下面是它们的摘要。
第一章 设B是C~n中的单位球,S是单位球面。H~p(B)为B上的Hardy空间。对于ξ∈S,0<δ≤2,令Q_δ(ξ)={η∈S:d(ξ,η)=|1-<ξ,η>|<δ}。S上的函数空间BMO(S)定义为设f∈H~1(B),f~*是f的径向极限,若f~*∈BMO(S),则称f∈BMOA。
S上的non-isotropic Lipschitz空间Lip_α(S)(0<α<1)定义如下:
In this thesis, we will mainly study some function problems in unit ball, bounded strongly pseudoconvex domain and finite type convex domain in C". This thesis consists of three chapters. In chapter 1, we introduce the definitions of area integral S_β(f) and invariant g-function on unit sphere, and study their boundedness on BMO(S) and non-isotropic Lipschitz spaces. In chapter 2, we give some equivalent charcterizations of Bloch space and little Bloch space. We also give some characterizations of compactness of composition operator on Bergman spaces. In chapter 3, for finite type convex domain, we solove Gleason's problem in weighted Bergman space. Using holomorphic support function, we give some characterizations of BMOA and multiplier on BMOA.Chapter 1. Let B be the unit ball in C~n, S be the unit sphere. H~p(B) be the Hardy space. For ξ∈S and α < δ ≤ 2, let Q_δ(ξ) = {η ∈ S : d(ξ, η) = |1 - <ξ,η> | < δ}. The BMO(S) space on 5 is defined byLet f ∈ H~1(B), f* be the normal limit of f. If f* ∈ BMO(S) then we say f ∈ BMOA.
The non-isotropic Lipschitz spaces Lipa(S) (0 < a < 1) on S is defined by Lipa(S) = {/ 6 L1^) : \\f\\LiPa = sup ^.^(^ < +oo}.For / G Cl{B), let V/ = (|￡, |￡, ?" ?, |￡) be the complex gradient of /, and Vf(z) = V(/ o (fz)(Q), where cpz be the involution automorphism of B. pa{z) — -------x_uA ------witn raz — j^rOj -n>* — u, i^a — j — .r0.We introduce the definitions of area integral Sp{f) and invariant ^—function on 5where D^(O = {^ ￡ B : |1 - (z,OI < f (1 ~ I?I2),)8 > g(f)(O =5,g (/) and ^(/) are the generalizations of Lusin area integral and Littlewood-Paley function on Rn to Unit Sphere. It is well known that Lusin area integral, Littlewood-Paley g—function and g\ function play important roles in higher dimensions harmonic analysis. The paper [66] proved the Lp(l < p < oo) boundedness of these three functions. [68] studied the BMO boundedness of g—functon. [69] studied the BMO boundedness of S—function and g\ function. For area integral Sp(f) and invariant g—function, the papers [16, 44, 47] proved their IP boundedness. For holomorphic function on B, the paper [1] studied the relations between Sa(f) and Hardy-Sobolev space. The aim of this chapter is to study the BMO(S), Lipa(S) boundedness of Sp(f) and g(f). We have the following theorems
Theorem 0.0.1 Let f € BMOA, if there exists a positive measure set E C S such that Sp(f) < +00 on E, then Sp(f) < +00 a.e. on S, S0(f) e BMO(S) and there is a constant C such thatTheorem 0.0.2 Let f € Lipa{S), if there exists a positive measure set E C S such that Sp(f) < +00 on E, then Sp(f) < +00 a.e. on S, Sp(f) E Lipa(S) and there is a constant C such that\\S0(f)\\LiMS) < C\\f\\LlMs).Theorem 0.0.3 Let f € BMOA, if there exists a positive measure set E C S such that g(f) < +00 on E, then g(f) < +co a.e. on S, g(f) € BMO(S) and there is a constant C such thatU < cil/ll..Theorem 0.0.4 Let f € Lipa(S), if there exists a positive measure set E C S such that g(f) < +00 on E, then g(f) < +00 a.e. on S, g(f) 6 Lipa(S) and there is a constant C such thatLipa(S) < C\\f\\Lipa(S).Chapter 2. Let Q be a smoothly bounded strongly pseudoconvex domain in Cn, p{z) be the plurisubharmonic defining function of Q, that is Q = {z € Cn : p(z) < 0} and Vp(z) ^ 0 (z E dQ,), denote K{z, w) the Bergman kernel function for
Q., /3(z,w) the Bergman distance between z and w, and B(z, r) the Bergman metric ball, i.e., B{z, r) = {w € ft : 0(z, u>) < r}.Let iif(ft) denote the sets of all holornorphic function on ft, Ap(￡l) be the Bergman space, J3(ft) be the Bloch space, B0(Q) be the little Bloch space, they are defined byH(Q) : jT |/(z)|"di/(z) < +00},= {/ € if(n) : sup|p(*)||V/(z)| < +00},z€Cl= {/ € /r(fl) : 2nmn|p(z)||V/(^)| = 0}.Let ip be a holomorphic mapping from Q to Q., we define composition operator as follows) (zen,ueH(n)).There have been a lot of studies about composition operator on some function spaces. For Disc, polydisc and unitball in C", the papers [37, 65, 56, 53] gave some characterizations of boundedness and compactness of composition operator on Bergman spaces.For bounded strongly pseudoconvex domain, article [51] gave a characterization of boundedness and compactness of composition operator on Hardy spaces. In the first part of this chapter, we study the compactness of composition operator on Bergman spaces, and have the following resultsTheorem 0.0.5 Let Q, be a bounded strongly pseudoconvex domain, (p be a holomorphic mapping, then the following conditions are equivalent
(i) C;: B(Q) -? B0(fi) ia bounded,(ii) C,p : ^(fi) —> ^(fi) is compact,(in) C,p : AP(Q) -> AP(Q) is compact (for some 0
0 there holdsfi(B(a,r))SUp 7P, s < +°O-then we call fj, a Carleson measure on ￡l.If /lj satisfies the condition linia-^n \bu't)\ = ^' *^en ^ 1S cs^e& a Vanishing Carleson measure on Q.For Disc, Unit ball in C"\ the articles [70, 53] give the characterizations of Carleson measure and Vanishing Carleson measure, using the results of [48], we give some equivalent characterizations of Carleson measure and Vanishing Carleson measure on bounded strongly pesudoconvex domain Q.Theorem 0.0.6 Let /j, be a positive Borel measure, then the followings are equivalent(i) pi is a Carleson measure on fi.(ii) th inclusion mapping i : Ap{Vl,dv) —> AP(Q, d/u.) is continuous that is there exists a constant C such that for all f 6 AP(Q, dt>)(0 < p < 00)f \f(z)\pd^
Denote A(Q) the space of continuous functions in Q. and holomorphic in fi. The original Gleason's problem for a domain D,, consist in, given a function in A(￡l) and a fixed point f € fi, to find n functions ^ 6 A(Cl) such thatThis problem is soloved by Leibenzon[34] for the ball and by Henkin[34], Kerz-man - Nagel[46], Lieb[52] and Ovrelid[61] in the strongly pseudoconvex case. P. Ahem, R. Schneider[4], Jakobezak[38] and J. M. Ortega[57, 58] have studied this problem for Lipschitz and Ck functions. Zhu[71], Ortega[59] and Ren-Shi[63] have studied this problem for Bergman space on the ball, Bergman -Sobolev space on a strongly pseudoconvex domain and weighted Bergman space on an egg domain. The first part of this chapter is to generalize this result to AP(Q) on a finite type convex domain.Theorem 0.0.10 Let Q be a smoothly bounded finite type convex domain in Cn. For every ￡ in Q, there exist continuous linear operators Tk,k = 1,2, ■ ■ ■ ,n from AP{Q) ->? AP(Q) (1 < p < oo, q > 0) such thatJt=l Let HP(Q) be the Hardy space of Q, it is defined as followingHp(Q) = {/ e H(SI) : ||/||^= sup / \f(w)fdae < +oo},0
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