A simple and consistent definition of homogeneous besov spaces on stratified lie groups authors. Siam journal on mathematical analysis siam society for. Local and global sobolev interpolation inequalities for the follandstein sobolev spaces on stratified groups, pdf. Analysis on lie groups with polynomial growth book also available. Sobolev spaces on metric measure spaces by juha heinonen. University of jyvaskyla records all publications to tutka database. Boundedness of certain commutators on triebellizorkin spaces. New type singular operators on product spaces tan, chaoqiang, taiwanese journal of mathematics. Homogeneous besov spaces on stratified lie groups and their. For holder coefficients we derive the wellposedness in the spaces of ultradistributions associated to rockland operators on graded groups.
Gevrey functions and ultradistributions on compact lie. A nonuniform estimate on the system of homogeneous balls is given. Mar 26, 2009 then the authors characterize the norms of inhomogeneous besov and triebel lizorkin spaces by the norms of homogeneous besov and triebel lizorkin spaces together with the norm of local hardy spaces in the sense of goldberg. G 0 stratified lie group g in terms of littlewoodpaley type decompositions, in analogy to the wellknown characterization of the euclidean case. This new function space covers the besov spaces and the triebel lizorkin spaces in the homogeneous case, and further the morrey spaces. Moreover, we discuss their nontriviality under a poincar\e inequality. Download pdf exceptional lie algebras and the structure. Zun wei fu, guang lan wang and qing yan wu subject. Let us also remark that the bessel potential spaces. Boundedness of gfunctions on triebellizorkin spaces zhang, chunjie and chen, jiecheng, taiwanese journal of mathematics, 2009. It is remarkable that so much about lie groups could be packed into this small book.
Littlewoodpaley characterization of holderzygmund spaces on stratified lie groups guorong hu. Boundedness and continuity of several integral operators with. On a metric measure space satisfying the doubling property, we establish several optimal characterizations of besov and triebellizorkin spaces, including a pointwise characterization. The natural setting for this kind of problem is that of homogeneous groups, in particular, including the cases of anisotropic structures on. Publications department of mathematics and statistics. Triebellizorkin, wiener and beurling spaces on compact homogeneous manifolds. Firstly,a perturbation method is used to obtain an initial solution for the case 0 by using a nonlinear time transform and rewriting the differential equation as an integral aqelation.
Lipschitz spaces on stratified groups its lie algebra. The results were rst published in pap, 1992a, 1992b, 1992c. In this paper we prove that the jacobian jf of a map ff 1. We show that such spaces can be well defined under some conditions. But after reading it, students will be wellprepared to continue with more advanced, graduatelevel topics in differential geometry or the theory of lie groups. Littlewoodpaley characterization of holderzygmund spaces on. One of the important examples of stratified lie groups is the heisenberg group h n. On a metric measure space satisfying the doubling property, we establish several optimal characterizations of besov and triebel lizorkin spaces, including a pointwise characterization. Littlewoodpaley characterizations of haj\laszsobolev and triebel lizorkin. Russ, the atomic decomposition for tent spaces on spaces of homogeneous type, cmaamsi research symposium asymptotic geometric analysis, harmonic analysis, and related topics 42 proc.
Suppose that g is a stratified lie group with homogeneous dimension q and that j. Our approach mainly relies on almost orthogonality estimates and can be used to study other function spaces such as besov and triebellizorkin spaces on stratified lie groups. Pdf homogeneous triebellizorkin spaces on stratified lie. In this paper we introduce a new function space which unifies and generalizes the besovtype and the triebel lizorkin type function spaces introduced by s. Pdf strong hypercontractivity and logarithmic sobolev. Then the exponential map is a diffeomorphism of g onto g, and we can transfer, via the exponential map, the dilations 8r to g. The inhomogeneous versions of triebel lizorkin spaces and besov spaces, which are denoted by and, respectively, are obtained by adding the term to the right hand side of 11 or 12 with replaced by, where. Triebellizorkin, wiener and beurling spaces on compact homogeneous manifolds, ann. The goal of the centralized recording is to ease researchers activities. New properties of besov and triebellizorkin spaces on. We also present a nikolskiitype inequality and the littlewoodpaley theorem that play a role. Vectorvalued functions, closed operators, integration in sequentially complete locally convex spaces, complex powers of almost csectorial operators 1. F s,q p vextenddouble vextenddouble vextenddouble vextenddouble vextenddouble parenleftbigg. Spin wavelets on the sphere, journal of fourier analysis and.
In this paper we study the cauchy problem for the wave equations for hypoelliptic homogeneous leftinvariant operators on graded lie groups when the timedependent nonnegative propagation speed is regular, holder, and distributional. Abstract let lie groups in terms of littlewoodpaleytype decomposition. Compensated compactness and the stratified lie group. Drinfeld showed that if g is a poisson lie group with corresponding lie bialgebra g, then the isomorphism classes of poisson homogeneous g spaces are essentially in a 11 correspondence with the g. Research article homogeneous triebellizorkin spaces on. Interpolation of hardysobolevbesovtriebellizorkin spaces. Compactness of riesz transform commutator on stratified. Homogeneous besov spaces on stratified lie groups and. Coarse models of homogeneous spaces and translationslike. These function spaces on stratified lie groups were introduced earlier by folland in by using sublaplacians. Abstract in this note, we give embeddings and other properties of besov spaces, as well as spectral and fourier multiplier theorems, in the setting of graded lie groups. The semi stable limit cycles and bifurcations of lienard equation are studied by the perturbation incremental method.
Lizorkin, wiener and beurling spaces on compact homogeneous manifolds. The proof for the characterisation of gevrey spaces will rely on the harmonic. Homogeneous triebellizorkin spaces on stratified lie groups guorong hu graduate school of ma thematical sciences, e uni versity o f t okyo, 381 komaba, m ekuroku, t o kyo 1538914, japa n. Weighted lipschitz estimates for commutators of onesided operators on onesided triebel lizorkin spaces author. Gevrey functions and ultradistributions on compact lie groups and homogeneous spaces. Multipliers for besov spaces on graded lie groups sciencedirect. Characterizations of besov and triebellizorkin spaces on. The space of in general twisted representations of the fundamental group of a closed surface in a compact lie group or, equivalently, a moduli space of central yangmills connections on a principal bundle over a surface, inherits a positive normal stratified kahler structure. The pointwise kernel lower bound obtained in is as follows. One of the key ideas in the theory of lie groups is to replace the global object, the group, with its local or linearized version, which lie himself called its infinitesimal group and which has since become known as its lie algebra. In this paper we introduce a new function space which unifies and generalizes the besovtype and the triebellizorkintype function spaces introduced by s. Topics discussed include the classification of compact and connected lie groups, lie algebras, geometrical aspects of compact lie groups and reductive homogeneous spaces, and important classes of homogeneous spaces, such as symmetric spaces and flag manifolds.
This paper investigates the besov spaces on compact lie groups in a subelliptic setting, that is, associated with a family of vector fields, satisfying the hormander condition, and their corresponding sublaplacian. Homogeneous besov spaces on stratified lie groups and their wavelet. Pdf some equivalent definitions of high order sobolev spaces on. Homogeneous besov and triebellizorkin spaces associated to. In particular, as a special corollary of our much more general results, we have the following theorem see theorems a and e in section 1.
A nilpotent lie algebra and its associated lie group g is stratified if there is a direct sum vector space decomposition 1. Weighted lipschitz estimates for commutators of onesided. Ellis on extensions of the riemann and lebesgue integrals by nets. Nikolskii inequality and functional classes on compact lie groups. A simple and consistent definition of homogeneous besov. Researchers can send the publication to the library as a pdf file or send new publications by using the online form. These groups, equipped with the socalled homogeneous norm, were introduced by folland in the framework of subelliptic pde 14. Such decompositions are defined via the spectral measure of a sublaplacian on g, in place of the fourier transform in the classical setting. Abstract degenerate volterra integrodifferential equations. In recent years there were several efforts of extending besov and triebellizorkin spaces from euclidean spaces to other domains and nonisotropic settings. Continuous frames, function spaces, and the discretization problem massimo fornasier, holger rauhut october, 2004 abstract a continuous frame is a family of vectors in a hilber.
Journal of mathematical analysis and applications 4. In this article we consider the fractional integral operator i. This new function space covers the besov spaces and the triebellizorkin spaces in the homogeneous case, and further the morrey spaces. Homogeneous triebellizorkin spaces on stratified lie groups. Access full article top access to full text full pdf how to cite top. Localization principle for triebellizorkin spaces on. A note on weighted besovtype and triebel lizorkin type spaces this is an open access article distributed under the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. A stratified group is a connected, simply connected, nilpotent lie group g whose lie algebra g is stratified. Pdf homogeneous triebellizorkin spaces on stratified. Dec 14, 2019 3 besov and triebellizorkin spaces associated to operators. This section is devoted to the definition of the besov and triebellizorkin spaces associated to operators and their properties such as square function characterization, atomic decomposition, duality, and interpolation. Steinweiss inequalities for the fractional integral. On the other hand, in 14 the authors studied the boundedness properties of b,t on the triebellizorkin space.
Analysis and geometry in metric spaces directory of open. The main tool is the di erentiablity of lipschitz maps, proved by. Dec 01, 2010 in recent years, a rapidly growing literature has focussed on the construction of wavelet systems to analyze functions defined on the sphere. Riemannian manifolds, lie groups and lipschitz manifolds. Spectral multipliers of selfadjoint operators on besov and. Factorizations and hardyrellich inequalities on stratified groups, j. Land utilization developmental tendency and basic macro. Linearly bounded liars, adaptive covering codes, and deterministic. Dynamical yangbaxter equations, quasipoisson homogeneous spaces, and quantization article pdf available in letters in mathematical physics 7.
Download analysis on lie groups with polynomial growth in pdf and epub formats for free. These function spaces on stratified lie groups were introduced earlier by folland in 7 by using sublaplacians. We establish wavelet characterizations of homogeneous besov spaces on stratified lie groups, both in terms of continuous and. It is shown that the scale of these spaces is independent of the choice of littlewoodpaleytype decomposition and the sublaplacian used for the construction of the decomposition. Motivating the development of the analysis associated to homogeneous groups in, folland and stein raised an important question of determining which elements of the classical harmonic analysis do depend only on the group and the dilation structures. Multipliers for besov spaces on graded lie groups topic.
Lie group of real rank at least two, margulis established that these lattices are. Analysis on metric spaces emerged in the 1990s as an independent research field providing a unified treatment of firstorder analysis in diverse and potentially nonsmooth settings. Functional inequalities in stratified lie groups with sobolev, besov. Ren, applications of orlicz spaces dekker, new york, 2000. Homogeneous triebellizorkin spaces on stratified lie groups homogeneous triebellizorkin spaces on stratified lie groups. Type 1,1operators, triebellizorkin spaces, twisted diagonal, support rule.
Drinfeld showed that if g is a poisson lie group with corresponding lie bialgebra g, then the isomorphism classes of poisson homogeneous g spaces are. Littlewoodpaley theorem, nikolskii inequality, besov spaces. In this section we define homogeneous besov spaces on stratified lie groups via littlewoodpaley decompositions of distributions as. Asymptotic behavior for radial solutions for a certain class of semilinear equations, pdf. Carleson measure characterizations of the campanato type. The besov spaces for certain parameters p, q and r have been considered in in the. Triebellizorkin and besov spaces of arbitrary number of parameters.
Homogeneous triebellizorkin spaces with full range of parameters are introduced on stratified lie groups in terms of littlewoodpaleytype decomposition. For a stratified symplectic space, a suitable concept of stratified kaehler polarization encapsulates kaehler polarizations on the strata and the behaviour of the polarizations across the strata and leads to the notion of stratified kaehler space which establishes an intimate relationship between nilpotent orbits, singular reduction, invariant theory, reductive dual pairs, jordan triple. In particular, we propose needlettype spin wavelets as an extension of the needlet. You can find the publications of the department of mathematics and statistics below. Embedding properties between subelliptic besov spaces and besov spaces associated to the laplacian on the group are proved. Abstractcarnot groups are distinguished spaces that are rich of structure. Before we introduce the homogeneous triebellizorkin spaces on strati ed lie groups, we prove the following basic estimate,whichisageneralizationof,lemmab. The author provides several examples and computations. This paper gives a brief account on the main land utiliztion problems in pearl river delta after reformation and open as well as the land utiliztion development tendency and basic macro control ways according to the development directions and goals of future economy and society to achieve the rational and sustainable utilization of land resources,several problems should be cared. The theory of lie groups involves many areas of mathematics. The sharp poincare inequality for free vector fields. Also, the authors obtain the sharp locally integrability of elements in besov and triebel lizorkin spaces.
A note on weighted besovtype and triebellizorkintype spaces. Lie groups are smooth differentiable manifolds and as such can be studied using differential calculus, in contrast with the case of more general topological groups. Follands sobolev spaces coincide with those introduced in on graded lie groups in the setting of stratified groups. Jun 23, 2018 such decompositions are defined via the spectral measure of a sublaplacian on g, in place of the fourier transform in the classical setting.
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