseudoconcave CR manifolds

Let M₀ be a compact, regular q-pseudoconcave CR submanifold of a complex manifold G and - a holomorphic vector bundle on G such that dim for some fixed r<q. We prove a global homotopy formula with C^k estimates for r-cohomology of on arbitrary CR submanifold M close enough to M₀.Mathematics Subject Classification (2000):32F20, 32F10in final form: 15 October 2003     

Orlicz capacities and Hausdorff measures on metric spaces

In the setting of doubling metric measure spaces with a 1-Poincaré inequality, we show that sets of Orlicz Φ-capacity zero have generalized Hausdorff h-measure zero provided thatwhere Θ⁻¹ is the inverse of the function Θ(t)=Φ(t)/t, and s is the “upper dimension” of the metric measure space. This condition is a generalization of a well known condition in Rⁿ. For spaces satisfying the weaker q-Poincaré inequality, we obtain a similar but slightly more restrictive condition. Several examples are also provided.     

On the exact Hausdorff dimension of the set of Liouville numbers. II

Let denote the set of Liouville numbers. For a dimension function h, we write for the h-dimensional Hausdorff measure of . In previous work, the exact ``cut-point' at which the Hausdorff measure of drops from infinity to zero has been located for various classes of dimension functions h satisfying certain rather restrictive growth conditions. In the paper, we locate the exact ``cut-point' at which the Hausdorff measure of drops from infinity to zero for all dimension functions h. Namely, if h is a dimension function for which the function increases faster than any power function near 0, then , and if h is a dimension function for which the function increases slower than some power function near 0, then . This provides a complete characterization of all Hausdorff measures of without assuming anything about the dimension function h, and answers a question asked by R. D. Mauldin. We also show that if then does not have σ-finite measure. This answers another question asked by R. D. Mauldin. This work was done while Dave L. Renfro was at the Department of Mathematics at Central Michigan University.     

Hausdorff measures, capacities and compact composition operators

It is shown that there exist analytic self-maps ϕ of the unit disc inducing compact composition operators on the Hardy space , 1 ≤ p < ∞ such that the Hausdorff dimension of the set is one; sharpening a classical result due to Schwartz. Moreover, the same holds in the weighted Dirichlet spaces with 0 < α < 1. As a consequence, we deduce that there exist symbols ϕ inducing compact composition operators on such that the α-capacity of E_ϕ is positive, which is no longer true for those just inducing Hilbert-Schmidt composition operators on . First author is partially supported by Plan Nacional I+D grant no. BFM2003-00034, and Gobierno de Aragón research group Análisis Matemático y Aplicaciones, ref. DGA E-64 . Second author is partially supported by Plan Nacional I+D grant no. BFM2002-00571 and Junta de Andalucía RNM-314.     

Torus actions on homotopy complex projective spaces

We prove that an effective action of a torus T on a homotopy P^m is linear if m < 4 rk(T) – 1. Examples show that the bound is optimal. Combining this with a theorem of Hattori we conclude that the total Pontrjagin class of such a manifold is given by the usual formula (1 + x²)^{m + 1}.The second named author is an Alfred P. Sloan Research Fellow and was partly supported by an NSF grant.in final form: 10 May 2003     

Problèmes d'extension dans les domaines convexes de type fini

Résumé Soient donnés D un domaine borné de , convexe, de type fini, X une hypersurface complexe telle que soit non vide, connexe et transverse et . Nous nous intéressons au problème suivant. Sous quelle(s) condition(s) existe-t-il telle que Φ|X = φ. Nous allons donner une condition nécessaire à l'existence de l'extension Φ puis sous une condition un peu plus forte nous montrerons la continuité d'un opérateur d'extension de type Berndtsson-Andersson.