（基础数学专业论文）Bloch函数到QK型函数空间的距离.pdf

汕头大学 硕士学位论文 bloch函数到q型函数空间的距离 姓名 陈武福 申请学位级别 硕士 专业 基础数学 指导教师 娄增建 20090501 jones carleson bloch bmoa jones bloch m obius bloch qk bloch qk k carleson bloch qk bloch qk qk k carleson abstract we gave several distance formulas from bloch functions to some qk type spaces which generalize distance formulas from bloch functions to bmoa space by jones and to some m obius invariant spaces by ruhan zhao the fi rst part of the paper provides the defi nitions of some function spaces including bloch space and qk type spaces preliminaries and related known results the second part of the paper gives the distance formulas from bloch func tions to some qk type spaces which are characterized by using the k carleson measure the main theorems and their proofs are contained in this part key words bloch functions qkspaces qk type spaces k carleson measure distance 1 hardy hp 0 p bloch b b0 hp hp 5 26 2 john nirenberg bmo 31 feff erman h1 bmo bmo hardy h1 bmo 3 aulaskari lappan qp qp bmoa bloch dirichlet 1996 f p q s qp 2000 ess en qk qk p q f p q s h bmoa b 17 h bmoa h p 17 0 p 1 qp1 qp2 bmoa 19 0 p1 p2 0 1 qk b qk0 b 4 0 9 jones carleson bloch bmoa bloch 7 tjani bloch bloch carleson jones bloch f p p 2 s 0 s 1 1 p carleson 1 carleson bloch bloch carleson bloch qk qk bloch qk k bloch qk qk p q c a b a b c1 c2 c1a 6 b 6 c2a 2 1 1 d z z 1 d z z 1 d h d d a d g z a log 1 a z g z a a a z a z 1 az m obius 1 1 f h d f bloch b f kfkb sup z d 1 z 2 f z f h d lim z 1 1 z 2 f z 0 f bloch b0 b banach kfk b f 0 kfkb bloch bloch bloch bloch bloch 3 bloch 14 20 2 21 22 23 24 6 1 2 0 p 2 q 0 s 1 q s f h d f f p q s 14 f kfkp p q s sup a d z d f z p 1 z 2 qgs z a da z da z d lebesgue f p q s f lim a 1 z d f z p 1 z 2 qgs z a da z 0 f f0 p q s p 2 q 0 s 1 f p q s bmoa p 2 q 0 0 s 1 bmoa f p q s bmoa bmo 1 3 0 p 2 q k 0 f qk p q f kfkp qk p q sup a d z d f z p 1 z 2 qk 1 a z 2 da z f lim a 1 z d f z p 1 z 2 qk 1 a z 2 da z 0 f qk0 p q qk p q 10 p 2 q 0 qk p q qk0 p q qk qk0 6 qk qk0 qk m obius kf a kqk kfkqk 4 k k t ts 0 s 1 qk qk0 bloch bloch k a k b k t k 1 0 t 1 c k 2t k t t 0 k d r 1 0 k s ds s e r 1 k s ds s2 k s sup 0 t 1 k st k t 0 s k 6 k s k k k z 1 0 1 r2 qk log 1 r rdr qk p q 10 1 4 s i r d 1 i r 0 d s carleson sup i d s i i s i i i 1 s i d 5 carleson carleson 29 30 s carleson k carleson 1 5 d k carleson 6 sup i d z s i k 1 z i d z 0 1 distb f bmoa 2 inf f da z 1 z 2 carleson f z d f z 1 z 2 6 a carleson bloch bmoa b c carleson bloch f p p 2 s carleson bloch f0 p p 2 s b 0 s 1 1 p 0 q 0 f b 1 distb f f p p 2 s 2 inf f da z 1 z 2 2 s s carleson 3 inf supa d r f f z q 1 z 2 q 2 1 a z 2 sda z 4 inf supa d r f f z q 1 z 2 q 2gs z a da z c 0 s 1 1 p 0 q 0 f b 1 distb f f0 p p 2 s 2 distb f b0 3 inf f da z 1 z 2 2 s s carleson 4 inf lim a 1 r f f z q 1 z 2 q 2gs z a da z 0 5 inf lim a 1 r f f z q 1 z 2 q 2 1 a z 2 sda z 0 b a s 1 p 2 f p p 2 s bmoa a 1 2 c f0 p p 2 s b0 a b a bmoa qk b f p p 2 s qk p p 2 bloch qk qk p p 2 p 2 7 2 1 2 1 2 2 2 1 f b k d e k 2 0 q 0 1 distb f qk 2 inf f da z 1 z 2 2 k carleson 3 inf supa d r f f z q 1 z 2 q 2k g z a da z 4 inf supa d r f f z q 1 z 2 q 2k 1 a z 2 da z 2 2 2 p 0 q 0 k d e k 2 f b 1 distb f qk p p 2 2 inf f da z 1 z 2 2 k carleson 3 inf supa d r f f z q 1 z 2 q 2k g z a da z 4 inf supa d r f f z q 1 z 2 q 2k 1 a z 2 da z k t ts 0 s 1 qk f 2 0 s b 2 2 1 bloch qk0 c bloch qk0 b0 2 3 0 q 0 k d e k 2 f b 9 1 distb f qk0 2 distb f b0 3 inf f da z 1 z 2 2 k carleson 4 inf lim a 1 r f f z q 1 z 2 q 2k g z a da z 0 5 inf lim a 1 r f f z q 1 z 2 q 2k 1 a z 2 da z 0 2 2 2 1 f b 1 k1 k2 d e k1 2 k2 2 p1 2 k d e k 2 distb f qk p2 p2 2 distb f qk p1 p1 2 2 2 0 q k d e k 2 f h d 1 f qk bloch 2 f da z 1 z 2 2 k carleson 0 3 supa d r f f z q 1 z 2 q 2k g z a da z 0 4 supa d r f f z q 1 z 2 q 2k 1 a z 2 da z p1 2 k d e k 2 qk p1 p1 2 qk p2 p2 2 bloch 10 2 4 k d e k 2 1 s 0 c 0 z d z d 1 w tda w 1 zw 2 t s c 1 z s 2 1 2 4 2 2 2 2 2 3 6 3 1 3 2 2 2 k d f qk f z 2da z d k carleson 11 2 3 k d k carleson sup a d z d k 1 a z 2 d z 27 s carleson sup a d z d 1 a 2 1 az 2 sd z 0 k carleson 1 z 2 sd z s carleson 2 4 2 5 8 2 4 k carleson qk p p 2 2 5 qk 2 4 p 0 k d f qk p p 2 f z p 1 z 2 p 2da z k carleson 2 5 n k d k 2 f qk sup a d z d f n 1 z 2 1 z 2 2nk 1 a z 2 da z f3 2 6 f b f3 b f 3 z ckfkb 1 z 2 2 2 1 f3 f 3 z z f f w 1 w 2 1 zw 3 da w kfkb z d 1 1 zw 3 da w ckfkb 1 z 2 12 f 3 z 1 z 2 ckfkb f3 b 2 1 f 3 z z f wf w 1 w 2 1 zw 4 da w kfkb z d 1 1 zw 4 da w ckfkb 1 z 2 2 2 6 1 1 28 5 2 7 k e a w d z d k 1 a z 2 1 zw 4 da z c k 1 a w 2 1 w 2 2 u w d z w u w a 1 ei a z ei u 13 i z d k 1 a z 2 1 zw 4 da z k 1 a w 2 z d k 1 a z 2 k 1 a w 2 1 z w 4 da z k 1 a w 2 z d k 1 u 2 k 1 2 1 w w u 4 w u 2da u k 1 a w 2 z d k 1 u 2 1 2 1 u 2 k 1 2 1 w w u 4 w u 2da u k 1 a w 2 z d k 1 u 2 1 u 2 1 wu 4 1 w 2 2 1 w 2 4 1 wu 4 da u k 1 a w 2 1 w 2 2 z d k 1 u 1 u da u 2 k 1 a w 2 1 w 2 2 z 1 0 k 1 r 1 r r dr u r s 1 r 1 r i c k 1 a w 2 1 w 2 2 z 1 k s s 1 2 ds c k 1 a w 2 1 w 2 2 z 1 k s s2 ds c k 1 a w 2 1 w 2 2 2 7 14 2 4 k carleson 2 1 d1 d2 d3 d4 2 1 1 2 3 4 d1 d2 d2 d4 d3 d4 d1 cd2 f3 qk 2 5 sup a d z d f 3 z 2 1 z 2 2k 1 a z 2 da z 2 6 2 7 fubini sup a d z d f 3 z 2 1 z 2 2k 1 a z 2 da z ckfkbsup a d z d f 3 z k 1 a z 2 da z ckfkbsup a d z d z f wf w 1 w 2 1 zw 4 da w k 1 a z 2 da z ckfk2 bsup a d z f z d k 1 a z 2 1 zw 4 da z da w ckfk2 bsup a d z f k 1 a w 2 1 w 2 2 da w f da w 1 w 2 2 k carleson 2 3 sup a d z d f 3 z 2 1 z 2 2k 1 a z 2 da z f3 qk kf4k b c 15 f4 0 0 kf4k b kf4kb kf2kb 2 1 f 2 z 2 z d f f w 1 w 2 1 zw 3 da w 2 z d 1 1 zw 3 da w 2c 1 z 2 kf2kb sup z d f 2 z 1 z 2 2c distb f qk inf g qk kf gk b kf f3k b kf4k b 2c d2 cd1 d1 1 0 f qk f da z 1 z 2 2 k carleson kf f 1k b 1 1 z 2 f 1 z f z 1 z 2 kf f 1k b f z 1 z 2 1 1 f 1 f da z 1 z 2 2 f 1 z 2da z 1 2 f 1 qk 2 2 f 1 z 2da z k carleson f da z 1 z 2 2 k carleson d4 d2 f da z 1 z 2 2 k carleson sup a d z f k 1 a z 2 1 z 2 2 da z 16 2 3 sup a d z f f z q 1 z 2 q 2k 1 a z 2 da z ckfkq bsup a d z f k 1 a z 2 1 z 2 2 da z 0 k 2t k t d4 d3 d4 d3 d1 z d a 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