The first <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-cohomology of some groups with one end

Let p be a real number greater than one. In this paper we study the vanishing and nonvanishing of the first L ^p -cohomology space of some groups that have one end. We also make a connection between the first L ^p -cohomolgy space and the Floyd boundary of the Cayley graph of a group. We apply the result about Floyd boundaries to show that there exists a real number p such that the first L ^p -cohomology space of a nonelementary hyperbolic group does not vanish. Received: 4 August 2006 Revised: 2 November 2006     

Weighted L<Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-inequalities for multi-parameter Riesz type potentials and strong fractional maximal operators

We prove weighted L ^p -inequalities for multi-parameter Riesz type potentials, strong fractional maximal operators and their dyadic counterparts. Our proofs avoid the Good-λ inequalities used earlier in the R ^m -case and are based on our integrated multi-parameter summation by parts lemma, that might be of independent interest.     

An analogue of the Nielsen-Schreier formula for pro-<Emphasis Type="Italic">p</Emphasis>-groups

We introduce a condition analogous to the Nielsen-Schreier formula, and investigate basic properties of pro-p-groups satisfying the condition. Partly supported by the Grant-in Aid for Scientific Research (C), Japan Society for the Promotion of Science. Received: 23 January 2006     

<Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript>-Betti numbers of one-relator groups

We determine the L ²-Betti numbers of all one-relator groups and all surface-plus-one-relation groups. We also obtain some information about the L ²-cohomology of left-orderable groups, and deduce the non-L ² result that, in any left-orderable group of homological dimension one, all two-generator subgroups are free. Warren Dicks was Funded by the DGI (Spain) through Project BFM2003-06613.     

<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-maximal regularity for second order Cauchy problems

We introduce the concept of L^p-maximal regularity for second order Cauchy problems. We prove L^p-maximal regularity for an abstract model problem and we apply the abstract results to prove existence, uniqueness and regularity of solutions for nonlinear wave equations. The author acknowledges with thanks the support provided by the Department ofApplied Analysis, University of Ulm, and the travel grants provided by NBMH India and MSF Delhi, India.     

The Structure of the Closure of the Rational Functions in <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">q</Emphasis></Superscript>(μ)

Let K be a compact subset in the complex plane and let A(K) be the uniform closure of the functions continuous on K and analytic on K°. Let μ be a positive finite measure with its support contained in K. For 1 ≤ q < ∞, let A^q(K, μ) denote the closure of A(K) in L^q(μ). The aim of this work is to study the structure of the space A^q(K, μ). We seek a necessary and sufficient condition on K so that a Thomson-type structure theorem for A^q(K, μ) can be established. Our theorem deduces J. Thomson’s structure theorem for P^q(μ), the closure of polynomials in L^q(μ), as the special case when K is a closed disk containing the support of μ.     

<Emphasis Type="Bold">C</Emphasis><Superscript><Emphasis Type="Bold">1,</Emphasis><Emphasis Type="Italic">α</Emphasis></Superscript> regularity for infinity harmonic functions in two dimensions

We propose a new method for showing C ^{1, α} regularity for solutions of the infinity Laplacian equation and provide full details of the proof in two dimensions. The proof for dimensions n ≥ 3 depends upon some conjectured local gradient estimates for solutions of certain transformed PDE. LCE is supported in part by NSF Grant DMS-0500452. OS was supported in part by the Miller Institute for Basic Research in Science, Berkeley.     

Viscosity solutions of the <Emphasis Type="Italic">p</Emphasis>-Laplace equation with nonlinear source

In the present paper we consider the Dirichlet problem in a convex domain for the multidimensional p-Laplace equation with nonlinear source. We prove the existence of the unique continuous viscosity solution under quite general assumptions on the structure of the source. Received: 18 January 2006     

Spectral flexibility of symplectic manifolds <Emphasis Type="Italic">T</Emphasis><Superscript>2</Superscript> × <Emphasis Type="Italic">M</Emphasis>

We consider Riemannian metrics compatible with the natural symplectic structure on T ² × M, where T ² is a symplectic 2-torus and M is a closed symplectic manifold. To each such metric we attach the corresponding Laplacian and consider its first positive eigenvalue λ₁. We show that λ₁ can be made arbitrarily large by deforming the metric structure, keeping the symplectic structure fixed. The conjecture is that the same is true for any symplectic manifold of dimension ≥ 4. We reduce the general conjecture to a purely symplectic question.     

<Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript>-minimization methods for Hamilton–Jacobi equations: the one-dimensional case

A new approximation technique based on L ¹-minimization is introduced. It is proven that the approximate solution converges to the viscosity solution in the case of one-dimensional stationary Hamilton–Jacobi equation with convex Hamiltonian. This material is based upon work supported by the National Science Foundation grant DMS-0510650. J.-L. Guermond is on leave from LIMSI, UPRR 3251 CNRS, BP 133, 91403 Orsay Cedex, France.     

The obstacle problem for nonlinear elliptic equations with variable growth and <Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript>-data

The aim of this paper is twofold: to prove, for L ¹-data, the existence and uniqueness of an entropy solution to the obstacle problem for nonlinear elliptic equations with variable growth, and to show some convergence and stability properties of the corresponding coincidence set. The latter follow from extending the Lewy-Stampacchia inequalities to the general framework of L ¹. Current address: Manel Sanchón, Universitat de Barcelona, Departament de Matemàtica Aplicada i Anàlisi, Gran Via 585, 08007 Barcelona, Spain; e-mail: msanchon@maia.ub.es Authors’ addresses: J. F. Rodrigues, CMUC, Department of Mathematics, University of Coimbra, and FCUL/Universidade de Lisboa, Av. Prof. Gama Pinto 2, 1649-003 Lisboa, Portugal; M. Sanchón and J. M. Urbano, CMUC, Department of Mathematics, University of Coimbra, 3001-454 Coimbra, Portugal     

Global regularity for divergence form elliptic equations on quasiconvex domains

In this paper, we introduce a notion of quasiconvex domain, and show that the global W1,p regularity holds on such domains for a wide class of divergence form elliptic equations. The modified Vitali covering lemma, compactness method and the maximal function technique are the main analytical tools.     

Extensions of Well-Boundedness and <Emphasis Type="Italic">C</Emphasis><Superscript><Emphasis Type="Italic">m</Emphasis></Superscript>-Scalarity

We consider some extensions of well-boundedness and C^m-scalarity by using fractional calculus, and prove some theorems accordingly. These results are applied to the usual Laplacian on Rⁿ and sub-Laplacians on nilpotent Lie groups. The research of first and second authors has been partly supported by the Project MTM2004-03036 of the M.C.YT.-DGI/F.E.D.E.R., Spain, and the Project E-12/25, D. G. Aragón, Spain. Part of the research of second author was developed in the Christian-Albrechts Universit?t in Kiel, while he was enjoying a HARP-postdoctoral position in the European Harmonic Analysis Network, HARP, IHP 2002-06.     

Stable Grothendieck polynomials and <Emphasis Type="Italic">K</Emphasis>-theoretic factor sequences

We formulate a nonrecursive combinatorial rule for the expansion of the stable Grothendieck polynomials of Fomin and Kirillov (Proc Formal Power Series Alg Comb, 1994) in the basis of stable Grothendieck polynomials for partitions. This gives a common generalization, as well as new proofs of the rule of Fomin and Greene (Discret Math 193:565–596, 1998) for the expansion of the stable Schubert polynomials into Schur polynomials, and the K-theoretic Grassmannian Littlewood–Richardson rule of Buch (Acta Math 189(1):37–78, 2002). The proof is based on a generalization of the Robinson–Schensted and Edelman–Greene insertion algorithms. Our results are applied to prove a number of new formulas and properties for K-theoretic quiver polynomials, and the Grothendieck polynomials of Lascoux and Schützenberger (C R Acad Sci Paris Ser I Math 294(13):447–450, 1982). In particular, we provide the first K-theoretic analogue of the factor sequence formula of Buch and Fulton (Invent Math 135(3):665–687, 1999) for the cohomological quiver polynomials.     

Pseudodifferential operators on<Emphasis Type="Italic">SU</Emphasis>(2)

We construct an algebra of left-invariant pseudodifferential operators on SU(2). We require only that the symbols be homogeneous and C². For Fourier-bandlimited symbols, we derive the expected formulae for composition and commutators and construct an orthonormal basis of common approximate eigenvectors that could be used to study spectral theory. Some remarks on applications to matrices of operators are made.     

Time-dependent diffusion operators on <Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript>

We study the Cauchy problem for time-dependent diffusion operators with singular coefficients on L¹-spaces induced by infinitesimal invariant measures. We give sufficient conditions on the coefficients such that the Cauchy-Problem is well-posed. We construct associated diffusion processes with the help of the theory of generalized Dirichlet forms. We apply our results in particular to construct a large class of Nelson-diffusions that could not been constructed before.     

Connected components in the space of composition operators on<Emphasis Type="Italic">H∞</Emphasis> functions of many variables

LetE be a complex Banach space with open unit ballB _e. The structure of the space of composition operators on the Banach algebra H∞, of bounded analytic functions onB _e with the uniform topology, is studied. We prove that the composition operators arising from mappings whose range lies strictly insideB _e form a path connected component. WhenE is a Hilbert space or aC _o(X)- space, the path connected components are shown to be the open balls of radius 2. The research of this author was supported by grant number SAB1999-0214 from the Ministerio de Educación, Cultura y Deporte during his stay at the Universidad de Valencia. The research of this author was partially supported DGES(Spain) pr. 96-0758. The research of this author was partially supported by Magnus Ehrnrooths stiftelse.     

Countably generated prime ideals in <Emphasis Type="Italic">H</Emphasis><Superscript>∞</Superscript>

We confirm a twenty year old conjecture by showing that a nonzero prime ideal P in the algebra H^∞ of bounded analytic functions in the open unit disk is countably generated if and only if it is either a principal ideal generated by the polynomial z−z₀, |z₀|<1, or if P is generated by the n-th roots of an atomic inner function. The case of the algebra H^∞+C is also dealt with. Dedicated to the 70th birthday of Joseph Cima Research supported by the RIP-program Oberwolfach 2004.