Classical Fourier Analysis over Homogeneous Spaces of Compact Groups

This paper introduces a unified operator theory approach to the abstract Fourier analysis over homogeneous spaces of compact groups. Let G be a compact group and H be a closed subgroup of G. Let G/H be the left coset space of H in G and μ be the normalized G-invariant measure on G/H associated to the Weil＇s formula. Then, we present a generalized abstract framework of Fourier analysis for the Hilbert function space L^2 （G / H, μ）.

Boundedness of Some Operators and Commutators in Morrey-Herz Spaces on Non-Homogeneous Spaces

The authors introduce the homogeneous Morrey-Herz spaces and the weak homo- geneous Morrey-Herz spaces on non-homogeneous spaces and establish the boundedness in ho- mogeneous Morrey-Herz spaces for a class of sublinear operators including Hardy-Littlewood maximal operators,Calderón-Zygmund operators and fractional integral operators.Further- more,some weak estimate of these operators in weak homogeneous Morrey-Herz spaces are also obtained.Moreover,the authors discuss the boundedness in homogeneous Morrey-Herz spaces of the maximal commutators associated with Hardy-Littlewood maximal operators and multilinear commutators generated by Calderón-Zygmund operators or fractional integral operators with RBMO（μ）functions.

Boundedness of Linear Commutators on Herz-Morrey Spaces on Spaces of Homogeneous Type

Under certain weak local size conditions, the boundedness of linear commutators on Herz-Morrey spaces on spaces of homogeneous type are studied. In addition, the boundedness of Hardy-Littlewood maximum commutators on Herz-Morrey spaces on spaces of homogeneous type are obtained.

The p-Harmonic Heat Flow with Potential into Homogeneous Spaces

Abstract l11 tl1is papel' xte I7rove tl1e existellce of gIol>al wcak so1utiolls of the I)--11i1r11loliic flow with potelltial bett'eel1 Rit)mal1nian lnallifOlds AI an(l N fbr arbitrary iuitial data 1la\-i11g fl11ite P--e11erg}: ill the case wI1e11 the targct N is a l1on1ogeneous spact. witll a left invariant ln(3tri<'.

Topics on the Geometry of Rational Homogeneous Spaces

This is a survey paper about a selection of results in complex algebraic geometry that appeared in the recent and less recent litterature,and in which rational homogeneous spaces play a prominent role.This selection is largely arbitrary and mainly refiects the interests of the author.

Equidistribution of Expanding Translates of Curves in Homogeneous Spaces with the Action of(SO(n,1))^(k)

Let X=G/Γbe a homogeneous space with ambient group G containing the group H=(SO(n,1))^(k)and x∈X be such that Hx is dense in X.Given an analytic curve?:I=[a,b]→H,we will show that ifφsatisfies certain geometric condition,then for a typical diagonal subgroup A={a(t):t∈R}■H the translates{a(t)?(I)x:t>0}of the curve?(I)x will tend to be equidistributed in X as t→+∞.The proof is based on Ratner's theorem and linearization technique.

COMMUTATORS OF BMO FUNCTIONS AND SINGULAR INTEGRAL OPERATORS WITH NON-SMOOTH KERNELS ON SPACES OF HOMOGENEOUS TYPE OF FINITE MEASURE

Let X be a space of homogeneous type with finite measure. Let T be a singular integral operator which is bounded on L^p （X）, 1 〈 p 〈 ∞. We give a sufficient condition on the kernel k（x,y） of Tso thatwhen a function b ∈ BMO （X）,the commutator [b,T] （f）=T （b f）- bT （f） is bounded on spaces L^p for all p, 1 〈 p 〈 ∞.

Embeddings of Spherical Homogeneous Spaces

We review in these notes the theory of equivariant embeddings of spherical homogeneous spaces. Given a spherical homogeneous space G/H, the normal equivariant embeddings of G/H are classified by combinatorial objects called colored fans, which generalize the fans appearing in the classification of toric varieties and which encode several geometric properties of the corresponding variety.

BOUNDEDNESS AND COMPACTNESS FOR THE COMMUTATOR OF THEω-TYPE CALDER′ON-ZYGMUND OPERATOR ON LORENTZ SPACE

In this paper,the authors consider theω-type Calder´on-Zygmund operator T_(ω)and the commutator[b,T_(ω)]generated by a symbol function b on the Lorentz space L^(p,r)(X)over the homogeneous space(X,d,μ).The boundedness and the compactness of the commutator[b,T_(ω)]on Lorentz space L^(p,r)(X)are founded for any p∈(1,∞)and r∈[1,∞).

CHARACTERIZATION OF MAXIMAL OPERATORS IN ORLICZ-MORREY SPACES OF HOMOGENEOUS TYPE

. In this paper,the characterization of boundedness of Hardy-Littlewood maximal operators in Orlicz-Morrey spaces LΦφ（X,μ） of homogeneous type is founded.

BOUNDEDNESS OF COMMUTATORS ON HOMOGENEOUS MORREY-HERZ SPACES OVER LOCALLY COMPACT VILENKIN GROUPS

This paper studies some boundedness results of commutators on a class of new spaces MKp,q^αλ （G） named as homogenous Morrey-Herz spaces over locally compact Vilenkin groups

Characterizations of realized homogeneous Besov and Triebel-Lizorkin spaces via differences

Based on the role of the polynomial functions on the homogeneous Besov spaces, on the homogeneous Triebel-Lizorkin spaces and on their realized versions, we study and obtain characterizations of these spaces via difference operators in a certain sense.

A MAXIMAL FUNCTION CHARACTERIZATION 0F HARDY SPACES ON SPACES OF HOMOGENEOUS TYPE

A new maximal function is introduced in the dual spaces of test function spaces on spaces of homogeneous type. Using this maximal function, we get new characterization of atomic H^p spaces.

ROUGH OPERATORS AND COMMUTATORS ON HOMOGENEOUS WEIGHTED HERZ SPACES

The authors establish the baundedness on homogeneous weighted Herz spaces for a large class of rough operators and their commutators with BMO functions. In particular, the Calderon-Zygmund singular integrals and the rough R. Fefferman singular integral operators and the rough Ricci-Stein oscillatory singular integrals and the corresponding commutators are considered.

INTERPOLATION SPACES BETWEEN H^1 AND L~∞ ON SPACES OF HOMOGENEOUS TYPE

Using the maximal function characterization of Hardy spaces, we study the interpolation spaces between H1 and L∞ on spaces of homogeneous type.

FRACTIONAL INTEGRAL OPERATOR ON SPACES OF HOMOGENEOUS TYPE

An equivalent definition of fractional integral on spaces of homogeneous type is given. The behavior of the fractional integral operator in Triebel-Lizorkin space is discussed.

WEIGHTED ESTIMATES FOR COMMUTATORS OF POTENTIAL OPERATORS ON SPACES OF HOMOGENEOUS TYPE

We derive some strong type and weak type weighted norm estimates which re- late the commutators of potential integral operators to the corresponding maximal operators in the context of spaces of homogeneous type.

Weighted Estimates for the Maximal Commutator of Singular Integral Operator on Spaces of Homogeneous Type

Weighted estimates with general weights are established for the maximal operator associated with the commutator generated by singular integral operator and BMO function on spaces of homogeneous type, where the associated kernel satisfies the HSlder condition on the first variable and some condition which is fairly weaker than the Holder condition on the second variable.

Composition of Maximal Operators on Space of Homogeneous Type

Let Ф be a Young function and MФ be the maximal operator associated with in the space of homogeneous type. In this paper, the composition of the maximal operators of type MФ is considered, and the result established by Carrozza and Passarelli Di Napoli is generalized to the space of homogeneous type.

On Some Recent Progress in Complex Geometry-the Area Related to Homogeneous Manifolds

In this article,we give a survey of some progress of the complex geometry,mostly related to the Lie group actions on compact complex manifolds and complex homogeneous spaces in the last thirty years.In particular,we explore some works in the special area in Di erential Geometry,Lie Group and Complex Homogeneous Space.Together with the special area in nonlinear analysis on complex manifolds,they are the two major aspects of my research interests.