A characterization of

This work is devoted to the relationship between topological properties of a space and those of (= the space of continuous real-valued functions on , with the topology of pointwise convergence). The emphasis is on -compactness of and on location of in . In particular, -compact cosmic spaces are characterized in this way.

     

Operator ideal norms on

Let be a real number such that and its conjugate exponent . We prove that for an operator defined on with values in a Banach space, the image of the unit ball determines whether belongs to any operator ideal and its operator ideal norm. We also show that this result fails to be true in the remaining cases of . Finally we prove that when the result holds in finite dimension, the map which associates to the image of the unit ball the operator ideal norm is continuous with respect to the Hausdorff metric.

     

Distributions supported in a hypersurface and local

We give a necessary condition for a distribution with compact support in a hypersurface to be in the local Hardy space . We apply this condition to prove a result distinguishing two types of Hardy spaces of distributions on a smooth domain .

     

-algebras generated by a subnormal operator

Using the functional calculus for a normal operator, we provide a result for generalized Toeplitz operators, analogous to the theorem of Axler and Shields on harmonic extensions of the disc algebra. Besides that result, we prove that if is an injective subnormal weighted shift, then any two nontrivial subspaces invariant under cannot be orthogonal to each other. Then we show that the -algebra generated by and the identity operator contains all the compact operators as its commutator ideal, and we give a characterization of that -algebra in terms of generalized Toeplitz operators. Motivated by these results, we further obtain their several-variable analogues, which generalize and unify Coburn's theorems for the Hardy space and the Bergman space of the unit ball.

     

Fourier multipliers on weighted -spaces

In his 1986 paper in the Rev. Mat. Iberoamericana, A. Carbery proved that a singular integral operator is of weak type on if its lacunary pieces satisfy a certain regularity condition. In this paper we prove that Carbery's result is sharp in a certain sense. We also obtain a weighted analogue of Carbery's result. Some applications of our results are also given.

     

approximations of inertial manifolds via finite differences

We construct an inertial manifold for the evolution equation as a limit of the inertial manifolds for the difference approximations of the Trotter-Kato type and show that this limit is taken in a topology.

     

The problem on domains with piecewise smooth boundaries with applications

Let be a bounded domain in such that has piecewise smooth boudnary. We discuss the solvability of the Cauchy-Riemann equation

where is a smooth -closed form with coefficients up to the bundary of , and . In particular, Equation (0.1) is solvable with smooth up to the boundary (for appropriate degree if satisfies one of the following conditions:

i)

is the transversal intersection of bounded smooth pseudoconvex domains.

ii)

where is the union of bounded smooth pseudoconvex domains and is a pseudoconvex convex domain with a piecewise smooth boundary.

iii)

where is the intersection of bounded smooth pseudoconvex domains and is a pseudoconvex domain with a piecewise smooth boundary.

The solvability of Equation (0.1) with solutions smooth up to the boundary can be used to obtain the local solvability for on domains with piecewise smooth boundaries in a pseudoconvex manifold.

     

Positive differentials, theta functions and Hardy kernels

Let be a planar regular region whose Schottky double has genus and set . For fixed we determine the range of the function where is the Riemann theta function on . Also we introduce two weighted Hardy spaces to study the problem when the matrix is positive definite. The proof relies on new theta identities using Fay's trisecants formula.

     

Solving the -Laplacian on manifolds

We prove in this paper that the equation on a -hyperbolic manifold has a solution with -integrable gradient for any bounded measurable function with compact support.

     

An definition of interpolating Blaschke products

We give a new characterization of interpolating Blaschke products in terms of -norms of their reciprocals. We also obtain a characterization of finite unions of interpolating sequences.

     

A convolution estimate for a measure on a curve in . II

This paper contains almost-sharp convolution estimates for measures on the curve in .

     

On the optimal local regularity for the Yang-Mills equations in

The aim of the paper is to develop the Fourier Analysis techniques needed in the study of optimal well-posedness and global regularity properties of the Yang-Mills equations in Minkowski space-time , for the case of the critical dimension . We introduce new functional spaces and prove new bilinear estimates for solutions of the homogeneous wave equation, which can be viewed as generalizations of the well-known Strichartz-Pecher inequalities.

     

Level sets of a typical function

We determine the level set structure of a typical function.

     

Existence and uniqueness for a degenerate parabolic equation with -data

In this paper we study existence and uniqueness of solutions for the boundary-value problem, with initial datum in ,

where a is a Carathéodory function satisfying the classical Leray-Lions hypothesis, is the Neumann boundary operator associated to , the gradient of and is a maximal monotone graph in with .

     

On the approximation of stochastic partial differential equations i

The stability of abstract stochastic partial differential equations with respect to the simultaneous perturbation of the driving processes and of the differential operators is investigated. The results obtained here will be applied to concrete stochastic partial differential equations in the continuation of this paper     

Normality criteria for families of holomorphic mappings of several complex variables into

By applying the heuristic principle in several complex variables obtained by Aladro and Krantz, we shall prove some normality criteria for families of holomorphic mappings of several complex variables into , the complex N-dimensional projective space, related to Green's and Nochka's Picard type theorems. The equivalence of normality to being uniformly Montel at a point will be obtained. Some examples will be given to complement our theory in this paper.

     

The closing lemma for nonsingular endomorphisms equivariant under free actions of finite groups

In this paper a closing lemma for nonsingular endomorphisms equivariant under free actions of finite-groups is proved. Hence a recurrent trajectory, as well as all of its symmetric conjugates, of a nonsingular endomorphism equivariant under a free action of a finite group can be closed up simultaneously by an arbitrarily small equivariant perturbation.

     

Gradient type noises II - Systems of stochastic partial differential equations

The present paper is the second and main part of a study of partial differential equations under the influence of noisy perturbations. Existence and uniqueness of function solutions in the mild sense are obtained for a class of deterministic linear and semilinear parabolic boundary initial value problems. If the noise data are random, the results may be seen as a pathwise approach to SPDE's. For typical examples, such as spatially one-dimensional stochastic heat equations with additive or multiplicative perturbations of fractional Brownian type, we recover and extend known results. In addition, we propose to consider partial noises of low order.     

A priori error estimates for Galerkin approximations to porous medium and fast diffusion equations

Galerkin approximations to solutions of a Cauchy-Dirichlet problem governed by the generalized porous medium equation

on bounded convex domains are considered. The range of the parameter includes the fast diffusion case . Using an Euler finite difference approximation in time, the semi-discrete solution is shown to converge to the exact solution in norm with an error controlled by for and for . For the fully discrete problem, a global convergence rate of in norm is shown for the range . For , a rate of is shown in norm.