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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 for the 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 our earlier results 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. [ABSTRACT FROM AUTHOR]

In this paper we show how the higher index theory can be used to prove results concerning the non-existence of a complete Riemannian metric with uniformly positive scalar curvature at infinity. By improving some classical results due to M. Gromov and B. Lawson we show the efficiency of these methods to prove such non-existence theorems. [ABSTRACT FROM AUTHOR]

In this paper we answer in the negative the question raised by G. Costakis and A. Manoussos whether there exists a J-class operator on every non-separable Banach space. In particular we show that there exists a non-separable Banach space constructed by S. Argyros, A. Arvanitakis and A. Tolias such that the J-set of every operator on this space has empty interior for each non-zero vector. On the other hand, on non-separable spaces which are reflexive there always exists a J-class operator. [ABSTRACT FROM AUTHOR]

A characterization of convex functions in RN states that an upper semicontinuous function u is convex if and only if u(Ax) is subharmonic (with respect to the usual Laplace operator) for every symmetric positive definite matrix A. The aim of this paper is to prove that an analogue of this result holds for free Carnot groups G when considering convexity in the viscosity sense. In the subelliptic context of Carnot groups, the linear maps x … Ax of the Euclidean case must be replaced by suitable group isomorphisms x … TA(x), whose differential preserves the first layer of the stratification of Lie(G). [ABSTRACT FROM AUTHOR]

Several useful results in the theory of p-summing operators, such as Pietsch's composition theorem and Grothendieck's theorem, share a common form: for certain values q and p, there is an operator such that whenever it is followed by a q-summing operator, the composition is p-summing. This is precisely the concept of (q, p)-mixing operators, defined and studied by A. Pietsch. On the other hand, J. Farmer and W. B. Johnson recently introduced the notion of a Lipschitz p-summing operator, a nonlinear generalization of p-summing operators. In this paper, a corresponding nonlinear concept of Lipschitz (q, p)-mixing operators is introduced, and several characterizations of it are proved. An interpolation-style theorem relating different Lipschitz (q, p)-mixing constants is obtained, and it is used to show reversed inequalities between Lipschitz p-summing norms. [ABSTRACT FROM AUTHOR]

It is shown that in uniformly convex Banach spaces, Stepanov almost-periodic functions with Stepanov almost-periodic derivatives are uniformly almost-periodic in the sense of Bohr. This in natural situations yields, jointly with the derived properties of the associated Nemytskii operators, the nonexistence of purely (i.e. nonuniformly continuous) Stepanov almost-periodic solutions of ordinary differential equations. In particular, the existence problem of such solutions, considered in a series of five papers of Z. Hu and A. B. Mingarelli, is answered in a negative way. [ABSTRACT FROM AUTHOR]

In 2001, E. M. Stein and S. Wainger gave the Lp boundedness of the Carleson type maximal operator Τ*, which is defined by Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed. In this paper, the authors show that if K is a homogeneous kernel, i.e. K(y) = Ω(y!)|y|--n, then Stein-Wainger's result still holds on the weighted Lp spaces when Ω satisfies only an Lq-Dini condition for some 1 < q ≤ ∞. [ABSTRACT FROM AUTHOR]

When is an entire function of finite order a solution to a complex 2nd order homogeneous linear differential equation with polynomial coefficients? In this paper we will give two (equivalent) answers to this question. The starting point of both answers is the Hadamard product representation of a given entire function of finite order. While the first answer involves certain Stieltjes-like relations associated to the function, the second one requires the vanishing of all but finitely many suitable expressions constructed via the Gil' sums of the zeros of the function. Applications of these results will also be given, most notably to the spectral theory of one-dimensional Schrödinger operators with polynomial potentials. [ABSTRACT FROM AUTHOR]

In this paper we prove a discrete version of Tanaka's theorem for the Hardy-Littlewood maximal operator in dimension n = 1, both in the noncentered and centered cases. For the non-centered maximal operator Μ M we prove that, given a function f : Z → R of bounded variation, Var(Mf) ≤ Var(f), where Var(f) represents the total variation of f. For the centered maximal operator M we prove that, given a function f : Z → R such that f ϵ Μ1(Z), Var(Mf) ≤ C‖f‖Μе1(Z). This provides a positive solution to a question of HajΜlasz and Onninen in the discrete one-dimensional case. [ABSTRACT FROM AUTHOR]

The main goal of this paper is to extend in Rn a result of Seeley on eigenfunction expansions of real analytic functions on compact manifolds. As a counterpart of an elliptic operator in a compact manifold, we consider in Rn a selfadjoint, globally elliptic Shubin type differential operator with spectrum consisting of a sequence of eigenvalues λj, j ϵ N, and a corresponding sequence of eigenfunctions uj, j ϵ N, forming an orthonormal basis of L²(Rn). Elements of Schwartz S(Rn), resp. Gelfand-Shilov S½½ spaces, are characterized through expansions Σj ajuj and the estimates of coefficients aj by the power function, resp. exponential function of λj . [ABSTRACT FROM AUTHOR]

In this paper we study an operator $ T$ which is $ M$ of $ E$-hypercyclic and use it to answer negatively two questions asked by Madore and Martínez-Avendaño. We also give a sufficient condition for $ T$-hypercyclic for all finite co-dimensional subspaces $ M$. [ABSTRACT FROM AUTHOR]

In this paper, a nonlinear version of the Extrapolation Theorem is proved and, as a corollary, a nonlinear version of Grothendieck's Theorem is presented. Finally, we prove that if $ T:X\to H$ being a pointed metric space and $ T(0)=0$ $ T^\char93 \vert _{H^*}$-summing $ (1\le q<\infty)$ is Lipschitz 1-summing. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to prove that there exist measures $ d\mu(x)=\gamma(x)dx$ $ \gamma(x)=\gamma_{0}(\vert x\vert)$ $ \gamma_{0}$ $ \mathcal{M}_{\mu}$ does not map $ L^{p}_{\mu}(\mathbb{R}^{n})$ $ L^{p}_{\mu}(\mathbb{R}^{n})$. This result answers an open question of P. Sjögren and F. Soria. [ABSTRACT FROM AUTHOR]

This paper gives an asymptotic expansion for the solution of the boundary problem with respect to the Laplace-Beltrami operator. We also consider some examples in the domain whose boundary is real ellipsoid, where the boundary problem does not have a $ C^n$ [ABSTRACT FROM AUTHOR]

In this paper, we identify a large collection of open subsets of the complex plane for which the core of corresponding unbounded Bergman operators is densely defined. This result gives the necessary background to investigate the concept of invariant subspaces, index, and cyclicity in the unbounded case. [ABSTRACT FROM AUTHOR]

In this paper we construct the main ingredients of a discrete function theory in higher dimensions by means of a new ``skew'' type of Weyl relations. We will show that this new type overcomes the difficulties of working with standard Weyl relations in the discrete case. A Fischer decomposition, Euler operator, monogenic projection, and basic homogeneous powers will be constructed. [ABSTRACT FROM AUTHOR]

In this paper, we prove some of Rubio de Francia's extrapolation results for the class $B_p$ of weights for which the Hardy operator is bounded on $L^p(w)$ restricted to decreasing functions. Applications to the boundedness of operators on $L^p_{{rm dec}}(w)$ are given. We also present an extension to the $B_infty $ case and some connections with classical $A_p$ theory. [ABSTRACT FROM AUTHOR]

In this paper we present necessary and sufficient conditions for the existence of a unique solution to the relaxed commutant lifting problem. The obtained conditions are more complicated than those for the classical commutant lifting setting, and earlier obtained sufficient conditions turn out not to be necessary conditions. It is also shown that these conditions simplify in certain special cases. [ABSTRACT FROM AUTHOR]

In this paper we study the cyclic properties of the infinite continuous Cesàro operator defined on $L^2(0,infty )$ by $(C_infty f)(x)=frac {1}{x}int _0^x f(s) ds $. Despite this operator being cyclic, we show that it is not supercyclic; even more, it is not weakly supercyclic. These results complement some recent ones on the cyclic behavior of Cesàro operators. [ABSTRACT FROM AUTHOR]

par Polynomial solutions to the Heine-Stieltjes equation, textit {the Stieltjes polynomials}, and the associated textit {Van Vleck polynomials} have been studied since the 1830's in various contexts including the solution of the Laplace equation on an ellipsoid. Recently there has been renewed interest in the distribution of the zeros of Van Vleck polynomials as the degree of the corresponding Stieltjes polynomials increases. In this paper we show that the zeros of Van Vleck polynomials corresponding to Stieltjes polynomials of successive degrees interlace. We also show that the spectral polynomials formed from the Van Vleck zeros are not orthogonal with respect to any measure. This furnishes a counterexample, coming from a second order differential equation, to the converse of the well-known theorem that the zeros of orthogonal polynomials interlace. [ABSTRACT FROM AUTHOR]

Typographical errors which changed important definitions in the paper ``Vector measures and the strong operator topology'' (textit {Proc. Amer. Math. Soc.} textbf {137} (2009), 2345--2350) are addressed. [ABSTRACT FROM AUTHOR]