Multilinear estimates for periodic KdV equations, and applications

We prove an endpoint multilinear estimate for the X^s,b spaces associated to the periodic Airy equation. As a consequence we obtain sharp local well-posedness results for periodic generalized KdV equations, as well as some global well-posedness results below the energy norm.     

Summability and estimates for polynomials and multilinear mappings

In this paper we extend and generalize several known estimates for homogeneous polynomials and multilinear mappings on Banach spaces. Applying the theory of absolutely summing nonlinear mappings, we prove that estimates which are known for mappings on ?_p spaces in fact hold true for mappings on arbitrary Banach spaces.     

On coefficient estimates for a class of holomorphic mappings

Let X be a complex Banach space with norm ‖ · ‖, B be the unit ball in X, D ⁿ be the unit polydisc in ℂ ⁿ . In this paper, we introduce a class of holomorphic mappings on B or D ⁿ . Let f(x) be a normalized locally biholomorphic mapping on B such that (Df(x))⁻¹ f(x) ∈ and f(x) − x has a zero of order k + 1 at x = 0. We obtain coefficient estimates for f(x). These results unify and generalize many known results. This work was supported by National Natural Science Foundation of China (Grant No. 10571164), Specialized Research Fund for the Doctoral Program of Higher Education (Grant No. 20050358052), the Jiangxi Provincial Natural Science Foundation of China (Grant No. 2007GZS0177) and Specialized Research Fund for the Doctoral Program of Jiangxi Normal University.     

Quantitative estimates for periodic points of reversible and symplectic holomorphic mappings

We show that reversible holomorphic mappings of have periodic points accumulating at an elliptic fixed point of general type. On the contrary, we also show the existence of holomorphic symplectic mappings that have no periodic points of certain periods in a sequence of deleted balls about an elliptic fixed point of general type. The radii of the balls are carefully chosen in terms of the periods, which allows us to show the existence of holomorphic mappi ngs of that are not reversible with respect to any involution with a holomorphic linear part, and that admit no invariant totally real and real surfaces. Received: 5 December 2000 / Published online: 4 April 2002     

多重奇异积分的多线性交换子

引进了由多重奇异积分和BMO函数生成的多线性交换子,然后得到了此类算子从Lebesgue积空间到Lebesgue空间的有界性,最后也给出了此类算子的加权和向量值不等式.     

Multilinear eigenfunction estimates and global existence for the three dimensional nonlinear Schrödinger equations

We study nonlinear Schrödinger equations, posed on a three dimensional Riemannian manifold M. We prove global existence of strong H¹ solutions on M=S³ and M=S²×S¹ as far as the nonlinearity is defocusing and sub-quintic and thus we extend results of Ginibre, Velo and Bourgain who treated the cases of the Euclidean space R³ and the torus T³=R³/Z³ respectively. The main ingredient in our argument is a new set of multilinear estimates for spherical harmonics.     

Gradient estimates for potentials of invertible gradient–mappings on the sphere

McCann showed that, if the potential of a gradient-mapping, on a compact riemannian manifold, is c-convex, the length of its gradient cannot exceed the diameter of the manifold. We improve this bound in two different manners on the constant curvature spheres, under assumptions on the relative density of the image-measure of the riemannian volume. One proof, with the standard metric, relies on the Brenier–McCann optimal measure-transport property; the other, purely pde, ignores it. This work is thought of as a preliminary step toward a second derivatives estimate which would yield smooth optimal measure-transport (open). Mathematics Subject Classification (2000) Primary 35J60, 35B45; Secondary 28C99, 49Q99     

The sharp estimates of homogeneous expansions for the generalized class of close-to-quasi-convex mappings

In this paper, the class of strongly close-to-quasi-convex mappings of type α and order β is introduced in the unit ball of complex Banach space or unit polydisk in ${\mathbb{C}}^n$, and we obtain the sharp estimates of homogeneous expansions for this class. These results generalize many known results.     

Schwarz-Pick estimates for holomorphic mappings from the polydisk to the unit ball

In this paper, Schwarz-Pick estimates of arbitrary order partial derivatives for holomorphic mappings from the polydisk to the unit ball are presented. We generalize the early work on Schwarz-Pick estimates of higher order derivatives for bounded holomorphic functions on the unit disk and the polydisk.     

Linear and Multilinear Algebra

An international journal publishing articles, reviews and problems