A note on Besov-Morrey spaces and Triebel-Lizorkin-Morrey spaces

In the present paper, we obtain three independent results on the Besov-Morrey spaces and the Triebel-Lizorkin-Morrey spaces. That is, we are going to obtain the characterization of local means, the boundedness of pseudo-differential operators and the characterization of the Hardy-Morrey spaces. By using the maximal estimate and the molecular decomposition, we shall integrate and extend the known results on these spaces.     

Conditions for existence of a global strong solution to one class of nonlinear evolution equations in Hilbert space

We obtain a criterion of global strong solvability for one class of nonlinear evolution equations in Hilbert space.     

多值半流理论及其在三维Navier-Stokes方程中的应用

利用多值半流方法研究三维有界区域上Navier-Stokes方程的全局吸引子,证得了多值半流的一些性质,并将这些性质应用于三维Navier-Stokes方程,得出了弱解的几种全局吸引子.从而表明在三维情形,通过多值半流来研究Navier-Stokes方程的全局吸引子是可行的.     

Conditions for existence of a global strong solution to one class of nonlinear evolution equations in Hilbert space. II

We obtain a criterion of global strong solvability for one class of nonlinear evolution equations in Hilbert space.     

Generalized coorbit space theory and inhomogeneous function spaces of Besov-Lizorkin-Triebel type

Coorbit space theory is an abstract approach to function spaces and their atomic decompositions. The original theory developed by Feichtinger and Gröchenig in the late 1980ies heavily uses integrable representations of locally compact groups. Their theory covers, in particular, homogeneous Besov-Lizorkin-Triebel spaces, modulation spaces, Bergman spaces and the recent shearlet spaces. However, inhomogeneous Besov-Lizorkin-Triebel spaces cannot be covered by their group theoretical approach. Later it was recognized by Fornasier and Rauhut (2005) [24] that one may replace coherent states related to the group representation by more general abstract continuous frames. In the first part of the present paper we significantly extend this abstract generalized coorbit space theory to treat a wider variety of coorbit spaces. A unified approach towards atomic decompositions and Banach frames with new results for general coorbit spaces is presented. In the second part we apply the abstract setting to a specific framework and study coorbits of what we call Peetre spaces. They allow to recover inhomogeneous Besov-Lizorkin-Triebel spaces of various types of interest as coorbits. We obtain several old and new wavelet characterizations based on explicit smoothness, decay, and vanishing moment assumptions of the respective wavelet. As main examples we obtain results for weighted spaces (Muckenhoupt, doubling), general 2-microlocal spaces, Besov-Lizorkin-Triebel-Morrey spaces, spaces of dominating mixed smoothness and even mixtures of the mentioned ones. Due to the generality of our approach, there are many more examples of interest where the abstract coorbit space theory is applicable.     

Linear functionals on nonlinear spaces and applications to problems from viscoplasticity theory

A classical result in the theory of monotone operators states that if C is a reflexive Banach space, and an operator A: C→C^* is monotone, semicontinuous and coercive, then A is surjective. In this paper, we define the ‘dual space’ C^* of a convex, usually not linear, subset C of some Banach space X (in general, we will have C^*⊃X^*) and prove an analogous result. Then, we give an application to problems from viscoplasticity theory, where the natural space to look for solutions is not linear. Copyright © 2007 John Wiley & Sons, Ltd.     

Plancherel-Pôlya type inequality on spaces of homogeneous type and its applications

In this paper, using the discrete Calderon reproducing formula on spaces of homogeneous type obtained by the author, we obtain the Plancherel-Pôlya type inequalities on spaces of homogeneous type. These inequalities give new characterizations of the Besov spaces and the Triebel-Lizorkin spaces on spaces of homogeneous type introduced earlier by the author and E. T. Sawyer and also allow us to generalize these spaces to the case where . Moreover, using these inequalities, we can easily show that the Littlewood-Paley -function and -function are equivalent on spaces of homogeneous type, which gives a new characterization of the Hardy spaces on spaces of homogeneous type introduced by Macias and Segovia.

     

Metaplectic representation on Wiener amalgam spaces and applications to the Schrödinger equation

We study the action of metaplectic operators on Wiener amalgam spaces, giving upper bounds for their norms. As an application, we obtain new fixed-time estimates in these spaces for Schrödinger equations with general quadratic Hamiltonians and Strichartz estimates for the Schrödinger equation with potentials V(x)=±²|x|.     

Rubio de Francía's weighted extrapolation in mixed-norm spaces and applications

Weighted extrapolation for pairs of functions in mixed-norm Banach function spaces defined on the product of quasi-metric measure spaces $(X,d,\mu )$ and $(Y,\rho ,\nu )$ are derived. As special cases, we have appropriate results for mixed-norm Lebesgue, Lorentz, and Orlicz spaces. Some of the derived results are applied to get weighted extrapolation in mixed-norm grand Lebesgue spaces.     

Deformation theorems on non‐metrizable vector spaces and applications to critical point theory

Let E be a Banach space and Φ : E → ? a ??¹‐functional. Let ?? be a family of semi‐norms on E which separates points and generates a (possibly non‐metrizable) topology ??_?? on E weaker than the norm topology. This is a special case of a gage space, that is, a topological space where the topology is generated by a family of semi‐metrics. We develop some critical point theory for Φ : (E, ??) → ?. In particular, we prove deformation lemmas where the deformations are continuous with respect to ??_??. In applications this yields a gain in compactness when Φ does not satisfy the Palais–Smale condition because one can work with the weak topology. We also prove some foundational results on gage spaces. In particular, we introduce the concept of Lipschitz continuity in this setting and prove the existence of Lipschitz continuous partitions of unity. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Weak Morrey spaces and strong solutions to the Navier-Stokes equations

We consider the Cauchy problem of Navier-Stokes equations in weak Morrey spaces. We first define a class of weak Morrey type spaces Mp*,λ(Rn) on the basis of Lorentz space Lp,∞ = Lp*(Rn)(in particular, Mp*,0(Rn) = Lp,∞, if p ＞ 1), and study some fundamental properties of them; Second,bounded linear operators on weak Morrey spaces, and establish the bilinear estimate in weak Morrey spaces. Finally, by means of Kato's method and the contraction mapping principle, we prove that the Cauchy problem of Navier-Stokes equations in weak Morrey spaces Mp*,λ(Rn) (1＜p≤n) is time-global well-posed, provided that the initial data are sufficiently small. Moreover, we also obtain the existence and uniqueness of the self-similar solution for Navier-Stokes equations in these spaces, because the weak Morrey space Mp*,n-p(Rn) can admit the singular initial data with a self-similar structure. Hence this paper generalizes Kato's results.     

Decomposition of refinable spaces and applications to operator equations

This paper presents a brief survey of several recent applications of multilevel techniques, in particular, in connection with the solution of periodic pseudodifferential equations. It is pointed out that these applications naturally lead to certain decompositions of refinable spaces which are induced by a class of linear projectors. Then recent results on the construction of such nonorthogonal wavelets are reviewed and extended to the particular needs of the present context.     

Weak Orlicz space and its applications to the martingale theory

In this paper,the weak Orlicz space wL Φ is introduced and its applications to the martingale theory are discussed.In particular,a series of martingale inequalities including the maximal function inequality in weak Orlicz spaces are established;the relationships between these spaces are investigated.Moreover,the boundedness of several sublinear operators from one weak Orlicz space to another is proved;their vector-valued analogues are also considered.     

Well-posedness and scattering of solutions to the sixth-order Boussinesq type equation in the framework of modulation spaces

In this paper, we investigate the initial value problem for the sixth order Boussinesq type equation in the framework of modulation spaces. Under suitable conditions, we first prove that the problem has a unique local solutions and global solutions. Then scattering and stability of solutions are also discussed. The proof is mainly based on the decay properties of the solutions operator in modulation spaces and the contraction mapping principle.     

Local Hardy spaces of Musielak-Orlicz type and their applications

Letφ:R n × [0,∞) → [0,∞) be a function such that φ(x,·) is an Orlicz function and (·,t) ∈ A ∞loc (Rn) (the class of local weights introduced by Rychkov).In this paper,the authors introduce a local Musielak-Orlicz Hardy space hφ(Rn) by the local grand maximal function,and a local BMO-type space bmoφ(Rn) which is further proved to be the dual space of hφ(Rn).As an application,the authors prove that the class of pointwise multipliers for the local BMO-type space bmo φ (Rn),characterized by Nakai and Yabuta,is just the dual of L 1 (Rn) + h Φ 0 (Rn),where φ is an increasing function on (0,∞) satisfying some additional growth conditions and Φ 0 a Musielak-Orlicz function induced by φ.Characterizations of hφ(Rn),including the atoms,the local vertical and the local nontangential maximal functions,are presented.Using the atomic characterization,the authors prove the existence of finite atomic decompositions achieving the norm in some dense subspaces of hφ(Rn),from which,the authors further deduce some criterions for the boundedness on hφ(Rn) of some sublinear operators.Finally,the authors show that the local Riesz transforms and some pseudo-differential operators are bounded on hφ(Rn).     

Remarks on Fourier multipliers and applications to the wave equation

Exploiting continuity properties of Fourier multipliers on modulation spaces and Wiener amalgam spaces, we study the Cauchy problem for the NLW equation. Local wellposedness for rough data in modulation spaces and Wiener amalgam spaces is shown. The results formulated in the framework of modulation spaces refine those in [A. Bényi, K.A. Okoudjou, Local well-posedness of nonlinear dispersive equations on modulation spaces, preprint, April 2007 (available at ArXiv:0704.0833v1)]. The same arguments may apply to obtain local wellposedness for the NLKG equation.     

Relatively compact families of functionals on abstract Wiener space and applications

In this paper we prove two relatively compact criterions in some L^p-spaces(p>1) for the set of functionals on abstract Wiener space in terms of the compact embedding theorems in finite dimensional Sobolev spaces. Then, as applications we study several relatively compact families of random fields for the solutions to SDEs (and SPDEs) with coefficients satisfying some bounded assumptions, some stochastic integrals, and local times of diffusion processes.