Nonsmooth Characterizations of Asplund Spaces and Smooth Variational Principles

We show that the Asplund property of Banach spaces is not only sufficient but also a necessary condition for the fulfillment of some basic results in nonsmooth analysis involving Fréchet-like normals and subdifferentials as well as their sequential limits. In this way we obtain new characterizations of Asplund spaces within the framework of nonsmooth analysis. Then we study several versions of smooth variational principles in Asplund spaces, provide necessary and sufficient conditions for the validity of such principles, and establish their relationships with certain subdifferential properties of lower semicontinuous functions.     

Two Points Blow-up in Solutions of the Nonlinear Schrödinger Equation with Quartic Potential on \mathbb{R}

We consider the blow-up problem for the nonlinear Schrödinger equation with quartic self-interacting potential on . We exhibit a class of initial data leading to the blow-up solutions which have at least two L ²-concentration points.     

Inequalities for the gamma function

Some inequalities for the gamma function are given. These results refine the classical Stirling approximation and its many recent improvements. Received: 2 May 2008, Revised: 2 September 2008     

New identities for 7-cores with prescribed BG-rank

Let $\pi$ be a partition. BG-rank$(\pi )$ is defined as an alternating sum of parities of parts of $\pi$ [A. Berkovich, F.G. Garvan, On the Andrews-Stanley refinement of Ramanujan's partition congruence modulo 5 and generalizations, Trans. Amer. Math. Soc. 358 (2006) 703–726. [1]]. Berkovich and Garvan [The BG-rank of a partition and its applications, Adv. in Appl. Math., to appear in $〈$http://arxiv.org/abs/math/0602362$〉$] found theta series representations for the t-core generating functions ${\sum }_{n?0}{a}_{t,j}(n)q^n$, where $a_{t,j}(n)$ denotes the number of t-cores of n with $\text{BG-rank}=j$. In addition, they found positive eta-quotient representations for odd t-core generating functions with extreme values of BG-rank. In this paper we discuss representations of this type for all 7-cores with prescribed BG-rank. We make an essential use of the Ramanujan modular equations of degree seven [B.C. Berndt, Ramanujan's Notebooks, Part III, Springer, New York, 1991] to prove a variety of new formulas for the 7-core generating function$\prod _{j?1}\frac{(1-q^{7j}{)}^7}{(1-q^j)}\text{.}$These formulas enable us to establish a number of striking inequalities for $a_{7,j}(n)$ with $j=-1,0,1,2$ and $a_7(n)$, such as$a_7(2n+2)?2a_7(n),a_7(4n+6)?10a_7(n)\text{.}$Here $a_7(n)$ denotes a number of unrestricted 7-cores of n. Our techniques are elementary and require creative imagination only.     

Alternative characterisations of Lorentz-Karamata spaces

We present new formulae providing equivalent quasi-norms on Lorentz-Karamata spaces. Our results are based on properties of certain averaging operators on the cone of non-negative and non-increasing functions in convenient weighted Lebesgue spaces. We also illustrate connections between our results and mapping properties of such classical operators as the fractional maximal operator and the Riesz potential (and their variants) on the Lorentz-Karamata spaces.     

Pseudometrics, distances and multivariate polynomial inequalities

We discuss three natural pseudodistances and pseudometrics on a bounded domain in based on polynomial inequalities.     

Maximal ergodic theorems for some group actions

We prove maximal ergodic inequalities for a sequence of operators and for their averages in the noncommutative Lp-space. We also obtain the corresponding individual ergodic theorems. Applying these results to actions of a free group on a von Neumann algebra, we get noncommutative analogues of maximal ergodic inequalities and pointwise ergodic theorems of Nevo-Stein.     

Sharp Two Weight Inequalities for Commutators of Riemann-Liouville and Weyl Fractional Integral Operators

Let b be a BMO function, and the commutator of order k for the Weyl fractional integral. In this paper we prove weighted strong type (p, p) inequalities (p > 1) and weighted endpoint estimates (p = 1) for the operator and for the pairs of weights of the type (w, ), where w is any weight and is a suitable one-sided maximal operator. We also prove that, for weights, the operator is controlled in the L ^p (w) norm by a composition of the one-sided fractional maximal operator and the one-sided Hardy-Littlewood maximal operator iterated k times. These results improve those obtained by an immediate application of the corresponding two-sided results and provide a different way to obtain known results about the operators . The same results can be obtained for the commutator of order k for the Riemann-Liouville fractional integral This research has been partially supported by Spanish goverment Grant MTM2005-8350-C03-02. The first author was also supported by CONICET, ANPCyT and CAI+D-UNL. The second author was also supported by Junta de Andalucía Grant FQM 354.     

Monotone positive solution for three-point boundary value problem

In this paper, the existence of monotone positive solution for the following secondorder three-point boundary value problem is studied：
x″（t）＋f（t,x（t））=0,0〈t〈1,
x′（0）=0,x（1）＋δx′（η）=0,
where η ∈ （0, 1）, δ∈ [0, ∞）, f ∈ C（[0, 1] × [0, ∞）, [0, ∞））. Under certain growth conditions on the nonlinear term f and by using a fixed point theorem of cone expansion and compression of functional type due to Avery, Anderson and Krueger, sufficient conditions for the existence of monotone positive solution are obtained and the bounds of solution are given. At last, an example is given to illustrate the result of the paper.     

Moser Type Inequalities for Higher-Order Derivatives in Lorentz Spaces

Sharp constants are exhibited in exponential inequalities corresponding to the limiting case of the Sobolev inequalities in Lorentz-Sobolev spaces of arbitrary order.