Traces of functions in Hardy and Besov spaces on Lipschitz domains with applications to compensated compactness and the theory of Hardy and Bergman type spaces

In this paper we study conditions guaranteeing that functions defined on a Lipschitz domain Ω have boundary traces in Hardy and Besov spaces on ∂Ω. In turn these results are used to develop a new approach to the theory of compensated compactness and the theory of non-locally convex Hardy and Bergman type spaces.     

Atomic and molecular decompositions of anisotropic Besov spaces

In this work we develop the theory of weighted anisotropic Besov spaces associated with general expansive matrix dilations and doubling measures with the use of discrete wavelet transforms. This study extends the isotropic Littlewood- Paley methods of dyadic -transforms of Frazier and Jawerth [19, 21] to non-isotropic settings.Several results of isotropic theory of Besov spaces are recovered for weighted anisotropic Besov spaces. We show that these spaces are characterized by the magnitude of the -transforms in appropriate sequence spaces. We also prove boundedness of an anisotropic analogue of the class of almost diagonal operators and we obtain atomic and molecular decompositions of weighted anisotropic Besov spaces, thus extending isotropic results of Frazier and Jawerth [21].The author was partially supported by the NSF grant DMS-0441817.     

Differential inequalities and equations in Banach spaces with a cone

A class of second-order abstract dissipative evolution differential operators D$D$ with 0∈kerD$0\in kerD$ is shown for which the fact that a non-zero t?u(t)$t?u\left(t\right)$ belongs to a cone and −Du$-Du$ to a dual cone may hold only on time intervals whose length is less than or equal to a defined number. Then oscillatory functions are dealt with in the framework of Banach spaces with a cone and conditions for the existence of a uniform oscillatory time for solutions of the equation Du=0$Du=0$ are given.     

Invariants and spaces of zero solutions of linear partial differential operators

It is shown that for open convex , d > 1 and a nontrivial polynomial P the space does not have property . If P is elliptic or homogeneous, then this holds for every open Ω. For even cannot occur and if it occurs for some Ω, then P must be hypoelliptic. Received: 18 July 2005     

Stochastic partial differential equations in M-type 2 Banach spaces

We study abstract stochastic evolution equations in M-type 2 Banach spaces. Applications to stochastic partial differential equations inL ^p spaces withp2 are given. For example, solutions of such equations are Hölder continuous in the space variables.The author is an Alexander von Humboldt Stiftung fellow     

Compactly supported refinable distributions in Triebel-Lizorkin spaces and besov spaces

The aim of this article is to characterize compactly supported refinable distributions in Triebel-Lizorkin spaces and Besov spaces by projection operators on certain wavelet space and by some operators on a finitely dimensional space.Research partially supported by the National Natural Sciences Foundation of China # 69735020, the Tian Yuan Projection of the National Natural Sciences Foundation of China, the Doctoral Bases Promotion Foundation of National Educational Commission of China #97033519 and the Zhejiang Provincial Sciences Foundation of China # 196083, and by the Wavelets Strategic Research Program funded by the National Science and Technology Board and the Ministry of Education, Singapore.     

Continuity properties for modulation spaces, with applications to pseudo-differential calculus—I

Let M^p,q denote the modulation space with parameters p,q∈[1,∞]. If 1/p₁+1/p₂=1+1/p₀ and 1/q₁+1/q₂=1/q₀, then it is proved that . The result is used to get inclusions between modulation spaces, Besov spaces and Schatten classes in calculus of Ψdo (pseudo-differential operators), and to extend the definition of Toeplitz operators. We also discuss continuity of ambiguity functions and Ψdo in the framework of modulation spaces.     

New characterizations of Morrey spaces and their preduals with applications to fractional Laplace equations

In this article, the authors characterize the Morrey spaces as well as their preduals via quadratic functions related to the Taylor remainder of the kernel of the Riesz potential. As applications, the authors obtain some strong capacitary inequalities, which are then used to study the regularity of the duality/weak solution to the fractional Laplace equation with measure data.     

On the Besov regularity of conformal maps and layer potentials on nonsmooth domains

We study the Besov regularity of conformal mappings for domains with rough boundary based on the well-posedness for the Dirichlet problem with Besov data. Also, sharp invertibility results for the classical layer potential operators on Sobolev-Besov spaces on the boundary of curvilinear polygons are obtained.     

Periodic solutions of second order differential equations in Banach spaces

We use operator-valued Fourier multiplier theorems to study second order differential equations in Banach spaces. We establish maximal regularity results in L^p and C^s for strong solutions of a complete second order equation. In the second part, we study mild solutions for the second order problem. Two types of mild solutions are considered. When the operator A involved is the generator of a strongly continuous cosine function, we give characterizations in terms of Fourier multipliers and spectral properties of the cosine function. The results obtained are applied to elliptic partial differential operators. The first author is supported in part by Convenio de Cooperación Internacional (CONICYT) Grant # 7010675 and the second author is partially financed by FONDECYT Grant # 1010675     

The KKM principle in abstract convex spaces: Equivalent formulations and applications

The partial KKM principle for an abstract convex space is an abstract form of the classical KKM theorem. A KKM space is an abstract convex space satisfying the partial KKM principle and its “open” version. In this paper, we clearly derive a sequence of a dozen statements which characterize the KKM spaces and equivalent formulations of the partial KKM principle. As their applications, we add more than a dozen statements including generalized formulations of von Neumann minimax theorem, von Neumann intersection lemma, the Nash equilibrium theorem, and the Fan type minimax inequalities for any KKM spaces. Consequently, this paper unifies and enlarges previously known several proper examples of such statements for particular types of KKM spaces.     

Perturbation of the mapping and solvability theorems in Banach space

Here we consider perturbations of continuous mappings on Banach spaces, and investigate their images under various conditions. Consequently we study the solvability of some classes of equations and inclusions. For these we start by the investigation of local properties of the considered mapping and local comparisons of this mapping with certain smooth mappings. Moreover, we study different mixed problems.     

Extrapolation with weights, rearrangement-invariant function spaces, modular inequalities and applications to singular integrals

We present an extrapolation theory that allows us to obtain, from weighted L^p inequalities on pairs of functions for p fixed and all A_∞ weights, estimates for the same pairs on very general rearrangement invariant quasi-Banach function spaces with A_∞ weights and also modular inequalities with A_∞ weights. Vector-valued inequalities are obtained automatically, without the need of a Banach-valued theory. This provides a method to prove very fine estimates for a variety of operators which include singular and fractional integrals and their commutators. In particular, we obtain weighted, and vector-valued, extensions of the classical theorems of Boyd and Lorentz-Shimogaki. The key is to develop appropriate versions of Rubio de Francia's algorithm.     

On non-surjective coarse isometries between Banach spaces

Assume that X, Y are real Banach spaces, Y has uniform convexity of type p (≥ 1), and f: X → Y is a standard coarse isometry. In this paper, we show that if

then there is a linear isometry U : X → Y so that

where is defined by

Representation properties of coarse isometries in free ultrafilter limits on are also discussed.     

The splitting of exact sequences of PLS-spaces and smooth dependence of solutions of linear partial differential equations

We investigate the splitting of short exact sequences of the form

0→X→Y→E→0,     

The lattice of closed ideals in the Banach algebra of operators on certain Banach spaces

Very few Banach spaces E are known for which the lattice of closed ideals in the Banach algebra of all (bounded, linear) operators on E is fully understood. Indeed, up to now the only such Banach spaces are, up to isomorphism, Hilbert spaces and the sequence spaces c₀ and ?_p for 1?p<∞. We add a new member to this family by showing that there are exactly four closed ideals in for the Banach space E?(⊕?₂ⁿ)_c0, that is, E is the c₀-direct sum of the finite-dimensional Hilbert spaces ?₂¹,?₂²,…,?₂ⁿ,… .     

Oscillation of linear difference equations in Banach spaces

One can easily show that almost all solutions of the difference equation

     

The boundary-contact problem of elasticity for homogeneous anisotropic media with a contact on some part of the boundaries

The existence and uniqueness of solutions of the boundary-contact problem of elasticity for homogeneous anisotropic media with a contact on some part of their boundaries are investigated in the Besov and Bessel potential classes using the methods of the potential theory and the theory of pseudodifferential equations on manifolds with boundary. The smoothness of the solutions obtained is studied.     

Anisotropic Triebel-Lizorkin spaces with doubling measures

We introduce and study anisotropic Triebel-Lizorkin spaces associated with general expansive dilations and doubling measures on ℝⁿ with the use of wavelet transforms. This work generalizes the isotropic methods of dyadic ϕ-transforms of Frazier and Jawerth to nonisotropic settings. We extend results involving boundedness of wavelet transforms, almost diagonality, smooth atomic and molecular decompositions to the setting of doubling measures. We also develop localization techniques in the endpoint case of p = ∞, where the usual definition of Triebel-Lizorkin spaces is replaced by its localized version. Finally, we establish nonsmooth atomic decompositions in the range of 0 < p ≤ 1, which is analogous to the usual Hardy space atomic decompositions.     

Multisummability of formal solutions to the Cauchy problem for some linear partial differential equations

The paper considers the Cauchy problem for linear partial differential equations of non-Kowalevskian type in the complex domain. It is shown that if the Cauchy data are entire functions of a suitable order, the problem has a formal solution which is multisummable. The precise bound of the admissible order of entire functions is described in terms of the Newton polygon of the equation.