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We show that a relative entropy condition recently shown by Leger and Vasseur to imply uniqueness and stable $L^2$ dependence on initial data of Lax 1- or $n$-shock solutions of an $n\times n$ system of hyperbolic conservation laws with convex entropy implies Lopatinski stability in the sense of Majda. This means in particular that Leger and Vasseur's relative entropy condition represents a considerable improvement over the standard entropy condition of decreasing shock strength and increasing entropy along forward Hugoniot curves, which, in a recent example exhibited by Barker, Freist\"uhler and Zumbrun, was shown to fail to imply Lopatinski stability, even for systems with convex entropy. This observation bears also on the parallel question of existence, at least for small $BV$ or $H^s$ perturbations, Comment: to appear in Proceedings of the AMS

Concerns have been voiced about the correctness of certain technical points in DiPerna’s paper (Comm. Math. Phys. 91 (1983), 1–30) related to the vacuum state. In this note, we provide clarifications. Our conclusion is that these concerns mainly arise from the statement of a lemma for constructing the viscous approximate solutions and some typos; however, the gap can be either fixed by correcting the statement of the lemma and the typos or bypassed by employing the finite difference methods. In [Di], DiPerna found a global entropy solution of the isentropic Euler equations for the following exponents in the equation of state for the pressure: γ = 1 + 2/(2m+ 1), m ≥ 2 integer. (1) He divided his arguments into the following two steps. 1. Compactness framework Assume that a sequence of approximate solutions (ρ (x, t),m (x, t)), 0 ≤ t ≤ T , satisfies: (i). There exists a constant C(T ) > 0, independent of > 0, such that 0 ≤ ρ (x, t) ≤ C, |m (x, t)/ρ (x, t)| ≤ C; (ii). For all weak entropy pairs (η, q) of the isentropic Euler equations, the measure sequence η(ρ ,m )t + q(ρ ,m )x is contained in a compact subset of H −1 loc (R× [0, T ]). If γ satisfies (1), then the sequence (ρ (x, t),m (x, t)) is compact in Lloc(R× [0, T ]). The reason for the restriction on the number γ is that, in such a case, any weak entropy function is a polynomial function of the Riemann invariants (w, z). This is the key step in DiPerna’s arguments and is also his main contribution to the compensated compactness method in this aspect. Received by the editors May 16, 1996. 1991 Mathematics Subject Classification. Primary 35K55, 35L65; Secondary 76N15, 35L60, 65M06.

In this paper we point out an error in paper [2] and study the asymptotic behavior of the differential equation Lnx + f(t, x) = r(t). The results obtained are extensions of some of the results of [2].

A class of functions K(r) and the corresponding solutions are produced in the case of the differential equation d 2 u/dr 2 +1 du/r dr+K(r)e 2u(r) =1. The behavior of these solutions at an isolated singularity is also investigated

Recently Cameron and Storvick introduced and studied an operator valued function space integral related to the Feynman integral. The main theorems of their study establish the existence of the function space integral as a weak operator limit of operators defined at the first stage by finite-dimensional integrals. This paper provides a substantial strengthening of their existence theorem giving the function space integrals as strong operator limits rather than as weak operator limits.

We establish the relationship between certain invariant extension properties for linear functional of K. Fan when a semigroup S acts on a locally convex topological vector space and the existence of a left-invariant mean on the space of almost periodic and weakly almost periodic functions on S.

A subset A of a topological space is said to be semi-open if there exists an open set U such that U C A C Cl(U) where Cl(U) denotes the closure of U. The class of semi-open sets of a given topological space (X, J) is denoted S.O. (X, J). In the paper Semi-open sets and semi-continuity in topological spaces, Amer. Math. Monthly 70 (1963), 36-41, Norman Levine states in Theorem 10 that if J and V* are two topologies for a set X such that S.O.(X, 3) C S.O.(X, J*), then 'J C P. In a corollary to this theorem, Levine states that if S.O.(X, if) = S.O.(X,Jf*), then _T= f*. An example is given which shows the above-mentioned theorem and its corollary are false. This paper shows how different topologies on a set which determine the same class of semi-open subse,ts can arise from functions, and points out some of the implications of two topologies being related in this manner. In [6] Norman Levine defines a set A in a topological space X to be semi-open if there exists an open set U such that U C A C Cl (U), where Cl(U) denotes the closure of U. The class of semi-open sets for a given topological space (X, i) is denoted S.O. (X, sT). Levine states in Theorem 10 of [6] that if Jf and * are two topologies for a set X such that S.O. (X, i) C S.O. (X, J9 then iT C *. In a corollary to this theorem, Levine states that if S.O. (X, ) = S.O. (X, iT*), then Jf = -" The following example which is due to S. Gene Crossley and S. K. Hildebrand [1, Example 1.1] shows the above-mentioned theorem and its corollary are false. Example. Let X = la, b, cl, J; = t0, sal, la, bl, la, cl, XI, ;* = 10, sal, la, bl, XI. An exhaustion of all possibilities shows that S.O. (X, ) = S.O. (x, 5j*). Crossley and Hildebrand [3] defined two topologies iT and T* on a set X to be semi-correspondent if S.O. (X, J) = S.O. (X, 5f*). It is shown in [3] that semi-correspondence is an equivalence relation on the collection of Received by the editors March 3, 1974. AMS (MOS) subject classifications (1970). Primary 54B99; Secondary 54C10.

If BP denotes the space of smooth alternating p-forms on the C' manifold M, we are interested in finding the spaces of R-linear maps from iPP to be that commute with the diffeomorphisms of M. For a compact manifold M these spaces were found by R. S. Palais. In this note we find them for noncompact M. Let V, W be the spaces of sections of two tensor bundles over a connected Coo manifold M of dimension n. In a paper [3] appearing in Trans. Amer. Math. Soc. 92 (1959), R. S. Palais studied a number of the spaces 4 (V, W) of the R-linear maps V->W natural with respect to the action on V and W of the group G of diffeomorphisms on M. We sketch the definition of this action: If g-G we define R,:F(TM) -4F(TM) to be the differential of g, and Rg:r(T*M)-->F(T*M) to be the dual of the differential of g-1. For the other spaces of tensor fields Rg is defined as the obvious tensor product of these two R-linear isomorphisms. A map c: V->W is then called natural if cRg =R,c for all g G. Palais was particularly interested in the case when V and W are the space IP of p-forms and the space bq of q-forms respectively, 0 < p, q

Submitted by Vitor Silverio Rodrigues (vitorsrodrigues@reitoria.unesp.br) on 2014-05-27T11:25:28Z No. of bitstreams: 0Bitstream added on 2014-05-27T14:41:41Z : No. of bitstreams: 1 2-s2.0-79951846250.pdf: 494002 bytes, checksum: 56b6ee8beddda3e7dae971355d44a19f (MD5) Made available in DSpace on 2014-05-27T11:25:28Z (GMT). No. of bitstreams: 0 Previous issue date: 2011-03-01 Item merged in doublecheck by Felipe Arakaki (arakaki@reitoria.unesp.br) on 2015-12-11T17:28:11Z Item was identical to item(s): 71803, 21370 at handle(s): http://hdl.handle.net/11449/72321, http://hdl.handle.net/11449/21804 Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) The simplest necessary conditions for an entire function ψ(x) =∞ ∑ k=0 γk xk/k! to be in the Laguerre-Pólya class are the Turán inequalities γ2 k- γk+1γk-1 ≥ 0. These are in fact necessary and sufficient conditions for the second degree generalized Jensen polynomials associated with ψ to be hyperbolic. The higher order Turán inequalities 4(γ2 n - γn-1γn+1)(γ2n +1 - γnγn+2) - (γnγn+1 - γn-1γn+2) 2 ≥ 0 are also necessary conditions for a function of the above form to belong to the Laguerre-Pólya class. In fact, these two sets of inequalities guarantee that the third degree generalized Jensen polynomials are hyperbolic. Pólya conjectured in 1927 and Csordas, Norfolk and Varga proved in 1986 that the Turán inequalities hold for the coefficients of the Riemann ψ-function. In this short paper, we prove that the higher order Turán inequalities also hold for the ψ-function, establishing the hyperbolicity of the associated generalized Jensen polynomials of degree three. © 2010 American Mathematical Society. Departamento de Ciências de Computação e Estatística IBILCE, Universidade Estadual Paulista, 15054-000 São José do Rio Preto, SP Departamento de matemática Aplicada IMECC UNICAMP, 13083-859 Campinas, SP Departamento de Ciências de Computação e Estatística IBILCE, Universidade Estadual Paulista, 15054-000 São José do Rio Preto, SP FAPESP: 03/01874-2 FAPESP: 06/60420-0 CNPq: 305622/2009-9 CAPES: DGU-160

We are concerned with the study of the Lane-Emden equation with variable exponent and Dirichlet boundary condition. The feature of this paper is that the analysis that we develop does not assume any subcritical hypotheses and the reaction can fulfill a mixed regime (subcritical, critical and supercritical). We consider the radial and the nonradial cases, as well as a singular setting. The proofs combine variational and analytic methods with a version of the Palais principle of symmetric criticality., The final version this paper will be published in Proc. AMS

This paper extends the results from a work of Hytönen, Petermichl, and Volberg about sharp A 2 A_2 - A ∞ A_\infty estimates with matrix weights to the non-homogeneous situation.

In this paper we study the limit cycles bifurcating from a non-isolated zero-Hopf equilibrium of a differential system in R 3 \mathbb {R}^3 . The unfolding of the vector fields with a non-isolated zero-Hopf equilibrium is a family with at least three parameters. By using analysis techniques and the averaging theory of the second order, explicit conditions are given for the existence of one or two limit cycles bifurcating from such a zero-Hopf equilibrium. This result is applied to study three-dimensional generalized Lotka-Volterra systems in a paper by Bobieński and Żołądek (2005). The necessary and sufficient conditions for the existence of a non-isolated zero-Hopf equilibrium of this system are given, and it is shown that two limit cycles can be bifurcated from the non-isolated zero-Hopf equilibrium under a general small perturbation of three-dimensional generalized Lotka-Volterra systems.

We will show that two functions in the extended Selberg class satisfying the same functional equation must be identically equal if they have sufficiently many common zeros. This paper concerns the question of how L-functions are determined by their zeros. L-functions are Dirichlet series with the Riemann zeta function ζ(s) = ∑∞ n=1 1 ns as the prototype and are important objects in number theory. The Selberg class S of L-functions is the set of all Dirichlet series L(s) = ∑∞ n=1 a(n) ns of a complex variable s = σ + it with a(1) = 1, satisfying the following axioms (see [7]): (i) (Dirichlet series) For σ > 1, L(s) is an absolutely convergent Dirichlet series. (ii) (Analytic continuation) There is a nonnegative integer k such that (s − 1)L(s) is an entire function of finite order. (iii) (Functional equation) L satisfies a functional equation of type ΛL(s) = ωΛL(1− s), where ΛL(s) = L(s)Q s ∏K j=1 Γ(λjs+μj) with positive real numbers Q, λj and with complex numbers μj , ω with Reμj ≥ 0 and |ω| = 1. (iv) (Ramanujan hypothesis) a(n) n for every e > 0; (v) (Euler product) logL(s) = ∑∞ n=1 b(n) ns , where b(n) = 0 unless n is a positive power of a prime and b(n) n for some θ < 12 . The degree dL of an L-function L is defined to be dL = 2 ∑K j=1 λj , where K, λj are the numbers in axiom (iii). The Selberg class includes the Riemann zeta-function ζ and essentially those Dirichlet series where one might expect the analogue of the Riemann hypothesis. At the same time, there are a whole host of interesting Dirichlet series not possessing a Euler product (see e.g. [3], [8]). Throughout the paper, all L-functions are assumed to be functions from the extended Selberg class of those only satisfying the axioms (i)-(iii) (see [3]). Thus, the results obtained in the present paper particularly apply to L-functions in the Selberg class. Received by the editors October 5, 2011 and, in revised form, February 12, 2012. 2010 Mathematics Subject Classification. Primary 11M36, 30D30.

In 1974, S.-S. Chern and J. Simons published a paper where they defined a new type of characteristic class - one that depends not just on the topology of a manifold but also on the geometry. The goal of this paper is to investigate what kinds of geometric information is contained in these classes by studying their behavior under the Ricci flow. In particular, it is shown that the Chern- Simons class corresponding to the first Pontryagin class is invariant under the Ricci flow on the warped products $S^2\times_f S^1$ and $S^1 \times_f S^2$ but that this class is not invariant under the Ricci flow on a generalized Berger sphere., 8 pages

This paper deals with the long-time behavior of viscosity solutions of evolutionary contact Hamilton-Jacobi equations w t + H ( x , w , w x ) = 0 , \begin{equation*} w_t+H(x,w,w_x)=0, \end{equation*} where H ( x , u , p ) H(x,u,p) is strictly decreasing in u u and satisfies Tonelli conditions in p p . We show that each viscosity solution of the ergodic contact Hamilton-Jacobi equation H ( x , u , u x ) = 0 H(x,u,u_x)=0 can be reached by many different viscosity solutions of the above evolutionary equation in a finite time.

In this paper, we study the sign-changing Kazdan-Warner's problem on two dimensional closed Riemannian manifold with negative Euler number $\chi(M)

This paper deals with positive stationary solutions and spreading speeds of monostable equations with nonlocal dispersal in spatially periodic habitats. The existence and uniqueness of positive stationary solutions and the existence and characterization of spreading speeds of such equations with symmetric convolution kernels are established in the authors’ earlier work [41] for following cases: the nonlocal dispersal is nearly local; the periodic habitat is nearly globally homogeneous or it is nearly homogeneous in a region where it is most conducive to population growth. The above conditions guarantee the existence of principal eigenvalues of nonlocal dispersal operators associated to linearized equations at the trivial solution. In general, a nonlocal dispersal operator may not have a principal eigenvalue. In this paper, we extend the results in [41] to general spatially periodic nonlocal monostable equations. As a consequence, it is seen that the spatial spreading feature is generic for monostable equations with nonlocal dispersal.

In this paper we present two-sided estimates of Chernoff type for the weighted L w 2 L_{w}^{2} -distance of a smooth function to the k k -dimensional space of all polynomials of degree less than k k , whenever the weight function w w solves the Pearson differential equation and generates a finite or infinite sequence of classical orthogonal polynomials. These inequalities are simple corollaries of a unified general theorem, which is the main result of the paper.

In an unpublished paper, Hsu gives a proof of the convergence of the fundamental solutions. Since we had a problem understanding Hsu’s paper, in this paper we give a detailed proof of the convergence of the minimal positive fundamental solutions of the conjugate heat equation on complete non-compact manifolds under the Cheeger-Gromov convergence of Ricci flows.

Let M be an invariant subspace of the Bergman space L 2 a (D) and S M be the compression of the coordinate multiplication operator M z to the co-invariant subspace L 2 a (D) ⊖ M. The present paper determines when the identity representation of C* (S M ) is a boundary representation for the Banach subalgebra B(S M ). The paper also considers boundary representations on the co-invariant subspaces of L 2 a (B n ).

This paper is concerned with a parabolic-elliptic system, which was originally proposed to model the evolution of biological transport networks. Recent results show that the corresponding initial-boundary value problem possesses a global weak solution, which, in particular, is also classical in the one and two dimensional cases. In this work, we establish a Serrin-type blowup criterion for classical solutions in the three dimensional setting.

Derivatives of a solution of an ODE Cauchy problem can be computed inductively using the Faa di Bruno formula. In this paper, we exhibit a noninductive formula for these derivatives. At the heart of this formula is a combinatorial problem, which is solved in this paper. We also give a more tractable form of the Magnus expansion for the solution of a homogeneous linear ODE.

In a former paper we studied transformations on the set of all n n -dimensional subspaces of a Hilbert space H H which preserve the principal angles. In the case when dim ⁡ H ≠ 2 n \dim H\neq 2n , we could determine the general form of all such maps. The aim of this paper is to complete our result by considering the problem in the remaining case dim ⁡ H = 2 n \dim H=2n .

It is well known that the exceptional set in a resolution of a rational surface singularity is a tree of rational curves. We generalize the combinatoric part of this statement to higher dimensions and show that the highest cohomologies of the dual complex associated to a resolution of an isolated rational singularity vanish. We also prove that the dual complex associated to a resolution of an isolated hypersurface singularity is simply connected. As a consequence, we show that the dual complex associated to a resolution of a 3-dimensional Gorenstein terminal singularity has the homotopy type of a point., 10 pages, the structure of the paper has been slightly revised and a mistake in the proof of degeneration of spectral sequence in Lemma 2.3 (Lemma 2.4 of the published version of the paper) has been corrected

In this paper we study a nonlinear eigenvalue problem driven by the p-Laplacian. Assuming for the right-hand side nonlinearity only unilateral and sign conditions near zero, we prove the existence of three nontrivial solutions, two of which have constant sign (one is strictly positive and the other is strictly negative), while the third one belongs to the order interval formed by the two opposite constant sign solutions. The approach relies on a combination of variational and minimization methods coupled with the construction of upper-lower solutions. The framework of the paper incorporates problems with concave-convex nonlinearities.

In this paper, we prove a uniqueness theorem for a free boundary problem which is given in the form of a variational inequality. This free boundary problem arises as the limit of an equation that serves as a basic model in population biology. Apart from the interest in the problem itself, the techniques used in this paper, which are based on the regularity theory of variational inequalities and of harmonic functions, are of independent interest, and may have other applications.

The paper deals with asymptotic phase and invariant foliations near periodic orbits, extending for two-dimensional smooth vector fields results that have been obtained by Chicone and Liu (2004). The problem of the existence of asymptotic phase is completely solved for analytic vector fields and is reduced to a problem of infinite codimension for C ∞ C^{\infty } systems. Moreover it is proven that whenever asymptotic phase occurs, or in other words, when the periodic orbit is isochronous, then there also exists a C ∞ C^{\infty } foliation, with leaves transversally cutting the periodic orbit and invariant under the flow of the vector field. The paper also provides some results in three dimensions.

This paper provides a version of the transversality theorem for a class of Lipschitz functions of the form f : R n × C → R n f:\textbf {R}^n\times C\to \textbf {R}^n , where C C is a convex subset of a normed vector space Z Z indexing the parameters in the problem. The set C C may be infinite-dimensional, and the notion of generic used is the measure-theoretic notion of prevalence introduced by Hunt, Sauer and Yorke (1992) and Christensen (1974). This paper also provides some results on sensitivity analysis for solutions to locally Lipschitz equations.

In this paper, we introduce sequentially defined Besov spaces adapted to the scaling invariant property of the 3D incompressible Navier-Stokes equations. We first show that these spaces are Banach spaces. We then establish regularity conditions in these spaces.

We consider the initial value problem for the fourth order nonlinear Schrodinger type equation (4NLS) related to the theory of vortex filament. In this paper we prove the time local well-posedness for (4NLS) in the Sobolev space, which is an improvement of our previous paper.

The infinitesimal earthquake theorem gives a one-to-one correspondence between Thurston bounded earthquake measures and normalized Zygmund bounded functions. In this paper, we provide an intrinsic proof of a theorem given in an earlier paper by the author; that is, we show that the cross-ratio norm of a Zygmund bounded function is equivalent to the Thurston norm of the earthquake measure in the correspondence.

Weighted Opial-type inequalities are shown to be equivalent to weighted norm inequalities for sublinear operators and for nearly positive operators. Examples involving the Hardy-Littlewood maximal function and the non-increasing rearrange- ment are presented. Opial-type inequalities are related to norm inequalities much as quadratic forms are related to bilinear forms. A linear operator T on Hilbert space gives rise to the bilinear form (f,g) 7! hTf,gi and the quadratic form f 7! hTf,fi. Duality shows that the norm of T and the norm of the bilinear form coincide and a standard polarization argument shows that this norm is equivalent to but not necessarily equal to the norm of the quadratic form, called the numerical radius of T. In this paper, far from the luxuries of Hilbert spaces and linear operators, we show that the equivalence of operator norm and numerical radius persists. The work is in response to Richard Brown's suggestion that Steven Bloom's result (2, The- orem 1) which gives the equivalence for positive operators should apply in greater generality. Opial-type inequalities have been much studied since Opial's original paper in 1960 and the papers (2), (3) and (4) include many references. After the main theorem showing equivalence of Opial-type and norm inequali- ties, an example involving the Hardy-Littlewood maximal function is included to illustrate that the equivalence cannot be taken in a pointwise sense. To show that the method can be readily applied to generate non-trivial inequal- ities from known norm inequalities we give a simple weight characterization of an Opial-type inequality for the non-increasing rearrangement.

This paper investigates connections between the long-time asymptotics of heat distribution on a body Ω in Z n , and various geometric properties of Ω, starting from an initially constant heat distribution supported on Ω. We use combinatorial and differential geometric methods. We begin the paper with a result in R.

A topological space X is said to belong to the class of Stegall (weakly Stegall) spaces if for every Baire (complete metric) space B and minimal usco φ: B → 2 X , φ is single-valued at some point of B. In this paper we show that under some additional set-theoretic assumptions that are equiconsistent with the existence of a measurable cardinal there is a Banach space X whose dual, equipped with the weak* topology, is in the class of weakly Stegall spaces but not in the class of Stegall spaces. This paper also contains an example of a compact space K such that K belongs to the class of weakly Stegall spaces but (C(K)*, weak*) does not.

In this paper, the authors consider the boundary value problem (E) x (2m) (t) + (-1) m+1 f(x(t)) = 0, 0 < t < 1, (B) x (2i) (0) = x (2i) (1) = 0, i = 0, 1, 2, …, m - 1, and give sufficient conditions for the existence of any number of symmetric positive solutions of (E)-(B). The relationships between the results in this paper and some recent work by Henderson and Thompson (Proc. Amer. Math. Soc. 128 (2000), 2373-2379) are discussed.

The first section of this paper provides an improvement upon known finiteness theorems for Riemannian submersions; that is, theorems which conclude that there are only finitely many isomorphism types of fiber bundles among Riemannian submersions whose total spaces and base spaces both satisfy certain geometric bounds. The second section of this paper provides a sharpening of some recent theorems which conclude that, for an open manifold of nonnegative curvature satisfying certain geometric bounds near its soul, there are only finitely many possibilities for the isomorphism class of a normal bundle of the soul. A common theme to both sections is a reliance on basic facts about Riemannian submersions whose A A and T T tensors are both bounded in norm.

This paper deals with the spectra of the Laplace and stability operators of a constant mean curvature surface in the hyperbolic space. In a preceding work, the author described the essential spectra of these operators, assuming that the surface is of finite total curvature. In this paper, we prove that these two operators have a finite number of eigenvalues below their essential spectra.

Let B denote the unit ball in RI. This paper characterizes the subsets E of B with the property that supE h = SUPB h for all harmonic functions h on B which have finite Dirichlet integral. It also examines, in the spirit of a celebrated paper of Brelot and Doob, the associated question of the connection between non-tangential and fine cluster sets of functions on B at points of the boundary.

This paper deals with the existence of periodic solutions and attractiveness for some partial functional differential equations in Banach spaces. We assume that the first linear part generates a strongly continuous semigroup, while the delayed part is periodic with respect to the first argument. We prove that the existence of a bounded solution implies the existence of a periodic solution. Several results regarding uniqueness and global attractiveness of periodic solutions are also established. The analysis relies on a fixed point theorem of Chow and Hale’s type and uses some arguments of weak topology. Our theorems extend in a broad sense some new and classical related results. An application to a transport equation with delay is also presented.

In this paper, we study gradient estimates for a parabolic p p -Laplace equation with logarithmic nonlinearity, which is related to the L p L^p -log-Sobolev constant on Riemannian manifolds. We prove a global Li-Yau type gradient estimate and a Hamilton type gradient estimate for positive solutions to a parabolic p p -Laplace equation with logarithmic nonlinearity on compact Riemannian manifolds with nonnegative Ricci curvature. As applications, the corresponding Harnack inequalities are derived.

We prove that the homeomorphism group of each one of a collection of continua constructed in a paper by the first author (Trans. Amer. Math. Soc. 121 (1966), 516–548) is one dimensional. This answers a question posed in that paper.

We formulate and prove a topological Paley-Wiener theorem for the normalized spherical Laplace transform defined on the rank 1 causal symmetric spaces M = SO0(1,n)/SO(1,n -1), for n > 2. INTRODUCTION The spherical Laplace transform on causal symmetric spaces was introduced in [FHO, ?8] as a generalization of the spherical Fourier transform on Riemannian symmetric spaces defined by Helgason (see [Hi, Chapter 4]). Both transforms can be expressed in terms of (integrating against) spherical functions. It was furthermore shown in [01, ?5] that the spherical functions on the Riemannian dual of a causal symmetric space can be written as an expansion in spherical functions on the causal symmetric space. The inversion formula for the spherical Laplace transform easily follows (see [01, ?6]). One of the most important results on the spherical Fourier transform is the (topological) Paley-Wiener theorem (see [Hi, Chapter 4, ?7] and [H2, Chapter 3, ?5] for details) generalizing the classical Paley-Wiener theorem on Euclidean spaces. In this paper we generalize these results to the normalized spherical Laplace transform on causal symmetric spaces M of rank 1, thereby partially solving an open problem posed by the second author in [02, ?5]. The paper is divided into two sections: in the first section we recall some results on the spherical Fourier transform on the Riemannian dual Md of M, and in the second we consider the spherical Laplace transform defined on M. We define the Paley-Wiener space, the supposed image space of spherical Laplace transform, according to the growth and symmetry conditions satisfied by the spherical functions on M. The Paley-Wiener theorem for the normalized spherical Laplace transform follows by using Cauchy's theorem to change the path of integration in the inversion formula and from the Paley-Wiener theorem for the spherical Fourier transform on Md. Received by the editors December 9, 1998 and, in revised form, March 22, 1999. 2000 Mathematics Subject Classification. Primary 43A85, 22E30; Secondary 43A90, 33C60. The first author was supported by a postdoc fellowship from the European Commission within the European TMR Network "Harmonic Analysis" 1998-2001 (Contract ERBFMRX-CT97-0159). The second author was supported by LEQSF grant (1996-99)-RD-A-12. (?2000 American Mathematical Society

This paper proves the existence of solutions to the initial value problem \[ ( I V P ) { x ′ ( t ) = f ( t , x ( t ) ) ( 0 ≤ t ≤ 1 ) , x ( 0 ) = 0 , (\mathrm {IVP})\qquad \qquad \left \{\begin {array}{l} x’(t)=f(t,x(t))\qquad \quad (0\le t\le 1), x(0)=0,\end {array} \right . \] where f : [ 0 , 1 ] × R M → R M f:[0,1]\times \mathbb {R}^M\to \mathbb {R}^M may be discontinuous but is assumed to satisfy conditions of superposition-measurability, quasimonotonicity, quasisemicontinuity, and integrability. The set M M can be arbitrarily large (finite or infinite); our theorem is new even for card ( M ) = 2 \mbox {card}(M)=2 . The proof is based partly on measure-theoretic techniques used in one dimension under slightly stronger hypotheses by Rzymowski and Walachowski. Further generalizations are mentioned at the end of the paper.

We give conditions on a couple of ideal Banach spaces with weights which are both necessary and sufficient for the Hardy-Littlewood maximal function to satisfy the two-weighted estimations of weak type, and we consider a modification of the Hardy-Littlewood maximal function. We also give some conditions on weights in order for the Hardy-Littlewood maximal function and the modification under consideration to fulfil the two-weighted estimations of strong type. 0. INTRODUCTION The general problem of two-weighted estimations of weak type in LP-spaces for the Hardy-Littlewood maximal function has been solved in [9]. In the case of Lorentz spaces Lpq a corresponding result has been obtained in [2, 7] (see also [3, 4]). In order to obtain a positive answer the AP-condition for weights is required in both cases mentioned. The problem of two-weighted estimations of strong type for the Hardy-Littlewood maximal function in LP-spaces has been considered in [12]. The solution is boundedness of the operator of the Hardy-Littlewood maximal function on the set of the characteristic functions of cubes. Simpler conditions, necessary and sufficient on a certain class of weights, have been considered by many authors (see for example [3] or [4]). Among recent papers we should mention [11], where, for a large class of weights, there are given necessary and sufficient conditions for boundedness of the weighted fractional maximal operator. In the present paper we give necessary and sufficient conditions for two-weighed estimations of weak type for the Hardy-Littlewood maximal function and a modification in ideal Banach spaces. We also give some conditions on weights assuring two-weighted estimations of strong type for the Hardy-Littlewood maximal function and a modification. The obtained results are new even in the case of Orlicz spaces. 1. PRELIMINARIES Let S = S(R', ,u) be the space of all Lebesgue measurable real-valued functions on R' endowed with the topology of convergence in measure on each set of finite measure. Recall that a Banach subspace X in S is said to be ideal [8] if x E X, Received by the editors December 11, 1991 and, in revised form, September 19, 1996. 1991 Mathematics Subject Classification. Primary 42B20, 42B25. ?)1999 American Mathematical Society 79 This content downloaded from 157.55.39.144 on Mon, 25 Jul 2016 03:48:01 UTC All use subject to http://about.jstor.org/terms

In this paper, the authors establish a multiplier theorem for Herz type Hardy spaces. Let Tm be a multiplier operator defined in terms of Fourier transforms by Tmf Q) = M(W for suitable functions f. It is well-known that there is a multiplier theorem for H1 (Rn) (see [FS]): if a > n/2 and (1) J ID3m(e)12d d 0, let us denote m6QW) = M(6077(0) It is easy to check that (1) is equivalent to (2) sup |m6||Ka 2(Rn) < 0, 6 2 where K2'2 2(R) is a non-homogeneous Herz space (see [BS]). By using some embedding relations on Herz spaces, A. Baernstein II and E. T. Sawyer [BS] weakened (2) into (3) sup iM6i||Kf1l (Rn) < 0, 6 where 0 < E < an2. In fact, this is just a special case of their theorem. In [BS], Baernstein and Sawyer showed that m is a bounded multiplier of H1 (Rn) under an even weaker condition than (3); see Theorem 3b in [BS, page 21]. By using the technique of Herz type Hardy spaces developed by the authors in [LY1]-[LY3] and [Y], in this paper, we shall first establish a multiplier theorem for the homogeneous Herz type Hardy space Hk n(1-1/q), 1(IRn) which is introduced by y ~~~q the authors of this paper in [LY1]. Then as simple consequences of this theorem, a multiplier theorem for the corresponding non-homogeneous version of the space Received by the editors April 13, 1995 and, in revised form, April 5, 1997. 1991 Mathematics Subject Classification. Primary 42B15; Secondary 42B30.

If a smooth manifold has a Riemannian metric with positive scalar curvature, it follows immediately that the universal covering also has such a metric. The paper establishes a converse if the manifold in question is closed of dimension at least 5 and the fundamental group is an elementary abelian p-group of rank 2, where p is an odd prime. A conjecture of J. Rosenberg [Rs3] states that a closed smooth manifold with odd order fundamental group and dimension ≥ 5 admits a Riemannian metric with positive scalar curvature if and only if its universal covering admits such a metric. Standard considerations involving transfers (cf. [KS1]) reduce the conjecture to the special case of p-groups (where p is an odd prime). Results of Rosenberg [Rs1]– [Rs3] and S. Kwasik and the author [KS1] prove the conjecture if the fundamental group G is a cyclic p-group. In this work we study the conjecture when G is a finite abelian p-group. The following interim conclusion reflects many of the basic ideas and disposes of the first examples not covered by [KS1], [Rs1]–[Rs3]. Theorem. Let p be an odd prime, and let M be a closed smooth manifold of dimension n ≥ 5 with fundamental group Zp × Zp. Then M admits a Riemannian metric with positive scalar curvature if and only if its universal covering M does. In [RsS] J. Rosenberg and S. Stolz prove a stable result that yields a weaker conclusion for G ∼= Zp × Zp but applies to all finite groups. In particular, if G is a finite group of odd order and M is a closed smooth manifold with fundamental group G, their result states that the universal covering M has a Riemannian metric with positive scalar curvature if and only if some product M ×X × · · · ×X does, where X is an 8-dimensional Bott manifold; i.e., it is a spin manifold whose Â-genus is equal to 1. Although the methods should yield quantitative information on the number of factors of X that are needed, it is not clear how precise such estimates would be. Our result implies that if G ≈ Zp×Zp, then no copies of X are needed if n ≥ 5 and at most one copy of X is needed if n < 5. A more geometric proof of this result in dimensions ≤ 2p− 2 was obtained independently by Stolz (unpublished). It seems likely that the methods of this paper can be combined with the multiple Kunneth formula decomposition of [Hn] and finite induction to prove at least a semistable version of Rosenberg’s conjecture for all elementary abelian p-groups Received by the editors February 13, 1995 and, in revised form, September 13, 1995. 1991 Mathematics Subject Classification. Primary 53C21, 55N15, 57R75; Secondary 53C20, 57R85. c ©1997 American Mathematical Society

In this paper, we investigate the hyperbolic Gauss map of a complete CMC-1 surface in H3(−1), and prove that it cannot omit more than four points unless the surface is a horosphere. 0. Introduction In minimal surface theory, the value distribution of the Gauss map has been studied for a long time. An early result is the classical Bernstein Theorem: Any complete minimal graph in R is a plane. Replacing the graph by some geometric conditions, R. Osserman [3] showed that if the complete minimal surface is nonflat, then the Gauss map cannot omit a set of positive logarithmic capacity, which answered a conjecture of Nirenberg. Afterwards, it was generalized by F. Xavier [8] that the Gauss map of such a surface can omit at most six points. In 1989, H. Fujimoto [2] proved the best result finally that the Gauss map of the nonflat complete minimal surface in R can omit at most four points. It is natural to consider the similar properties of minimal surfaces in H. An interesting feature is that there exists a family of absolutely area-minimizing hypersurfaces in H and only one of them is totally geodesic (see [7]). The question seems to be how to raise an adequate Bernstein problem in hyperbolic space. Professor Y. L. Xin pointed out to me that the striking work done by R. Bryant [1] supplies a framework to solve the Bernstein problem in hyperbolic space. In this paper, we shall be concerned with the surfaces in hyperbolic space of constant mean curvature one. We abbreviate “constant mean curvature one” by CMC-1. These surfaces share many properties with minimal surfaces in R. They possess the “Weierstrass representation” in terms of holomorphic data. This formula was discovered by R. Bryant [1]. Many other properties may be found in papers by M. Umhara and K. Yamada ([4], [5]). Here we try to investigate the hyperbolic analogue of the Gauss map. It is a natural question how the values of the hyperbolic Gauss map distribute. Using Bryant’s representation formula we are able to answer this question as follows. Theorem. The hyperbolic Gauss map of nonflat complete CMC-1 surfaces in H3(−1) can omit at most four points. Received by the editors November 1, 1995 and, in revised form, April 2, 1996. 1991 Mathematics Subject Classification. Primary 53A10; Secondary 53C42.

We give an example as indicated in the title where the weight (i.e., the minimum cardinality of a base for the topology) of the product is the smallest uncountable cardinal. 0. INTRODUCTION AND THE MAIN THEOREM History. Throughout this paper the term "space" abbreviates "Hausdorff regular topological space". Is there a normal space and a metric space with a nonnormal product? The answer is yes in ZFC and is due to E. A. Michael ([6], 1963). (ZFC denotes the standard set of axioms for set theory.) Let IP and R denote respectively the irrationals and the reals with their usual topologies; and let M denote the Michael line which is the refinement of R obtained by isolating each irrational point. In Michael's example, M is the normal space (and moreover hereditarily paracompact) and P is the metric space. Note that the weight of M, and therefore the product, is c (c denotes the cardinality of the continuum of real numbers). Is there a ZFC example of a normal space with weight RI (the smallest uncountable cardinal) whose product with IP is nonnormal? (This question is due to Ronnie Levy.) It is not possible in ZFC to construct a Lindelbf space with both R, weight and a nonnormal product with IP (Lawrence, [5], 1990). Define a Michael space to be a Lindeldf space whose product with IP is nonnormal. The existence of a Michael space follows from both the Continuum Hypothesis (Michael, [7], 1971; first mentioned in a footnote of [6]) and Martin's Axiom (K. Alster, [1], 1990). Is there a Michael space in LZFC? This is Michael's problem and is equivalent to the existence in ZFC of a Lindeldf space and a separable completely metrizable space with a nonnormal product (see [5]). Suppose we drop the requirement for completeness on the metrizable factor. In the 1963 paper, Michael also gave a ZFC example of a Lindeldf space and a separable metric space with a nonnormal product. The Lindel6f space is the refinement of R obtained by isolating each point in a Bernstein set, and the metric space is the Bernstein set as a subspace of R. A subset of R is Bernstein if both it and its Received by the editors November 24, 1992 and, in revised form, March 14, 1994. 1991 Mathematics Subject Classification. Primary 54B10; Secondary 54D20, 54E50.