Pdf homogeneous triebellizorkin spaces on stratified lie. In this note we study besov, triebellizorkin, wiener, and beurling function. Boundedness of certain commutators on triebellizorkin spaces. Pdf strong hypercontractivity and logarithmic sobolev. Such decompositions are defined via the spectral measure of a sublaplacian on g, in place of the fourier transform in the classical setting. Lizorkin, wiener and beurling spaces on compact homogeneous manifolds.
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. Abstract degenerate volterra integrodifferential equations. Download pdf exceptional lie algebras and the structure. This new function space covers the besov spaces and the triebel lizorkin spaces in the homogeneous case, and further the morrey spaces. A stratified group is a connected, simply connected, nilpotent lie group g whose lie algebra g is stratified. Lipschitz spaces on stratified groups its lie algebra. Type 1,1operators, triebellizorkin spaces, twisted diagonal, support rule. The semi stable limit cycles and bifurcations of lienard equation are studied by the perturbation incremental method. In this section we define homogeneous besov spaces on stratified lie groups via littlewoodpaley decompositions of distributions as. 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. The proof for the characterisation of gevrey spaces will rely on the harmonic. Let us also remark that the bessel potential spaces. Riemannian manifolds, lie groups and lipschitz manifolds.
In this paper we prove that the jacobian jf of a map ff 1. 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. 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. Gevrey functions and ultradistributions on compact lie groups and homogeneous spaces. Download analysis on lie groups with polynomial growth in pdf and epub formats for free. A simple and consistent definition of homogeneous besov spaces on stratified lie groups authors. 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. 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. You can find the publications of the department of mathematics and statistics below. The theory of lie groups involves many areas of mathematics.
Pdf some equivalent definitions of high order sobolev spaces on. Littlewoodpaley characterizations of haj\laszsobolev and triebel lizorkin. One of the important examples of stratified lie groups is the heisenberg group h n. Sobolev spaces on metric measure spaces by juha heinonen. Triebellizorkin, wiener and beurling spaces on compact homogeneous manifolds, ann. Gevrey functions and ultradistributions on compact lie. Ren, applications of orlicz spaces dekker, new york, 2000.
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. The pointwise kernel lower bound obtained in is as follows. Compactness of riesz transform commutator on stratified. Publications department of mathematics and statistics. New type singular operators on product spaces tan, chaoqiang, taiwanese journal of mathematics. Littlewoodpaley characterization of holderzygmund spaces on stratified lie groups guorong hu. Asymptotic behavior for radial solutions for a certain class of semilinear equations, pdf. Vectorvalued functions, closed operators, integration in sequentially complete locally convex spaces, complex powers of almost csectorial operators 1. In particular, we propose needlettype spin wavelets as an extension of the needlet.
Ellis on extensions of the riemann and lebesgue integrals by nets. 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. Land utilization developmental tendency and basic macro. Access full article top access to full text full pdf how to cite top. On a metric measure space satisfying the doubling property, we establish several optimal characterizations of besov and triebel lizorkin spaces, including a pointwise characterization. Journal of mathematical analysis and applications 4. Factorizations and hardyrellich inequalities on stratified groups, j. Steinweiss inequalities for the fractional integral. Spin wavelets on the sphere, journal of fourier analysis and. These function spaces on stratified lie groups were introduced earlier by folland in by using sublaplacians.
Local and global sobolev interpolation inequalities for the follandstein sobolev spaces on stratified groups, pdf. Dynamical yangbaxter equations, quasipoisson homogeneous spaces, and quantization article pdf available in letters in mathematical physics 7. 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. Homogeneous besov spaces on stratified lie groups and their. Homogeneous besov spaces on stratified lie groups and. Suppose that g is a stratified lie group with homogeneous dimension q and that j. We establish wavelet characterizations of homogeneous besov spaces on stratified lie groups, both in terms of continuous and. 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. 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. In recent years there were several efforts of extending besov and triebellizorkin spaces from euclidean spaces to other domains and nonisotropic settings. Homogeneous besov spaces on stratified lie groups and their wavelet. 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.
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. Also, the authors obtain the sharp locally integrability of elements in besov and triebel lizorkin spaces. Characterizations of besov and triebellizorkin spaces on. The goal of the centralized recording is to ease researchers activities. 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. University of jyvaskyla records all publications to tutka database. Localization principle for triebellizorkin spaces on. 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.
On a metric measure space satisfying the doubling property, we establish several optimal characterizations of besov and triebellizorkin spaces, including a pointwise characterization. Weighted lipschitz estimates for commutators of onesided. 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. Coarse models of homogeneous spaces and translationslike. Linearly bounded liars, adaptive covering codes, and deterministic. 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. Multipliers for besov spaces on graded lie groups topic. Homogeneous triebellizorkin spaces on stratified lie groups homogeneous triebellizorkin spaces on stratified lie groups. 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. Dec 14, 2019 3 besov and triebellizorkin spaces associated to operators. 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. Siam journal on mathematical analysis siam society for. Triebellizorkin and besov spaces of arbitrary number of parameters.
For holder coefficients we derive the wellposedness in the spaces of ultradistributions associated to rockland operators on graded groups. Homogeneous triebellizorkin spaces with full range of parameters are introduced on stratified lie groups in terms of littlewoodpaleytype decomposition. 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. Research article homogeneous triebellizorkin spaces on. Zun wei fu, guang lan wang and qing yan wu subject. Moreover, we discuss their nontriviality under a poincar\e inequality.
Compensated compactness and the stratified lie group. A nonuniform estimate on the system of homogeneous balls is given. These groups, equipped with the socalled homogeneous norm, were introduced by folland in the framework of subelliptic pde 14. Pdf homogeneous triebellizorkin spaces on stratified. Follands sobolev spaces coincide with those introduced in on graded lie groups in the setting of stratified groups. Triebellizorkin, wiener and beurling spaces on compact homogeneous manifolds. Researchers can send the publication to the library as a pdf file or send new publications by using the online form. For and, the homogeneous triebel lizorkin spaces and besov spaces are defined by where for and satisfies the conditions. The sharp poincare inequality for free vector fields. In section 4 we study the rate of convergence in clt on strati ed nilpotent lie groups.
Boundedness of gfunctions on triebellizorkin spaces zhang, chunjie and chen, jiecheng, taiwanese journal of mathematics, 2009. 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. These function spaces on stratified lie groups were introduced earlier by folland in 7 by using sublaplacians. The results were rst published in pap, 1992a, 1992b, 1992c. Lie group of real rank at least two, margulis established that these lattices are. The besov spaces for certain parameters p, q and r have been considered in in the. Abstractcarnot groups are distinguished spaces that are rich of structure.
Our purpose in this paper is to generalize these constructions to situations where sections of line bundles, rather than ordinary scalarvalued functions, are considered. G 0 stratified lie group g in terms of littlewoodpaley type decompositions, in analogy to the wellknown characterization of the euclidean case. 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. Analysis on lie groups with polynomial growth book also available. Spectral multipliers of selfadjoint operators on besov and. 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. Functional inequalities in stratified lie groups with sobolev, besov. Kaehler spaces, nilpotent orbits, and singular reduction. In this paper we introduce a new function space which unifies and generalizes the besovtype and the triebellizorkintype function spaces introduced by s.
Littlewoodpaley theorem, nikolskii inequality, besov spaces. 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. In this article we consider the fractional integral operator i. New properties of besov and triebellizorkin spaces on. Homogeneous besov and triebellizorkin spaces associated to. Littlewoodpaley characterization of holderzygmund spaces on. 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. Pdf download analysis on lie groups with polynomial. On the other hand, in 14 the authors studied the boundedness properties of b,t on the triebellizorkin space.
A simple and consistent definition of homogeneous besov. We also present a nikolskiitype inequality and the littlewoodpaley theorem that play a role. Weighted lipschitz estimates for commutators of onesided operators on onesided triebel lizorkin spaces author. 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. The main tool is the di erentiablity of lipschitz maps, proved by. Then the exponential map is a diffeomorphism of g onto g, and we can transfer, via the exponential map, the dilations 8r to g. Boundedness and continuity of several integral operators with. Embedding properties between subelliptic besov spaces and besov spaces associated to the laplacian on the group are proved. Analysis and geometry in metric spaces directory of open. Before we introduce the homogeneous triebellizorkin spaces on strati ed lie groups, we prove the following basic estimate,whichisageneralizationof,lemmab. The natural setting for this kind of problem is that of homogeneous groups, in particular, including the cases of anisotropic structures on.
But after reading it, students will be wellprepared to continue with more advanced, graduatelevel topics in differential geometry or the theory of lie groups. F s,q p vextenddouble vextenddouble vextenddouble vextenddouble vextenddouble parenleftbigg. Nikolskii inequality and functional classes on compact lie groups. 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. A nilpotent lie algebra and its associated lie group g is stratified if there is a direct sum vector space decomposition 1. Abstract let lie groups in terms of littlewoodpaleytype decomposition. Integral transforms and special functions, 30 2019, 643655. Multipliers for besov spaces on graded lie groups sciencedirect. Carleson measure characterizations of the campanato type. We show that such spaces can be well defined under some conditions.
It is remarkable that so much about lie groups could be packed into this small book. 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. The author provides several examples and computations. Stratified lie groups and potential theory for their sub.
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